# Basic inequalities for statistical submanifolds in Golden-like statistical manifolds

Basic inequalities for statistical submanifolds in Golden-like statistical manifolds 1IntroductionThe comparison relationships between the intrinsic and extrinsic invariants are the basic problems in submanifold theory. In [1], Chen introduce some curvature invariants and for their usages, he derived optimal relationships between the intrinsic invariants (Chen invariants) and the extrinsic invariants, which become later an active and fruitful area of research (see, for instance, [1,2,3]).On the other hand, the notion of Casorati curvature (extrinsic invariant) for the surfaces was originally introduced in 1890 (see [4]). The Casorati curvature gives a better intuition of the curvature compared to the Gaussian curvature. The Gaussian curvature of a developable surface is zero. Thus, Casorati put forward the notion of Casorati curvature of a surface defined as C=1/2(1/κ12+1/κ22){\mathcal{C}}=1\hspace{-0.08em}\text{/}\hspace{-0.08em}2\left(1\hspace{-0.08em}\text{/}{\kappa }_{1}^{2}+1\text{/}\hspace{-0.08em}{\kappa }_{2}^{2}). For example, for developable surfaces (say, cylinder), the Gaussian curvature vanishes, while the Casorti curvature C{\mathcal{C}}surely does not vanish. The Casorati curvature of a submanifold in a Riemannian manifold is defined as the normalized square length of the second fundamental form [5].In the past decade, various geometers attracted toward the study of Chen-type comparison relationships between the Casorati curvature and the intrinsic invariants. For some references in this direction we refer to [6,7, 8,9,10, 11,12]. The submanifolds with equality case in the Chen-type inequalities are called ideal submanifolds and the name ideal is motivated by the fact that these submanifolds inherit the least possible tension from the ambient manifold (see [13]).In 1985, Amari introduced the notion of statistical manifolds via information geometry (see [14]). Statistical manifolds are endowed with a pair of dual torsion-free connections. This is analogous to conjugate connections in affine geometry (see [15]). The dual connections are not metric, thus it is very tough to give a notion of sectional curvature using the canonical definitions of Riemannian geometry. In [16], Opozda gave the definition of sectional curvature tensor on a statistical manifold. While studying the geometric properties of a submanifold, a very important problem is to obtain sharp relations between the intrinsic and the extrinsic invariants, and a vast number of such relations are revealed by certain inequalities. For example, let MMbe a surface in Euclidean 3-space, we know the Euler inequality: K≤∣H∣2K\le | H\hspace{-0.25em}{| }^{2}, where HHis the mean curvature (extrinsic property) and KKis the Gaussian curvature (intrinsic property). The equality holds at points where MMis congruent to an open piece of a plane or a sphere (umbilical points). Chen [17] obtained the same inequality for submanifolds of real space forms. Then in [18], Chen obtained the Chen-Ricci inequality, which is a sharp relation between the squared mean curvature and the Ricci curvature of a Riemannian submanifold of a real space form.In recent years, statistical manifolds have been studied very actively. In [19], Takano studied statistical manifolds with almost complex and almost contact structure. In 2015, Vîlcu and Vîlcu [20] studied statistical manifolds with quaternionic settings and proposed several open problems. While answering one of those open problems, Aquib [21] obtained some of the curvature properties of submanifolds and a couple of inequalities for totally real statistical submanifolds of quaternionic Kaehler-like statistical space forms. In 2019, Chen et al. derived a Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature [22]. In the same year, following the same paper of Chen et al., Atimur et al. [23] obtained Chen-type inequalities for statistical submanifolds of Kaehler-like statistical manifolds. Very recently, in 2020, Decu et al. obtained inequalities for the Casorati curvature of statistical manifolds in holomorphic statistical manifolds of constant holomorphic curvature [24]. For some of the recent works, we refer to [15, 19, 25, 26,27,28].Motivated by the aforementioned studies, we define Golden-like statistical manifolds and obtain certain interesting inequalities. The structure of this paper is as follows. In Section 2, we first give the definition of Golden-like statistical manifolds. We also construct an example for the Golden-like statistical manifolds. In the next section, we obtain the main inequalities. We also prove the results for their equality cases.2Golden-like statistical manifoldLet MMbe a smooth manifold. A (1,1)\left(1,1)tensor field TTon MMis said to be polynomial structure if TTsatisfies an algebraic equation [29,30]P(x)=xn+bnxn−1+⋯+b2x+b1I=0,P\left(x)={x}^{n}+{b}_{n}{x}^{n-1}+\cdots +{b}_{2}x+{b}_{1}I=0,where IIis the (1,1)\left(1,1)identity tensor field and Tn−1(q),Tn−2(q),…,T(q),I{T}^{n-1}\left(q),{T}^{n-2}\left(q),\ldots ,T\left(q),Iare linearly independent at every point q∈M.q\in M.The polynomial P(x)P\left(x)is called the structure polynomial. For P(x)=x2+IP\left(x)={x}^{2}+Iand P(x)=x2−IP\left(x)={x}^{2}-I, we obtain an almost complex structure and an almost product structure, respectively. It has to be noted here that the existence of almost complex structure implies the even dimensions of the manifold. For P(x)=x2P\left(x)={x}^{2}, we obtain the notion of an almost tangent structure.Definition 1[29,31,32] Let (M,g)\left(M,g)be the a semi-Riemannian manifold and let ϕ\phi be the (1,1)\left(1,1)tensor field on MMsatisfying the following equation: ϕ2=ϕ+I.{\phi }^{2}=\phi +I.Then the tensor field ϕ\phi is called a Golden structure on MM. If the Riemannian metric ggis ϕ\phi compatible, the (M,g,ϕ)\left(M,g,\phi )is called a Golden semi-Riemannian manifold.For ϕ\phi compatible metric gg, we have the following: (2.1)g(ϕX,Y)=g(X,ϕY),g\left(\phi X,Y)=g\left(X,\phi Y),(2.2)g(ϕX,ϕY)=g(ϕ2X,Y)=g(ϕX,Y)+g(X,Y),X,Y∈Γ(TM).g\left(\phi X,\phi Y)=g\left({\phi }^{2}X,Y)=g\left(\phi X,Y)+g\left(X,Y),\hspace{1.0em}X,Y\in \Gamma \left(TM).A remarkable fact about Golden structures is its appearance in pairs, i.e., if ϕ\phi is Golden structure, the ϕˆ=I−ϕ\hat{\phi }=I-\phi is also a Golden structure. But same is the case with almost tangent (RRand −R-R) and almost complex structure (JJand −J-J). So it is natural to ask the connection between Golden and product structures.Let MMbe a Riemannian manifold. Denote a torsion-free affine connection by ∇\nabla . The triple (M,∇,g)\left(M,\nabla ,g)is called a statistical manifold if ∇g\nabla gis symmetric. We define another affine connection ∇∗{\nabla }^{\ast }by (2.3)Xg(Y,Z)=g(∇XY,Z)+g(∇X∗Z,Y)Xg\left(Y,Z)=g\left({\nabla }_{X}Y,Z)+g\left({\nabla }_{X}^{\ast }Z,Y)for vector fields EE, FF, and GGon MM. The affine connection ∇∗{\nabla }^{\ast }is called conjugate (or dual) to ∇\nabla with respect to gg. The affine connection ∇∗{\nabla }^{\ast }is torsion-free, ∇∗g{\nabla }^{\ast }gis symmetric and satisfies ∇0=∇+∇∗2{\nabla }^{0}=\frac{\nabla +{\nabla }^{\ast }}{2}. Clearly, the triple (M,∇∗,g)\left(M,{\nabla }^{\ast },g)is statistical. We denote by RRand R∗{R}^{\ast }the curvature tensors on MMwith respect to the affine connection ∇\nabla and its conjugate ∇∗{\nabla }^{\ast }, respectively. Also the curvature tensor field R0{R}^{0}associated with the ∇0{\nabla }^{0}is called Riemannian curvature tensor. Then we find g(R(X,Y)Z,W)=−g(Z,R∗(X,Y)W)g\left(R\left(X,Y)Z,W)=-g\left(Z,{R}^{\ast }\left(X,Y)W)for vector fields XX, YY, ZZ, and WWon MM, where R(X,Y)Z=[∇X,∇Y]Z−∇[X,Y]ZR\left(X,Y)Z=\left[{\nabla }_{X},{\nabla }_{Y}]Z-{\nabla }_{\left[X,Y]}Z.In general, the dual connections are not metric, one cannot define the sectional curvature in statistical environment as in the case of semi-Riemannian geometry. Thus, Opozda proposed two notions of sectional curvature on statistical manifolds (see [16,33]).Let MMbe a statistical manifold and π\pi a plane section in TMTMwith orthonormal basis {X,Y}\left\{X,Y\right\}, then the sectional KK-curvature is defined in [16] as K(π)=12[g(R(X,Y)Y,X)+g(R∗(X,Y)Y,X)−g(R0(X,Y)Y,X)].K\left(\pi )=\frac{1}{2}{[}g\left(R\left(X,Y)Y,X)+g\left({R}^{\ast }\left(X,Y)Y,X)-g\left({R}^{0}\left(X,Y)Y,X)].Definition 2Let (M,g,ϕ)\left(M,g,\phi )be a Golden semi-Riemannian manifold endowed with a tensor field ϕ∗{\phi }^{\ast }of type (1,1) satisfying (2.4)g(ϕX,Y)=g(X,ϕ∗Y)g\left(\phi X,Y)=g\left(X,{\phi }^{\ast }Y)\hspace{3.3em}for vector fields XXand YY. In view of (2.4), we easily derive (2.5)(ϕ∗)2X=ϕ∗X+X,\hspace{-10.3em}{\left({\phi }^{\ast })}^{2}X={\phi }^{\ast }X+X,(2.6)g(ϕX,ϕ∗Y)=g(ϕX,Y)+g(X,Y).g\left(\phi X,{\phi }^{\ast }Y)=g\left(\phi X,Y)+g\left(X,Y).Then (M,g,ϕ)\left(M,g,\phi )is called Golden-like statistical manifold.According to (2.5) and (2.6), the tensor fields ϕ+ϕ∗\phi +{\phi }^{\ast }and ϕ−ϕ∗\phi -{\phi }^{\ast }are symmetric and skew symmetric with respect to gg, respectively. The equations (2.4), (2.5), and (2.6) imply the following proposition.Proposition 1(M,g,ϕ)\left(M,g,\phi )is a Golden-like statistical manifold if and only if it is (M,g,ϕ∗)\left(M,g,{\phi }^{\ast }).We remark that if we choose ϕ=ϕ∗\phi ={\phi }^{\ast }in a Golden-like statistical manifold, then we have a Golden semi-Riemannian manifold.We first present an example of a Golden-Riemannian manifold.Example 1[34] Consider the Euclidean 6-space R6{{\mathbb{R}}}^{6}with standard coordinates (x1,x2,x3,x4,x5,x6)\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6}). Let ϕ\phi be an (1,1)\left(1,1)tensor field on R6{{\mathbb{R}}}^{6}defined by ϕ(x1,x2,x3,x4,x5,x6)=(ψx1,ψx2,ψx3,(1−ψ)x4,(1−ψ)x5,(1−ψ)x6)\phi \left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})=\left(\psi {x}_{1},\psi {x}_{2},\psi {x}_{3},\left(1-\psi ){x}_{4},\left(1-\psi ){x}_{5},\left(1-\psi ){x}_{6})for any vector field (x1,x2,x3,x4,x5,x6)∈R6\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})\in {{\mathbb{R}}}^{6}, where ψ=1+52\psi =\frac{1+\sqrt{5}}{2}and 1−ψ=1−521-\psi =\frac{1-\sqrt{5}}{2}are the roots of the equation x2=x+1{x}^{2}=x+1. Then we obtain ϕ2(x1,x2,x3,x4,x5,x6)=(ψ2x1,ψ2x2,ψ2x3,(1−ψ)2x4,(1−ψ)2x5,(1−ψ)2x6)=(ψx1,ψx2,ψx3,(1−ψ)x4,(1−ψ)x5,(1−ψ)x6)+(x1,x2,x3,x4,x5,x6).\begin{array}{rcl}{\phi }^{2}\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})& =& \left({\psi }^{2}{x}_{1},{\psi }^{2}{x}_{2},{\psi }^{2}{x}_{3},{\left(1-\psi )}^{2}{x}_{4},{\left(1-\psi )}^{2}{x}_{5},{\left(1-\psi )}^{2}{x}_{6})\\ & =& \left(\psi {x}_{1},\psi {x}_{2},\psi {x}_{3},\left(1-\psi ){x}_{4},\left(1-\psi ){x}_{5},\left(1-\psi ){x}_{6})+\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6}).\end{array}Thus, we have ϕ2=ϕ+I{\phi }^{2}=\phi +I. Moreover, we can easily see that standard metric ⟨,⟩\langle \hspace{0.33em},\rangle on R6{{\mathbb{R}}}^{6}is ϕ\phi compatible. Hence, (R6,⟨,⟩,ϕ{{\mathbb{R}}}^{6},\langle \hspace{0.33em},\rangle ,\phi ) is a Golden Riemannian manifold.Next, we construct an example of a Golden-like statistical manifold in the following example.Example 2Consider the semi-Euclidean space R13{{\mathbb{R}}}_{1}^{3}with standard coordinates (x1,x2,x3)\left({x}_{1},{x}_{2},{x}_{3})and the semi-Riemannian metric ggwith the signature (−,+,+)\left(-,\hspace{0.33em}+,\hspace{0.33em}+). Let ϕ\phi be an (1,1)\left(1,1)tensor field on R13{{\mathbb{R}}}_{1}^{3}defined by ϕ(x1,x2,x3)=12(x1+5x2,x2+5x1,2ψx3)\phi \left({x}_{1},{x}_{2},{x}_{3})=\frac{1}{2}({x}_{1}+\sqrt{5}{x}_{2},{x}_{2}+\sqrt{5}{x}_{1},2\psi {x}_{3})for any vector field (x1,x2,x3)∈R13\left({x}_{1},{x}_{2},{x}_{3})\in {{\mathbb{R}}}_{1}^{3}, where ψ=1+52\psi =\frac{1+\sqrt{5}}{2}is the Golden mean. Then we obtain ϕ2=ϕ+I{\phi }^{2}=\phi +I, this implies that ϕ\phi is a Golden structure on R13{{\mathbb{R}}}_{1}^{3}.Now we define an (1,1)\left(1,1)tensor field ϕ∗{\phi }^{\ast }on R13{{\mathbb{R}}}_{1}^{3}by ϕ∗(x1,x2,x3)=12(x1−5x2,x2−5x1,2ψx3).{\phi }^{\ast }\left({x}_{1},{x}_{2},{x}_{3})=\frac{1}{2}({x}_{1}-\sqrt{5}{x}_{2},{x}_{2}-\sqrt{5}{x}_{1},2\psi {x}_{3}).Thus, we have ϕ∗2=ϕ∗+I{{\phi }^{\ast }}^{2}={\phi }^{\ast }+I. Moreover, we have the equation (2.4). Hence, (R13,g,ϕ{{\mathbb{R}}}_{1}^{3},g,\phi ) is a Golden-like simplified statistical manifold.Now we give a generalized example of the above example.Example 3Let Rn{{\mathbb{R}}}_{n}be a (2n+m)\left(2n+m)-dimensional affine space with the coordinate system (x1,⋯,xn,y1,…,yn,z1,…,zm)({x}_{1},\cdots \hspace{0.33em},{x}_{n},{y}_{1},\ldots ,{y}_{n},{z}_{1},\ldots ,{z}_{m}). Assume we define a semi-Riemannian metric ggwith the signature (−,…,−n−times,+,…,+(n+m)−times)\left(\mathop{-,\ldots ,-}\limits_{n-times},\mathop{+,\ldots ,+}\limits_{\left(n+m)-times})and the tensor field ϕ\phi as follows: ϕ=12δij5δij05δijδij000ψ,\phi =\frac{1}{2}\left[\begin{array}{ccc}{\delta }_{ij}& \sqrt{5}{\delta }_{ij}& 0\\ \sqrt{5}{\delta }_{ij}& {\delta }_{ij}& 0\\ 0& 0& \psi \end{array}\right],where ψ\psi is the Golden mean. Then ϕ\phi is golden structure on Rn2n+m{{\mathbb{R}}}_{n}^{2n+m}. Moreover, if the conjugate tensor field ϕ∗{\phi }^{\ast }is defined as ϕ∗=12δij−5δij05δij−δij000ψ.{\phi }^{\ast }=\frac{1}{2}\left[\begin{array}{ccc}{\delta }_{ij}& -\hspace{-0.25em}\sqrt{5}{\delta }_{ij}& 0\\ \sqrt{5}{\delta }_{ij}& -\hspace{-0.25em}{\delta }_{ij}& 0\\ 0& 0& \psi \end{array}\right].Then we can easily see that (Rn2n+m,g,ϕ)\left({{\mathbb{R}}}_{n}^{2n+m},g,\phi )and (Rn2n+m,g,ϕ∗)\left({{\mathbb{R}}}_{n}^{2n+m},g,{\phi }^{\ast })are Golden-like statistical manifolds. Also, this verifies Proposition 1.Let (M=Mp(cp)×Mq(cq),g,ϕ)\left(M={M}_{p}\left({c}_{p})\times {M}_{q}\left({c}_{q}),g,\phi )be a Golden product space form. Then the Riemannian curvature tensor RRof MMis given by [32]: (2.7)R(X,Y)Z=−(1−ψ)cp−ψcq25{g(Y,Z)X−g(X,Z)Y+g(ϕY,Z)ϕX−g(ϕX,Z)ϕY}+−(1−ψ)cp+ψcq4{g(ϕY,Z)X−g(ϕX,Z)Y+g(Y,Z)ϕX−g(X,Z)ϕY},\begin{array}{rcl}R\left(X,Y)Z& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left\{g\left(Y,Z)X-g\left(X,Z)Y+g\left(\phi Y,Z)\phi X-g\left(\phi X,Z)\phi Y\right\}\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)\left\{g\left(\phi Y,Z)X-g\left(\phi X,Z)Y+g\left(Y,Z)\phi X-g\left(X,Z)\phi Y\right\},\end{array}where Mp{M}_{p}and Mq{M}_{q}are space forms with constant sectional curvatures cp{c}_{p}and cq{c}_{q}, respectively. We can obtain the curvature tensor R∗{R}^{\ast }with respect to dual connection just by replacing ϕ\phi by ϕ∗.{\phi }^{\ast }.Let Mn{M}^{n}be statistical submanifold of (Nm,g,ϕ)\left({N}^{m},g,\phi ). The Gauss and Weingarten formulae are ∇XY=∇XY+σ(X,Y),∇Xξ=−AξX+∇X⊥ξ∇X∗Y=∇X∗Y+σ∗(X,Y),∇X∗ξ=−Aξ∗X+∇X∗⊥ξ\begin{array}{lcl}{\nabla }_{X}Y& =& {\nabla }_{X}Y+\sigma \left(X,Y),\hspace{1.51em}{\nabla }_{X}\xi =-{A}_{\xi }X+{\nabla }_{X}^{\perp }\xi \\ {\nabla }_{X}^{\ast }Y& =& {\nabla }_{X}^{\ast }Y+{\sigma }^{\ast }\left(X,Y),\hspace{1.0em}{\nabla }_{X}^{\ast }\xi =-{A}_{\xi }^{\ast }X+{\nabla }_{X}^{\ast \perp }\xi \end{array}for all X,Y∈TMX,Y\in TMand ξ∈T⊥M\xi \in {T}^{\perp }M, respectively. Moreover, we have the following equations: Xg(Y,Z)=g(∇XY,Z)+g(Y,∇X∗Z)g(σ(X,Y),ξ)=g(Aξ∗X,Y),g(σ∗(X,Y),ξ)=g(AξX,Y)Xg(ξ,η)=g(∇X⊥ξ,η)+g(ξ,∇X∗⊥η).\begin{array}{l}Xg\left(Y,Z)=g\left({\nabla }_{X}Y,Z)+g\left(Y,{\nabla }_{X}^{\ast }Z)\\ g\left(\sigma \left(X,Y),\xi )=g\left({A}_{\xi }^{\ast }X,Y),\hspace{1.0em}g\left({\sigma }^{\ast }\left(X,Y),\xi )=g\left({A}_{\xi }X,Y)\\ Xg\left(\xi ,\eta )=g\left({\nabla }_{X}^{\perp }\xi ,\eta )+g\left(\xi ,{\nabla }_{X}^{\ast \perp }\eta ).\end{array}The mean curvature vector fields for an orthonormal tangent frame {e1,e2,…,en}\left\{{e}_{1},{e}_{2},\ldots ,{e}_{n}\right\}and a normal frame {en+1,…,em}\left\{{e}_{n+1},\ldots ,{e}_{m}\right\}, respectively, are defined as H=1n∑i=1nσ(ei,ei)=1n∑γ=n+1m∑i=1nσiiγξγ,σijγ=g(σ(ei,ej),eγ)\hspace{1em}H=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}\sigma \left({e}_{i},{e}_{i})=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\gamma }\right){\xi }_{\gamma },\hspace{2.0em}{\sigma }_{ij}^{\gamma }=g\left(\sigma \left({e}_{i},{e}_{j}),{e}_{\gamma })\hspace{1.65em}and H∗=1n∑i=1nσ∗(ei,ei)=1n∑γ=n+1m∑i=1nσii∗γξγ,σij∗γ=g(σ∗(ei,ej),eγ){H}^{\ast }=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\sigma }^{\ast }\left({e}_{i},{e}_{i})=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\ast \gamma }\right){\xi }_{\gamma },\hspace{1.0em}{\sigma }_{ij}^{\ast \gamma }=g\left({\sigma }^{\ast }\left({e}_{i},{e}_{j}),{e}_{\gamma })for 1≤i,j≤n1\le i,j\le n, and 1≤l≤m1\le l\le m. Moreover, we have 2h0=h+h∗2{h}^{0}=h+{h}^{\ast }and 2H0=H+H∗2{H}^{0}=H+{H}^{\ast }, where the second fundamental form h0{h}^{0}and the mean curvature H0{H}^{0}are calculated with respect to Levi-Civita connection ∇0{\nabla }^{0}on MM.The squared mean curvatures are defined as ‖H‖2=1n2∑γ=n+1m∑i=1nσiiγ2,‖H∗‖2=1n2∑γ=n+1m∑i=1nσii∗γ2.\Vert H{\Vert }^{2}=\frac{1}{{n}^{2}}\mathop{\sum }\limits_{\gamma =n+1}^{m}{\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\gamma }\right)}^{2},\hspace{1.0em}\Vert {H}^{\ast }{\Vert }^{2}=\frac{1}{{n}^{2}}\mathop{\sum }\limits_{\gamma =n+1}^{m}{\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\ast \gamma }\right)}^{2}.The Casorati curvatures are defined as C=1n∑γ=n+1m∑i,j=1n(σijγ)2,C∗=1n∑γ=n+1m∑i,j=1n(σij∗γ)2.{\mathcal{C}}=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\mathop{\sum }\limits_{i,j=1}^{n}{({\sigma }_{ij}^{\gamma })}^{2},\hspace{1.0em}{{\mathcal{C}}}^{\ast }=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\mathop{\sum }\limits_{i,j=1}^{n}{({\sigma }_{ij}^{\ast \gamma })}^{2}.If we suppose that W{\mathcal{W}}is a dd-dimensional subspace of TMTM, d≥2d\ge 2, and {e1,e2,…,ed}\left\{{e}_{1},{e}_{2},\ldots ,{e}_{d}\right\}is an orthonormal basis of W{\mathcal{W}}, then the scalar curvature of the dd-plane section is given as τ(W)=∑1≤u<v≤dK(eu∧ev),\tau \left({\mathcal{W}})=\sum _{1\le u\lt v\le d}K\left({e}_{u}\wedge {e}_{v}),and the normalized scalar curvature ρ\rho is defined as ρ=2τs(s−1).\rho =\frac{2\tau }{s\left(s-1)}.Also, the Casorati curvature of the subspace W{\mathcal{W}}is given by C(W)=1d∑γ=r+1m∑i,j=1d(σijγ)2,C∗(W)=1d∑γ=r+1m∑i,j=1d(σij∗γ)2.{\mathcal{C}}\left({\mathcal{W}})=\frac{1}{d}\mathop{\sum }\limits_{\gamma =r+1}^{m}\mathop{\sum }\limits_{i,j=1}^{d}{({\sigma }_{ij}^{\gamma })}^{2},\hspace{1.0em}{{\mathcal{C}}}^{\ast }\left({\mathcal{W}})=\frac{1}{d}\mathop{\sum }\limits_{\gamma =r+1}^{m}\mathop{\sum }\limits_{i,j=1}^{d}{({\sigma }_{ij}^{\ast \gamma })}^{2}.A point x∈Mx\in Mis called as quasi-umbilical point, if at xxthere exist m−nm-nmutually orthogonal unit normal vectors ei{e}_{i}, i∈{n+1,…,m}i\in \left\{n+1,\ldots ,m\right\}in a way the shape operators with respect to all vectors ei{e}_{i}have an eigenvalue with multiplicity n−1n-1and for each ei{e}_{i}the distinguished eigen vector is the same.The normalized δ\delta -Casorati curvatures δc(n−1){\delta }_{c}\left(n-1)and δ^c(n−1){\widehat{\delta }}_{c}\left(n-1)of the submanifold Ms{M}^{s}are, respectively, given by [δc(n−1)]x=12Cx+n+12sinf{C(W)∣Wa hyperplane ofTxM}{{[}{\delta }_{c}\left(n-1)]}_{x}=\frac{1}{2}{{\mathcal{C}}}_{x}+\frac{n+1}{2s}{\rm{\inf }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\}and [δ^c(n−1)]x=2Cx−2n−12nsup{C(W)∣Wa hyperplane ofTxM}.{{[}{\widehat{\delta }}_{c}\left(n-1)]}_{x}=2{{\mathcal{C}}}_{x}-\frac{2n-1}{2n}{\rm{\sup }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\}.In [5], Decu et al. generalized the notion of normalized δ\delta -Casorati curvature to the generalized normalized δ\delta -Casorati curvatures δC(k;n−1){\delta }_{C}\left(k;\hspace{0.33em}n-1)and δ^C(k;n−1).{\widehat{\delta }}_{C}\left(k;\hspace{0.33em}n-1).For a submanifold Mn{M}^{n}and for any positive real number k≠n(n−1)k\ne n\left(n-1), the generalized normalized δ\delta -Casorati curvature is given by: [δC(k;n−1)]x=kCx+(n−1)(n+k)(n2−n−k)kninf{C(W)∣Wa hyperplane ofTxM},{{[}{\delta }_{C}\left(k;n-1)]}_{x}=k{{\mathcal{C}}}_{x}+\frac{\left(n-1)\left(n+k)\left({n}^{2}-n-k)}{kn}{\rm{\inf }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\},if 0<k<n2−n0\lt k\lt {n}^{2}-n, and [δ^C(k;n−1)]x=kCx−(n−1)(n+k)(k−n2+n)knsup{C(W)∣Wa hyperplane ofTxM},{{[}{\widehat{\delta }}_{C}\left(k;n-1)]}_{x}=k{{\mathcal{C}}}_{x}-\frac{\left(n-1)\left(n+k)\left(k-{n}^{2}+n)}{kn}{\rm{\sup }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\},if k>n2−nk\gt {n}^{2}-n.The generalized normalized δ\delta -Casorati curvatures δC(k:n−1){\delta }_{C}\left(k:n-1)and δˆC(k:n−1){\hat{\delta }}_{C}\left(k:n-1)are generalizations of normalized δ\delta -Casorati curvatures δC(n−1){\delta }_{C}\left(n-1)and δˆC(n−1){\hat{\delta }}_{C}\left(n-1). In fact, we have the following relations (see [5]): (2.8)δCn(n−1)2;n−1x=n(n−1)[δC(n−1)]x,{\left[{\delta }_{C}\left(\frac{n\left(n-1)}{2};n-1\right)\right]}_{x}=n\left(n-1){\left[{\delta }_{C}\left(n-1)]}_{x},(2.9)[δˆC(2n(n−1);n−1)]x=n(n−1)[δˆC(n−1)]x.\hspace{0.5em}{{[}{\hat{\delta }}_{C}(2n\left(n-1);n-1)]}_{x}=n\left(n-1){\left[{\hat{\delta }}_{C}\left(n-1)]}_{x}.In the same way, the dual Casorati curvatures are obtained just by replacing δ\delta and δ∗{\delta }^{\ast }and C{\mathcal{C}}by C∗{{\mathcal{C}}}^{\ast }.Now, we state the following fundamental results on statistical manifolds.Proposition 2[20] Let MMbe statistical submanifold of (M,g,ϕ)\left(M,g,\phi ). Let RRand R∗{R}^{\ast }be the Riemannian curvature tensors on MMfor ∇\nabla and ∇∗{\nabla }^{\ast }, respectively. Then we have the following.g(R(X,Y)Z,W)=g(R(X,Y)Z,W)+g(σ(X,Z),σ∗(Y,W))−g(σ∗(X,W),σ(Y,Z)),g(R∗(X,Y)Z,W)=g(R∗(X,Y)Z,W)+g(σ∗(X,Z),σ(Y,W))−g(σ(X,W),σ∗(Y,Z)),g(R⊥(X,Y)ξ,η)=g(R(X,Y)ξ,η)+g([Aξ∗,Aη]X,Y),g(R∗⊥(X,Y)ξ,η)=g(R∗(X,Y)ξ,η)+g([Aξ,Aη∗]X,Y),\begin{array}{l}g\left(R\left(X,Y)Z,W)=g\left(R\left(X,Y)Z,W)+g\left(\sigma \left(X,Z),{\sigma }^{\ast }\left(Y,W))-g\left({\sigma }^{\ast }\left(X,W),\sigma \left(Y,Z)),\\ g\left({R}^{\ast }\left(X,Y)Z,W)=g\left({R}^{\ast }\left(X,Y)Z,W)+g\left({\sigma }^{\ast }\left(X,Z),\sigma \left(Y,W))-g\left(\sigma \left(X,W),{\sigma }^{\ast }\left(Y,Z)),\\ g\left({R}^{\perp }\left(X,Y)\xi ,\eta )=g\left(R\left(X,Y)\xi ,\eta )+g\left(\left[{A}_{\xi }^{\ast },{A}_{\eta }]X,Y),\\ g\left({{R}^{\ast }}^{\perp }\left(X,Y)\xi ,\eta )=g\left({R}^{\ast }\left(X,Y)\xi ,\eta )+g\left(\left[{A}_{\xi },{A}_{\eta }^{\ast }]X,Y),\end{array}where [Aξ,Aη∗]=AξAη∗−Aη∗Aξ\left[{A}_{\xi },{A}_{\eta }^{\ast }]={A}_{\xi }{A}_{\eta }^{\ast }-{A}_{\eta }^{\ast }{A}_{\xi }and [Aξ∗,Aη]=Aξ∗Aη−AηAξ∗\left[{A}_{\xi }^{\ast },{A}_{\eta }]={A}_{\xi }^{\ast }{A}_{\eta }-{A}_{\eta }{A}_{\xi }^{\ast }, for X,Y,Z,W∈TMX,Y,Z,W\in TMand ξ,η∈T⊥M\xi ,\eta \in {T}^{\perp }M.Now, we state two important lemmas which we use to prove the main results in the upcoming sections.Lemma 1Let n≥3n\ge 3be an integer and a1,a2,…,an{a}_{1},{a}_{2},\ldots ,{a}_{n}are nnreal numbers. Then, we have∑1≤i<j≤nnaiaj−a1a2≤n−22(n−2)∑i=1nai2.\mathop{\sum }\limits_{1\le i\lt j\le n}^{n}{a}_{i}{a}_{j}-{a}_{1}{a}_{2}\le \frac{n-2}{2\left(n-2)}{\left(\mathop{\sum }\limits_{i=1}^{n}{a}_{i}\right)}^{2}.Moreover, the equality holds if and only if a1+a2=a3=⋯=an{a}_{1}+{a}_{2}={a}_{3}=\cdots ={a}_{n}.The optimization techniques have a pivotal role in improving inequalities involving Chen invariants. Oprea [35] applied the constrained extremum problem to prove Chen-Ricci inequalities for Lagrangian submanifolds of complex space forms. In the characterization of our main result, we will use the following lemma.Let MMbe a Riemannian submanifold in a Riemannian manifold (Mˆ,g)\left(\hat{M},g)and y:Mˆ→Ry:\hat{M}\to {\mathbb{R}}be a differentiable function. If we have the constrained extremum problem (2.10)minx∈M[y(x)].{{\rm{\min }}}_{x\in M}[y\left(x)].Then we have the following lemma.Lemma 2[35] If x0∈M{x}_{0}\in Mis a solution of the problem (2.10), then(1)(grady)(x0)∈Tx0⊥M\left({\rm{grad}}\hspace{0.33em}y)\left({x}_{0})\in {T}_{{x}_{0}}^{\perp }M;(2)The bilinear form Ω:Tx0M×Tx0M→R\Omega :{T}_{{x}_{0}}M\times {T}_{{x}_{0}}M\to {\mathbb{R}}defined byΩ(X,Y)=Hessy(X,Y)+g(σ(X,Y),(grady)(x0))\Omega \left(X,Y)={{\rm{Hess}}}_{y}\left(X,Y)+g\left(\sigma \left(X,Y),\left({\rm{grad}}\hspace{0.33em}y)\left({x}_{0}))is positive semi-definite, where σ\sigma is the second fundamental form of MMin Mˆ\hat{M}and grady{\rm{grad}}\hspace{0.33em}yis the gradient of yy.In principle, the bilinear form Ω\Omega is Hessy∣M(x0){}_{y| M}\left({x}_{0}). Therefore, if Ω\Omega is positive semi-definite on MM, then the critical points of y∣My| M, which coincide with the points where grad yyis normal to MM, are global optimal solutions of the problem (2.6) (for instance see [36, Remark 3.2]).3Main inequalitiesLet π\pi be a two plane spanned by {e1,e2}\left\{{e}_{1},{e}_{2}\right\}and denote g(ϕe1,e1)g(ϕ2,e2)=Ψ(π)g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})=\Psi \left(\pi ). Also, as in [37], for an orthonormal basis {e1,e2}\left\{{e}_{1},{e}_{2}\right\}of two-plane section, we denote Θ(π)=g(ϕe1,e2)g(ϕ∗e1,e2),\Theta \left(\pi )=g\left(\phi {e}_{1},{e}_{2})g\left({\phi }^{\ast }{e}_{1},{e}_{2}),where Θ(π)\Theta \left(\pi )is a real number in [0,1]\left[0,1].Theorem 1Let (N,g,ϕ)\left(N,g,\phi )be a Golden-like statistical manifold of dimension mmand MMbe its statistical submanifold of dimension nn. Then, we have the following: (τ−K(π))−(τ0−K0(π))≥−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]−n2(n−2)4(n−1)[‖H‖2+‖H∗‖2]+2Kˆ0(π)−2τˆ0.\begin{array}{l}\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))\\ \hspace{1.0em}\ge \left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )\\ \hspace{2.0em}+\left(\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]-\frac{{n}^{2}\left(n-2)}{4\left(n-1)}{[}\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2}]+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.\end{array}\hspace{2.55em}ProofLet {e1,e2,…,en}\left\{{e}_{1},{e}_{2},\ldots ,{e}_{n}\right\}and {en+1,…,em}\left\{{e}_{n+1},\ldots ,{e}_{m}\right\}be the orthonormal frames of TMTMand T⊥M{T}^{\perp }M, respectively.The scalar curvature corresponding to the sectional KK-curvature isτ=12∑1≤i<j≤n[g(R(ei,ej)ej,ei)+g(R∗(ei,ej)ej,ei)−2g(R0(ei,ej)ej,ei)].\tau =\frac{1}{2}\sum _{1\le i\lt j\le n}{[}g(R\left({e}_{i},{e}_{j}){e}_{j},{e}_{i})+g({R}^{\ast }\left({e}_{i},{e}_{j}){e}_{j},{e}_{i})-2g({R}^{0}\left({e}_{i},{e}_{j}){e}_{j},{e}_{i})].Using (2.7) and Gauss equation for RRand R∗{R}^{\ast }and doing some simple calculations, we obtain τ=−(1−ψ)cp−ψcq25[n(n−1)+tr2(ϕ)−tr(ϕ∗2)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−τ0+12∑γ=n+1m∑1≤i<j≤n[σii∗γσjj∗γ+σiiγσjj∗γ−2σij∗γσijγ].\tau =\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-1)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({{\phi }^{\ast }}^{2})]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )-{\tau }_{0}+\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}\hspace{0.25em}\sum _{1\le i\lt j\le n}{[}{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }+{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\ast \gamma }-2{\sigma }_{ij}^{\ast \gamma }{\sigma }_{ij}^{\gamma }].\hspace{20.4em}In view of (2.5), we obtain τ=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−τ0+12∑γ=n+1m∑1≤i<j≤n[σii∗γσjj∗γ+σiiγσjj∗γ−2σij∗γσijγ],\tau =\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )-{\tau }_{0}+\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}\hspace{0.25em}\sum _{1\le i\lt j\le n}{[}{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }+{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\ast \gamma }-2{\sigma }_{ij}^{\ast \gamma }{\sigma }_{ij}^{\gamma }],\hspace{20.35em}which can be written as τ=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−τ0+2∑γ=n+1m∑1≤i<j≤n[σii0γσjj0γ−(σij0γ)2]−12∑γ=n+1m∑1≤i<j≤n[{σiiγσjjγ+(σijγ)2}+{σii∗γσjj∗γ−(σij∗γ)2}].\hspace{-46em}\begin{array}{rcl}\tau & =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )\\ & & -{\tau }_{0}+2\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\hspace{0.16em}\displaystyle \sum _{1\le i\lt j\le n}\left[{\sigma }_{ii}^{0\gamma }{\sigma }_{jj}^{0\gamma }-{\left({\sigma }_{ij}^{0\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\hspace{0.25em}\displaystyle \sum _{1\le i\lt j\le n}\left[\left\{{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\gamma }+{\left({\sigma }_{ij}^{\gamma })}^{2}\right\}+\left\{{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\ast \gamma })}^{2}\right\}].\end{array}By using Gauss equation for the Levi-Civita connection, we have (3.1)τ=τ0+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−2τˆ0−12∑γ=n+1m∑1≤i<j≤n[{σiiγσjj∗γ−(σijγ)2}+{σii∗γσjj∗γ−(σij∗γ)2}].\begin{array}{rcl}\tau & =& {\tau }_{0}+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )-2{\hat{\tau }}_{0}\\ & & -\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\hspace{0.16em}\displaystyle \sum _{1\le i\lt j\le n}{[}\left\{{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\gamma })}^{2}\right\}+\left\{{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\ast \gamma })}^{2}\right\}].\end{array}Now, the sectional KK-curvature K(π)K\left(\pi )of the plane section π\pi is(3.2)K(π)=12[g(R(e1,e2)e2,e1)+g(R∗(e1,e2)e2,e1)−2g(R0(e1,e2)e2,e1)].K\left(\pi )=\frac{1}{2}{[}g(R\left({e}_{1},{e}_{2}){e}_{2},{e}_{1})+g({R}^{\ast }\left({e}_{1},{e}_{2}){e}_{2},{e}_{1})-2g({R}^{0}\left({e}_{1},{e}_{2}){e}_{2},{e}_{1})].Using (2.7) and Gauss equation for RRand R∗{R}^{\ast }and putting the values in (3.2), we obtain K(π)=−(1−ψ)cp−ψcq25[1+g(ϕe1,e1)g(ϕ2,e2)−g(ϕe1,e2)g(ϕe2,e1)]−K0(π)+12∑γ=n+1m{[σ11γσ22∗γ+σ11∗γσ22γ−2σ12∗γσ12γ]}.\begin{array}{rcl}K\left(\pi )& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})-g\left(\phi {e}_{1},{e}_{2})g\left(\phi {e}_{2},{e}_{1})]-{K}_{0}\left(\pi )\\ & & +\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\{{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\ast \gamma }+{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\gamma }-2{\sigma }_{12}^{\ast \gamma }{\sigma }_{12}^{\gamma }]\}.\end{array}\hspace{2.75em}Using σ+σ∗=2σ0\sigma +{\sigma }^{\ast }=2{\sigma }^{0}, we obtain K(π)=−(1−ψ)cp−ψcq25[1+g(ϕe1,e1)g(ϕ2,e2)−g(ϕe1,e2)g(ϕ2,e1)]−K0(π)+2∑γ=n+1m[σ110γσ220γ−(σ120γ)2]−12∑γ=n+1m{[σ11γσ22γ−(σ12γ)2]+[σ11∗γσ22∗γ−(σ12∗γ)2]}.\begin{array}{rcl}K\left(\pi )& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})-g\left(\phi {e}_{1},{e}_{2})g\left({\phi }_{2},{e}_{1})]\\ & & -{K}_{0}\left(\pi )+2\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{0\gamma }{\sigma }_{22}^{0\gamma }-{\left({\sigma }_{12}^{0\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\{{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]+{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}]\}.\end{array}\hspace{0.3em}Using Gauss equation with respect to Levi-Civita connection, we have K(π)=K0(π)+−(1−ψ)cp−ψcq25[1+g(ϕe1,e1)g(ϕ2,e2)+g(ϕe1,e2)g(ϕ∗e1,e2)]−2Kˆ0(π)−12∑γ=n+1m[σ11γσ22γ−(σ12γ)2]−12∑γ=n+1m[σ11∗γσ22∗γ−(σ12∗γ)2].\begin{array}{rcl}K\left(\pi )& =& {K}_{0}\left(\pi )+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})+g\left(\phi {e}_{1},{e}_{2})g\left({\phi }^{\ast }{e}_{1},{e}_{2})]\\ & & -2{\hat{K}}_{0}\left(\pi )-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}].\end{array}\hspace{2.45em}The above equation can be written in the form(3.3)K(π)=K0(π)+−(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]−2Kˆ0(π)−12∑γ=n+1m[σ11γσ22γ−(σ12γ)2]−12∑γ=n+1m[σ11∗γσ22∗γ−(σ12∗γ)2].K\left(\pi )={K}_{0}\left(\pi )+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]-2{\hat{K}}_{0}\left(\pi )-\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]-\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}].From (3.1) and (3.3), we have (τ−K(π))−(τ0−K0(π))=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]\begin{array}{rcl}\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )+\left(\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]\end{array}−12∑γ=n+1m[σiiγσjjγ−(σijγ)2]−12∑γ=n+1m[σii∗γσjj∗γ−(σij∗γ)2]+12∑γ=n+1m∑α=13{[σ11γσ22γ−(σ12γ)2]+[σ11∗γσ22∗γ−(σ12∗γ)2]}+2Kˆ0(π)−2τˆ0.\begin{array}{rcl}& & -\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\gamma }-{\left({\sigma }_{ij}^{\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\ast \gamma })}^{2}]\\ & & +\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\mathop{\displaystyle \sum }\limits_{\alpha =1}^{3}\{{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]+{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}]\}+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.\end{array}Using Lemma 1, we can obtain the above equation in simplified form as (τ−K(π))−(τ0−K0(π))≥−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]−n2(n−2)4(n−1)[‖H‖2+‖H∗‖2]+2Kˆ0(π)−2τˆ0.\begin{array}{rcl}\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))& \ge & \left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )+\left(\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]\\ & & -\frac{{n}^{2}\left(n-2)}{4\left(n-1)}{[}\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2}]+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.\end{array}This proves our claims.□Corollary 1Let (N,g,ϕ)\left(N,g,\phi )be a Golden-like statistical manifold of dimension mmand MMbe its totally real statistical submanifold of dimension nn. Then, we have the following(τ−K(π))−(τ0−K0(π))≥−(1−ψ)cp−ψcq25[n(n−2)−1]−n2(n−2)4(n−1)[‖H‖2+‖H∗‖2]+2Kˆ0(π)−2τˆ0.\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))\ge \left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)-1]-\frac{{n}^{2}\left(n-2)}{4\left(n-1)}{[}\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2}]+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.Theorem 2Let Mn{M}^{n}be a statistical submanifold of a Golden-like statistical manifold Nm{N}^{m}. Then for the generalized normalized δ\delta -Casorati curvature, we have the following optimal relationships: (i)For any real number kk, such that 0<k<n(n−1)0\lt k\lt n\left(n-1), (3.4)ρ≤δC0(k;n−1)n(n−1)+1(n−1)C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+1n(n−1)−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+2n−(1−ψ)cp+ψcq4tr(ϕ),\rho \le \frac{{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{\left(n-1)}{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\frac{1}{n\left(n-1)}\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right){[}n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ),where δC0(k;n−1)=12[δC(k;n−1)+δC∗(k;n−1)]{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\delta }_{C}\left(k;\hspace{0.33em}n-1)+{\delta }_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].(ii)For any real number k>n(n−1)k\gt n\left(n-1), (3.5)ρ≤δ^C0(k;n−1)n(n−1)+1(n−1)C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+1n(n−1)−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+2n−(1−ψ)cp+ψcq4tr(ϕ),\rho \le \frac{{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{\left(n-1)}{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\frac{1}{n\left(n-1)}\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right){[}n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ),where δ^C0(k;n−1)=12[δ^C(k;n−1)+δ^C∗(k;n−1)]{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\widehat{\delta }}_{C}\left(k;\hspace{0.33em}n-1)+{\widehat{\delta }}_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].ProofLet p∈Mp\in Mand {e1,…,en}\left\{{e}_{1},\ldots ,{e}_{n}\right\}, {en+1,…,em}\left\{{e}_{n+1},\ldots ,{e}_{m}\right\}be the orthonormal basis of TpM{T}_{p}Mand Tp⊥M{T}_{p}^{\perp }M, respectively. From Gauss equation, we obtain 2τ=n2g(H,H∗)−n∑1≤i,j≤ng(σ∗(ei,ej),σ(ei,ej))+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ).2\tau ={n}^{2}g\left(H,{H}^{\ast })-n\sum _{1\le i,j\le n}g\left({\sigma }^{\ast }\left({e}_{i},{e}_{j}),\sigma \left({e}_{i},{e}_{j}))+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi ).\hspace{24.85em}Denote H+H∗=2H0H+{H}^{\ast }=2{H}^{0}and C+C∗=2C0{\mathcal{C}}+{{\mathcal{C}}}^{\ast }=2{{\mathcal{C}}}^{0}. Then the above equation becomes 2τ=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+2n2‖H0‖2−n22(‖H‖2+‖H∗‖2)−2nC0+n2(C+C∗).2\tau =\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )+2{n}^{2}\Vert {H}^{0}{\Vert }^{2}-\frac{{n}^{2}}{2}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})-2n{{\mathcal{C}}}^{0}+\frac{n}{2}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast }).We define a polynomial P{\mathcal{P}}in the components of second fundamental form as: (3.6)P=kC0+a(k)C0(W)+n2(C+C∗)−n22(‖H‖2+‖H∗‖2)−2τ(p)+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ),{\mathcal{P}}=k{{\mathcal{C}}}^{0}+a\left(k){{\mathcal{C}}}^{0}\left({\mathcal{W}})+\frac{n}{2}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast })-\frac{{n}^{2}}{2}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})-2\tau \left(p)+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi ),where W{\mathcal{W}}is a hyperplane in TpM{T}_{p}M. Assuming W{\mathcal{W}}is spanned by {e1,…,en−1}\left\{{e}_{1},\ldots ,{e}_{n-1}\right\}, we have (3.7)P=∑α=n+1m2n+kn∑i,j=1n(σij0α)2+a(k)1n−1∑i,j=1n−1(σij0α)2−2∑i,j=1nσij0α2.{\mathcal{P}}=\mathop{\sum }\limits_{\alpha =n+1}^{m}\left[\frac{2n+k}{n}\mathop{\sum }\limits_{i,j=1}^{n}{\left({\sigma }_{ij}^{0\alpha })}^{2}+a\left(k)\frac{1}{n-1}\mathop{\sum }\limits_{i,j=1}^{n-1}{\left({\sigma }_{ij}^{0\alpha })}^{2}-2{\left(\mathop{\sum }\limits_{i,j=1}^{n}{\sigma }_{ij}^{0\alpha }\right)}^{2}\right].\hspace{31.75em}Equation (3.7) yields P=∑α=n+1n2(2n+k)n+2a(k)n−1∑1≤i<j≤n−1(σij0α)2+2(2n+k)n+2a(k)n−1∑i=1n−1(σin0α)22n+kn+a(k)n−1−2∑i=1n−1(σii0α)2−4∑1≤i<j≤nσii0ασjj0α+2n+kn−2(σnn0α)2≥∑α=n+1mk(n−1)+a(k)nn(n−1)∑i=1n−1(σii0α)2+km(σnn0α)2−4∑1≤i<j≤nσii0ασjj0α.\begin{array}{rcl}{\mathcal{P}}& =& \mathop{\displaystyle \sum }\limits_{\alpha =n+1}^{n}\left[\left(\frac{2\left(2n+k)}{n}+\frac{2a\left(k)}{n-1}\right)\displaystyle \sum _{1\le i\lt j\le n-1}{\left({\sigma }_{ij}^{0\alpha })}^{2}+\left(\frac{2\left(2n+k)}{n}+\frac{2a\left(k)}{n-1}\right)\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{\left({\sigma }_{in}^{0\alpha })}^{2}\left(\frac{2n+k}{n}+\frac{a\left(k)}{n-1}-2\right)\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{\left({\sigma }_{ii}^{0\alpha })}^{2}-4\displaystyle \sum _{1\le i\lt j\le n}{\sigma }_{ii}^{0\alpha }{\sigma }_{jj}^{0\alpha }+\left(\frac{2n+k}{n}-2\right){\left({\sigma }_{nn}^{0\alpha })}^{2}\right]\\ & \ge & \mathop{\displaystyle \sum }\limits_{\alpha =n+1}^{m}\left[\frac{k\left(n-1)+a\left(k)n}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{\left({\sigma }_{ii}^{0\alpha })}^{2}+\frac{k}{m}{\left({\sigma }_{nn}^{0\alpha })}^{2}-4\displaystyle \sum _{1\le i\lt j\le n}{\sigma }_{ii}^{0\alpha }{\sigma }_{jj}^{0\alpha }\right].\end{array}Let yα{y}_{\alpha }be a quadratic form defined by yα=Rn→R{y}_{\alpha }={{\mathbb{R}}}^{n}\to {\mathbb{R}}for any α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}, yα(σ110α,σ220α,…,σnn0α)=∑i=1n−1k(n−1)+a(k)nn(n−1)(σii0α)2+kn(σnn0α)2−4∑1≤i<j≤nσii0ασjj0α,{y}_{\alpha }\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })=\mathop{\sum }\limits_{i=1}^{n-1}\frac{k\left(n-1)+a\left(k)n}{n\left(n-1)}{\left({\sigma }_{ii}^{0\alpha })}^{2}+\frac{k}{n}{\left({\sigma }_{nn}^{0\alpha })}^{2}-4\sum _{1\le i\lt j\le n}{\sigma }_{ii}^{0\alpha }{\sigma }_{jj}^{0\alpha },and the optimization problem min{yα}{\rm{\min }}\{{y}_{\alpha }\}subject to G:σ110α+σ220α+…+σnn0α=pα{\mathcal{G}}:{\sigma }_{11}^{0\alpha }+{\sigma }_{22}^{0\alpha }+\ldots +{\sigma }_{nn}^{0\alpha }={p}^{\alpha }, where pα{p}^{\alpha }is a real constant.The partial derivatives of yα{y}_{\alpha }are (3.8)∂yα∂σii0α=2k(n−1)+a(k)nn(n−1)σii0α−4∑l=1nσll0α−σii0α=0∂yα∂σnn0α=2knσnn0α−4∑l=1n−1σll0α=0,\left\{\begin{array}{l}\frac{\partial {y}_{\alpha }}{\partial {\sigma }_{ii}^{0\alpha }}=2\frac{k\left(n-1)+a\left(k)n}{n\left(n-1)}{\sigma }_{ii}^{0\alpha }-4\left(\mathop{\displaystyle \sum }\limits_{l=1}^{n}{\sigma }_{ll}^{0\alpha }-{\sigma }_{ii}^{0\alpha }\right)=0\\ \frac{\partial {y}_{\alpha }}{\partial {\sigma }_{nn}^{0\alpha }}=\frac{2k}{n}{\sigma }_{nn}^{0\alpha }-4\mathop{\displaystyle \sum }\limits_{l=1}^{n-1}{\sigma }_{ll}^{0\alpha }=0,\end{array}\right.for every i∈{1,…,n−1}i\in \left\{1,\ldots ,n-1\right\}, α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}.For an optimal solution (σ110α,σ220α,…,σnn0α)\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })of the problem, the vector grad yα{y}_{\alpha }is normal at G{\mathcal{G}}. It is collinear with the vector (1,…,1)\left(1,\ldots ,1). From (3.8) and ∑i=1nσii0α=pα{\sum }_{i=1}^{n}{\sigma }_{ii}^{0\alpha }={p}^{\alpha }it follows that a critical point of the corresponding problem has the form σii0α=2n(n−1)(n−1)(2n+k)+na(k)pασnn0α=2n2n+kpα,\left\{\begin{array}{l}{\sigma }_{ii}^{0\alpha }=\frac{2n\left(n-1)}{\left(n-1)\left(2n+k)+na\left(k)}{p}^{\alpha }\\ {\sigma }_{nn}^{0\alpha }=\frac{2n}{2n+k}{p}^{\alpha },\end{array}\right.\hspace{8.25em}for any i∈{1,…,n−1}i\in \left\{1,\ldots ,n-1\right\}, α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}.For p∈Gp\in {\mathcal{G}}, let A{\mathcal{A}}be a 2-form, A:TpG×G→R{\mathcal{A}}:{T}_{p}{\mathcal{G}}\times {\mathcal{G}}\to {\mathbb{R}}defined by A(X,Y)=Hess(yα)(X,Y)+⟨σ′(X,Y),(grad(y)(p))⟩,{\mathcal{A}}\left(X,Y)={\rm{Hess}}({y}_{\alpha })\left(X,Y)+\langle {\sigma }^{^{\prime} }\left(X,Y),\left({\rm{grad}}(y)\left(p))\rangle ,where σ′{\sigma }^{^{\prime} }is the second fundamental form of G{\mathcal{G}}in Rn{{\mathbb{R}}}^{n}and ⟨⋅,⋅⟩\langle \cdot ,\hspace{0.33em}\cdot \rangle is the standard inner product on Rn{{\mathbb{R}}}^{n}.Moreover, it is easy to see that the Hessian matrix of yα{y}_{\alpha }has the form Hess(yα)=2(n−1)(k+2n)+na(k)n(n−1)−4⋯−4−4−42(n−1)(k+2n)+na(k)n(n−1)⋯−4−4⋮⋮⋱⋮⋮−4−4⋯2(n−1)(k+2n)+na(k)n(n−1)−4−4−4⋯−42kn.{\rm{Hess}}({y}_{\alpha })=\left(\begin{array}{ccccc}2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}& -\hspace{-0.25em}4& \cdots \hspace{0.33em}& -4& -\hspace{-0.25em}4\\ -\hspace{-0.25em}4& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}& \cdots \hspace{0.33em}& -4& -\hspace{-0.25em}4\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ -\hspace{-0.25em}4& -\hspace{-0.25em}4& \cdots \hspace{0.33em}& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}& -\hspace{-0.25em}4\\ -\hspace{-0.25em}4& -\hspace{-0.25em}4& \cdots \hspace{0.33em}& -4& \frac{2k}{n}\end{array}\right).As G{\mathcal{G}}is a totally geodesic hyperplane in Rn,{{\mathbb{R}}}^{n},considering a vector X=(X1,…,Xn)X=\left({X}_{1},\ldots ,{X}_{n})tangent to G{\mathcal{G}}at an arbitrary point xxon G{\mathcal{G}}, that is, verifying the relation ∑i=1n=0{\sum }_{i=1}^{n}=0, we obtain A(X,X)=2(n−1)(k+2n)+na(k)n(n−1)∑i=1n−1Xi2+2knXn2−8∑i,j=1nXiXj,i≠j=2(n−1)(k+2n)+na(k)n(n−1)∑i=1n−1Xi2+2knXn2+4∑i=1nXi2−8∑i,j=1nXiXj,i≠j=2(n−1)(k+2n)+na(k)n(n−1)∑i=1n−1Xi2+2knXn2+4∑i=1nXi2≥0.\begin{array}{rcl}{\mathcal{A}}\left(X,X)& =& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{X}_{i}^{2}+\frac{2k}{n}{X}_{n}^{2}-8\mathop{\displaystyle \sum }\limits_{i,j=1}^{n}{X}_{i}{X}_{j},\hspace{1.0em}i\ne j\\ & =& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{X}_{i}^{2}+\frac{2k}{n}{X}_{n}^{2}+4{\left(\mathop{\displaystyle \sum }\limits_{i=1}^{n}{X}_{i}\right)}^{2}-8\mathop{\displaystyle \sum }\limits_{i,j=1}^{n}{X}_{i}{X}_{j},\hspace{1.0em}i\ne j\\ & =& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{X}_{i}^{2}+\frac{2k}{n}{X}_{n}^{2}+4\mathop{\displaystyle \sum }\limits_{i=1}^{n}{X}_{i}^{2}\ge 0.\end{array}Hence, by Lemma 2, the critical point (σ110α,σ220α,…,σnn0α)\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })of yα{y}_{\alpha }is the global minimum point of the problem. Moreover, since yα(σ110α,σ220α,…,σnn0α)=0,{y}_{\alpha }\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })=0,we obtain P≥0.{\mathcal{P}}\ge 0.This implies that 2τ≤kC0+a(k)C0(W)+n2(C+C0)−n22(‖H‖2+‖H∗‖2)+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ).2\tau \le k{{\mathcal{C}}}^{0}+a\left(k){{\mathcal{C}}}^{0}\left({\mathcal{W}})+\frac{n}{2}\left({\mathcal{C}}+{{\mathcal{C}}}^{0})-\frac{{n}^{2}}{2}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi ).Thus, we have (3.9)ρ≤kn(n−1)C0+(n+k)(n2−n−k)n2kC0(W)+12(n−1)(C+C∗)−n2(n−1)(‖H‖2+‖H∗‖2)+−(1−ψ)cp−ψcq25(n−2)(n−1)+tr2(ϕ)n(n−1)−tr(ϕ∗)n(n−1)+2n−(1−ψ)cp+ψcq4tr(ϕ).\begin{array}{rcl}\rho & \le & \frac{k}{n\left(n-1)}{{\mathcal{C}}}^{0}+\frac{\left(n+k)\left({n}^{2}-n-k)}{{n}^{2}k}{{\mathcal{C}}}^{0}\left({\mathcal{W}})+\frac{1}{2\left(n-1)}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast })-\frac{n}{2\left(n-1)}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[\frac{\left(n-2)}{\left(n-1)}+\frac{{{\rm{tr}}}^{2}\left(\phi )}{n\left(n-1)}-\frac{{\rm{tr}}\left({\phi }^{\ast })}{n\left(n-1)}\right]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ).\end{array}Now taking the infimum over all tangent hyperplanes W{\mathcal{W}}of TpM{T}_{p}M, we obtain ρ≤δC(k;n−1)n(n−1)+12(n−1)(C+C∗)−n2(n−1)(‖H‖2+‖H∗‖2)+−(1−ψ)cp−ψcq25(n−2)(n−1)+tr2(ϕ)n(n−1)−tr(ϕ∗)n(n−1)+2n−(1−ψ)cp+ψcq4tr(ϕ).\begin{array}{rcl}\rho & \le & \frac{{\delta }_{C}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{2\left(n-1)}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast })-\frac{n}{2\left(n-1)}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[\frac{\left(n-2)}{\left(n-1)}+\frac{{{\rm{tr}}}^{2}\left(\phi )}{n\left(n-1)}-\frac{{\rm{tr}}\left({\phi }^{\ast })}{n\left(n-1)}\right]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ).\end{array}This gives us the inequality (3.4). Similarly on taking the supremum over all tangent hyperplanes W{\mathcal{W}}of TpM{T}_{p}Min (3.9), we obtain the inequality (3.5).□Corollary 2Let Mn{M}^{n}be a totally real statistical submanifold of a Golden-like statistical manifold Nm{N}^{m}. Then for the generalized normalized δ\delta -Casorati curvature, we have the following optimal relationships: (i)For any real number kk, such that 0<k<n(n−1)0\lt k\lt n\left(n-1), ρ≤δC0(k;n−1)n(n−1)+C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+−(1−ψ)cp−ψcq25n−2n−1,\rho \le \frac{{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left(\frac{n-2}{n-1}\right),where δC0(k;n−1)=12[δC(k;n−1)+δC∗(k;n−1)]{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\delta }_{C}\left(k;\hspace{0.33em}n-1)+{\delta }_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].(ii)For any real number k>n(n−1)k\gt n\left(n-1), ρ≤δ^C0(k;n−1)n(n−1)+1(n−1)C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+−(1−ψ)cp−ψcq25n−2n−1,\rho \le \frac{{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{\left(n-1)}{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left(\frac{n-2}{n-1}\right),where δ^C0(k;n−1)=12[δ^C(k;n−1)+δ^C∗(k;n−1)]{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\widehat{\delta }}_{C}\left(k;\hspace{0.33em}n-1)+{\widehat{\delta }}_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].3.1Equality caseThe submanifolds enjoying the equality for the Casorati curvature at every point are called Casorati ideal submanifolds (for instance see [38]). In this subsection, we investigate the Casorati ideal submanifolds for (3.4) and (3.5) in terms of their second fundamental form.Theorem 3The Casorati ideal Lagrangian submanifolds for (3.4) and (3.5) are totally geodesic submanifolds with respect to Levi-Civita connection.ProofFirst, we find out the critical points of P{\mathcal{P}}σc=(σ110n+1,σ120n+1,…,σnn0n+1,…,σ110m,…,σnn0m){\sigma }^{c}=\left({\sigma }_{11}^{0n+1},{\sigma }_{12}^{0n+1},\ldots ,{\sigma }_{nn}^{0n+1},\ldots ,{\sigma }_{11}^{0m},\ldots ,{\sigma }_{nn}^{0m})as the solutions of the following system of linear homogeneous equations ∂P∂σii0α=22n+kk+(k+n)(n2−n−k)nk−2σii0α−4∑l=1,l≠inσll0α=0,∂P∂σnn0α=2knσnn0α−4∑l=1n−1σll0α=0,∂P∂σij0α=42n+kk+(k+n)(n2−n−k)nkσij0α=0,i≠j,∂P∂σin0α=42n+kk+(k+n)(n2−n−k)nkσin0α=0.\left\{\begin{array}{l}\frac{\partial {\mathcal{P}}}{\partial {\sigma }_{ii}^{0\alpha }}=2\left[\frac{2n+k}{k}+\frac{\left(k+n)\left({n}^{2}-n-k)}{nk}-2\right]{\sigma }_{ii}^{0\alpha }-4\mathop{\displaystyle \sum }\limits_{l=1,l\ne i}^{n}{\sigma }_{ll}^{0\alpha }=0,\\ \frac{\partial {\mathcal{P}}}{\partial {\sigma }_{nn}^{0\alpha }}=2\frac{k}{n}{\sigma }_{nn}^{0\alpha }-4\mathop{\displaystyle \sum }\limits_{l=1}^{n-1}{\sigma }_{ll}^{0\alpha }=0,\\ \frac{\partial {\mathcal{P}}}{\partial {\sigma }_{ij}^{0\alpha }}=4\left[\frac{2n+k}{k}+\frac{\left(k+n)\left({n}^{2}-n-k)}{nk}\right]{\sigma }_{ij}^{0\alpha }=0,\hspace{1.0em}i\ne j,\\ \frac{\partial {\mathcal{P}}}{\partial {\sigma }_{in}^{0\alpha }}=4\left[\frac{2n+k}{k}+\frac{\left(k+n)\left({n}^{2}-n-k)}{nk}\right]{\sigma }_{in}^{0\alpha }=0.\end{array}\right.The critical points satisfy σij0α=0{\sigma }_{ij}^{0\alpha }=0, i,j=∈{1,…,n}i,j=\in \left\{1,\ldots ,n\right\}and α={n+1,…,m}\alpha =\left\{n+1,\ldots ,m\right\}. Moreover, we know that P≥0{\mathcal{P}}\ge 0and P(σc)=0,{\mathcal{P}}\left({\sigma }^{c})=0,then the critical point σc{\sigma }^{c}is a minimum point of P.{\mathcal{P}}.Consequently, the equality holds in (3.4) and (3.5) if and only if σijα+σij∗α=0,{\sigma }_{ij}^{\alpha }+{\sigma }_{ij}^{\ast \alpha }=0,for i,j∈{1,…,n}i,j\in \left\{1,\ldots ,n\right\}and α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}. In other words, the equalities hold identically at all points p∈Mp\in Mif and only if σ+σ∗=0,\sigma +{\sigma }^{\ast }=0,where σ\sigma and σ∗{\sigma }^{\ast }are the imbedding curvature tensors of the submanifold associated with the dual connection ∇\nabla and ∇∗{\nabla }^{\ast }, respectively. Hence, the equality in (3.4) and (3.5) holds at ppif and only if ppis totally geodesic point with respect to Levi-Civita connection.□Remark 1The results for normalized Casorati curvature can be easily obtained by using (2.8) and (2.9) in the inequalities (3.4) and (3.5).4ConclusionIn this paper, we introduced and studied Golden-like statistical manifolds. We obtained some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provided a couple of examples. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

# Basic inequalities for statistical submanifolds in Golden-like statistical manifolds

, Volume 20 (1): 14 – Jan 1, 2022
14 pages

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de Gruyter
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© 2022 Mohamd Saleem Lone et al., published by De Gruyter
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10.1515/math-2022-0017
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### Abstract

1IntroductionThe comparison relationships between the intrinsic and extrinsic invariants are the basic problems in submanifold theory. In [1], Chen introduce some curvature invariants and for their usages, he derived optimal relationships between the intrinsic invariants (Chen invariants) and the extrinsic invariants, which become later an active and fruitful area of research (see, for instance, [1,2,3]).On the other hand, the notion of Casorati curvature (extrinsic invariant) for the surfaces was originally introduced in 1890 (see [4]). The Casorati curvature gives a better intuition of the curvature compared to the Gaussian curvature. The Gaussian curvature of a developable surface is zero. Thus, Casorati put forward the notion of Casorati curvature of a surface defined as C=1/2(1/κ12+1/κ22){\mathcal{C}}=1\hspace{-0.08em}\text{/}\hspace{-0.08em}2\left(1\hspace{-0.08em}\text{/}{\kappa }_{1}^{2}+1\text{/}\hspace{-0.08em}{\kappa }_{2}^{2}). For example, for developable surfaces (say, cylinder), the Gaussian curvature vanishes, while the Casorti curvature C{\mathcal{C}}surely does not vanish. The Casorati curvature of a submanifold in a Riemannian manifold is defined as the normalized square length of the second fundamental form [5].In the past decade, various geometers attracted toward the study of Chen-type comparison relationships between the Casorati curvature and the intrinsic invariants. For some references in this direction we refer to [6,7, 8,9,10, 11,12]. The submanifolds with equality case in the Chen-type inequalities are called ideal submanifolds and the name ideal is motivated by the fact that these submanifolds inherit the least possible tension from the ambient manifold (see [13]).In 1985, Amari introduced the notion of statistical manifolds via information geometry (see [14]). Statistical manifolds are endowed with a pair of dual torsion-free connections. This is analogous to conjugate connections in affine geometry (see [15]). The dual connections are not metric, thus it is very tough to give a notion of sectional curvature using the canonical definitions of Riemannian geometry. In [16], Opozda gave the definition of sectional curvature tensor on a statistical manifold. While studying the geometric properties of a submanifold, a very important problem is to obtain sharp relations between the intrinsic and the extrinsic invariants, and a vast number of such relations are revealed by certain inequalities. For example, let MMbe a surface in Euclidean 3-space, we know the Euler inequality: K≤∣H∣2K\le | H\hspace{-0.25em}{| }^{2}, where HHis the mean curvature (extrinsic property) and KKis the Gaussian curvature (intrinsic property). The equality holds at points where MMis congruent to an open piece of a plane or a sphere (umbilical points). Chen [17] obtained the same inequality for submanifolds of real space forms. Then in [18], Chen obtained the Chen-Ricci inequality, which is a sharp relation between the squared mean curvature and the Ricci curvature of a Riemannian submanifold of a real space form.In recent years, statistical manifolds have been studied very actively. In [19], Takano studied statistical manifolds with almost complex and almost contact structure. In 2015, Vîlcu and Vîlcu [20] studied statistical manifolds with quaternionic settings and proposed several open problems. While answering one of those open problems, Aquib [21] obtained some of the curvature properties of submanifolds and a couple of inequalities for totally real statistical submanifolds of quaternionic Kaehler-like statistical space forms. In 2019, Chen et al. derived a Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature [22]. In the same year, following the same paper of Chen et al., Atimur et al. [23] obtained Chen-type inequalities for statistical submanifolds of Kaehler-like statistical manifolds. Very recently, in 2020, Decu et al. obtained inequalities for the Casorati curvature of statistical manifolds in holomorphic statistical manifolds of constant holomorphic curvature [24]. For some of the recent works, we refer to [15, 19, 25, 26,27,28].Motivated by the aforementioned studies, we define Golden-like statistical manifolds and obtain certain interesting inequalities. The structure of this paper is as follows. In Section 2, we first give the definition of Golden-like statistical manifolds. We also construct an example for the Golden-like statistical manifolds. In the next section, we obtain the main inequalities. We also prove the results for their equality cases.2Golden-like statistical manifoldLet MMbe a smooth manifold. A (1,1)\left(1,1)tensor field TTon MMis said to be polynomial structure if TTsatisfies an algebraic equation [29,30]P(x)=xn+bnxn−1+⋯+b2x+b1I=0,P\left(x)={x}^{n}+{b}_{n}{x}^{n-1}+\cdots +{b}_{2}x+{b}_{1}I=0,where IIis the (1,1)\left(1,1)identity tensor field and Tn−1(q),Tn−2(q),…,T(q),I{T}^{n-1}\left(q),{T}^{n-2}\left(q),\ldots ,T\left(q),Iare linearly independent at every point q∈M.q\in M.The polynomial P(x)P\left(x)is called the structure polynomial. For P(x)=x2+IP\left(x)={x}^{2}+Iand P(x)=x2−IP\left(x)={x}^{2}-I, we obtain an almost complex structure and an almost product structure, respectively. It has to be noted here that the existence of almost complex structure implies the even dimensions of the manifold. For P(x)=x2P\left(x)={x}^{2}, we obtain the notion of an almost tangent structure.Definition 1[29,31,32] Let (M,g)\left(M,g)be the a semi-Riemannian manifold and let ϕ\phi be the (1,1)\left(1,1)tensor field on MMsatisfying the following equation: ϕ2=ϕ+I.{\phi }^{2}=\phi +I.Then the tensor field ϕ\phi is called a Golden structure on MM. If the Riemannian metric ggis ϕ\phi compatible, the (M,g,ϕ)\left(M,g,\phi )is called a Golden semi-Riemannian manifold.For ϕ\phi compatible metric gg, we have the following: (2.1)g(ϕX,Y)=g(X,ϕY),g\left(\phi X,Y)=g\left(X,\phi Y),(2.2)g(ϕX,ϕY)=g(ϕ2X,Y)=g(ϕX,Y)+g(X,Y),X,Y∈Γ(TM).g\left(\phi X,\phi Y)=g\left({\phi }^{2}X,Y)=g\left(\phi X,Y)+g\left(X,Y),\hspace{1.0em}X,Y\in \Gamma \left(TM).A remarkable fact about Golden structures is its appearance in pairs, i.e., if ϕ\phi is Golden structure, the ϕˆ=I−ϕ\hat{\phi }=I-\phi is also a Golden structure. But same is the case with almost tangent (RRand −R-R) and almost complex structure (JJand −J-J). So it is natural to ask the connection between Golden and product structures.Let MMbe a Riemannian manifold. Denote a torsion-free affine connection by ∇\nabla . The triple (M,∇,g)\left(M,\nabla ,g)is called a statistical manifold if ∇g\nabla gis symmetric. We define another affine connection ∇∗{\nabla }^{\ast }by (2.3)Xg(Y,Z)=g(∇XY,Z)+g(∇X∗Z,Y)Xg\left(Y,Z)=g\left({\nabla }_{X}Y,Z)+g\left({\nabla }_{X}^{\ast }Z,Y)for vector fields EE, FF, and GGon MM. The affine connection ∇∗{\nabla }^{\ast }is called conjugate (or dual) to ∇\nabla with respect to gg. The affine connection ∇∗{\nabla }^{\ast }is torsion-free, ∇∗g{\nabla }^{\ast }gis symmetric and satisfies ∇0=∇+∇∗2{\nabla }^{0}=\frac{\nabla +{\nabla }^{\ast }}{2}. Clearly, the triple (M,∇∗,g)\left(M,{\nabla }^{\ast },g)is statistical. We denote by RRand R∗{R}^{\ast }the curvature tensors on MMwith respect to the affine connection ∇\nabla and its conjugate ∇∗{\nabla }^{\ast }, respectively. Also the curvature tensor field R0{R}^{0}associated with the ∇0{\nabla }^{0}is called Riemannian curvature tensor. Then we find g(R(X,Y)Z,W)=−g(Z,R∗(X,Y)W)g\left(R\left(X,Y)Z,W)=-g\left(Z,{R}^{\ast }\left(X,Y)W)for vector fields XX, YY, ZZ, and WWon MM, where R(X,Y)Z=[∇X,∇Y]Z−∇[X,Y]ZR\left(X,Y)Z=\left[{\nabla }_{X},{\nabla }_{Y}]Z-{\nabla }_{\left[X,Y]}Z.In general, the dual connections are not metric, one cannot define the sectional curvature in statistical environment as in the case of semi-Riemannian geometry. Thus, Opozda proposed two notions of sectional curvature on statistical manifolds (see [16,33]).Let MMbe a statistical manifold and π\pi a plane section in TMTMwith orthonormal basis {X,Y}\left\{X,Y\right\}, then the sectional KK-curvature is defined in [16] as K(π)=12[g(R(X,Y)Y,X)+g(R∗(X,Y)Y,X)−g(R0(X,Y)Y,X)].K\left(\pi )=\frac{1}{2}{[}g\left(R\left(X,Y)Y,X)+g\left({R}^{\ast }\left(X,Y)Y,X)-g\left({R}^{0}\left(X,Y)Y,X)].Definition 2Let (M,g,ϕ)\left(M,g,\phi )be a Golden semi-Riemannian manifold endowed with a tensor field ϕ∗{\phi }^{\ast }of type (1,1) satisfying (2.4)g(ϕX,Y)=g(X,ϕ∗Y)g\left(\phi X,Y)=g\left(X,{\phi }^{\ast }Y)\hspace{3.3em}for vector fields XXand YY. In view of (2.4), we easily derive (2.5)(ϕ∗)2X=ϕ∗X+X,\hspace{-10.3em}{\left({\phi }^{\ast })}^{2}X={\phi }^{\ast }X+X,(2.6)g(ϕX,ϕ∗Y)=g(ϕX,Y)+g(X,Y).g\left(\phi X,{\phi }^{\ast }Y)=g\left(\phi X,Y)+g\left(X,Y).Then (M,g,ϕ)\left(M,g,\phi )is called Golden-like statistical manifold.According to (2.5) and (2.6), the tensor fields ϕ+ϕ∗\phi +{\phi }^{\ast }and ϕ−ϕ∗\phi -{\phi }^{\ast }are symmetric and skew symmetric with respect to gg, respectively. The equations (2.4), (2.5), and (2.6) imply the following proposition.Proposition 1(M,g,ϕ)\left(M,g,\phi )is a Golden-like statistical manifold if and only if it is (M,g,ϕ∗)\left(M,g,{\phi }^{\ast }).We remark that if we choose ϕ=ϕ∗\phi ={\phi }^{\ast }in a Golden-like statistical manifold, then we have a Golden semi-Riemannian manifold.We first present an example of a Golden-Riemannian manifold.Example 1[34] Consider the Euclidean 6-space R6{{\mathbb{R}}}^{6}with standard coordinates (x1,x2,x3,x4,x5,x6)\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6}). Let ϕ\phi be an (1,1)\left(1,1)tensor field on R6{{\mathbb{R}}}^{6}defined by ϕ(x1,x2,x3,x4,x5,x6)=(ψx1,ψx2,ψx3,(1−ψ)x4,(1−ψ)x5,(1−ψ)x6)\phi \left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})=\left(\psi {x}_{1},\psi {x}_{2},\psi {x}_{3},\left(1-\psi ){x}_{4},\left(1-\psi ){x}_{5},\left(1-\psi ){x}_{6})for any vector field (x1,x2,x3,x4,x5,x6)∈R6\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})\in {{\mathbb{R}}}^{6}, where ψ=1+52\psi =\frac{1+\sqrt{5}}{2}and 1−ψ=1−521-\psi =\frac{1-\sqrt{5}}{2}are the roots of the equation x2=x+1{x}^{2}=x+1. Then we obtain ϕ2(x1,x2,x3,x4,x5,x6)=(ψ2x1,ψ2x2,ψ2x3,(1−ψ)2x4,(1−ψ)2x5,(1−ψ)2x6)=(ψx1,ψx2,ψx3,(1−ψ)x4,(1−ψ)x5,(1−ψ)x6)+(x1,x2,x3,x4,x5,x6).\begin{array}{rcl}{\phi }^{2}\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})& =& \left({\psi }^{2}{x}_{1},{\psi }^{2}{x}_{2},{\psi }^{2}{x}_{3},{\left(1-\psi )}^{2}{x}_{4},{\left(1-\psi )}^{2}{x}_{5},{\left(1-\psi )}^{2}{x}_{6})\\ & =& \left(\psi {x}_{1},\psi {x}_{2},\psi {x}_{3},\left(1-\psi ){x}_{4},\left(1-\psi ){x}_{5},\left(1-\psi ){x}_{6})+\left({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6}).\end{array}Thus, we have ϕ2=ϕ+I{\phi }^{2}=\phi +I. Moreover, we can easily see that standard metric ⟨,⟩\langle \hspace{0.33em},\rangle on R6{{\mathbb{R}}}^{6}is ϕ\phi compatible. Hence, (R6,⟨,⟩,ϕ{{\mathbb{R}}}^{6},\langle \hspace{0.33em},\rangle ,\phi ) is a Golden Riemannian manifold.Next, we construct an example of a Golden-like statistical manifold in the following example.Example 2Consider the semi-Euclidean space R13{{\mathbb{R}}}_{1}^{3}with standard coordinates (x1,x2,x3)\left({x}_{1},{x}_{2},{x}_{3})and the semi-Riemannian metric ggwith the signature (−,+,+)\left(-,\hspace{0.33em}+,\hspace{0.33em}+). Let ϕ\phi be an (1,1)\left(1,1)tensor field on R13{{\mathbb{R}}}_{1}^{3}defined by ϕ(x1,x2,x3)=12(x1+5x2,x2+5x1,2ψx3)\phi \left({x}_{1},{x}_{2},{x}_{3})=\frac{1}{2}({x}_{1}+\sqrt{5}{x}_{2},{x}_{2}+\sqrt{5}{x}_{1},2\psi {x}_{3})for any vector field (x1,x2,x3)∈R13\left({x}_{1},{x}_{2},{x}_{3})\in {{\mathbb{R}}}_{1}^{3}, where ψ=1+52\psi =\frac{1+\sqrt{5}}{2}is the Golden mean. Then we obtain ϕ2=ϕ+I{\phi }^{2}=\phi +I, this implies that ϕ\phi is a Golden structure on R13{{\mathbb{R}}}_{1}^{3}.Now we define an (1,1)\left(1,1)tensor field ϕ∗{\phi }^{\ast }on R13{{\mathbb{R}}}_{1}^{3}by ϕ∗(x1,x2,x3)=12(x1−5x2,x2−5x1,2ψx3).{\phi }^{\ast }\left({x}_{1},{x}_{2},{x}_{3})=\frac{1}{2}({x}_{1}-\sqrt{5}{x}_{2},{x}_{2}-\sqrt{5}{x}_{1},2\psi {x}_{3}).Thus, we have ϕ∗2=ϕ∗+I{{\phi }^{\ast }}^{2}={\phi }^{\ast }+I. Moreover, we have the equation (2.4). Hence, (R13,g,ϕ{{\mathbb{R}}}_{1}^{3},g,\phi ) is a Golden-like simplified statistical manifold.Now we give a generalized example of the above example.Example 3Let Rn{{\mathbb{R}}}_{n}be a (2n+m)\left(2n+m)-dimensional affine space with the coordinate system (x1,⋯,xn,y1,…,yn,z1,…,zm)({x}_{1},\cdots \hspace{0.33em},{x}_{n},{y}_{1},\ldots ,{y}_{n},{z}_{1},\ldots ,{z}_{m}). Assume we define a semi-Riemannian metric ggwith the signature (−,…,−n−times,+,…,+(n+m)−times)\left(\mathop{-,\ldots ,-}\limits_{n-times},\mathop{+,\ldots ,+}\limits_{\left(n+m)-times})and the tensor field ϕ\phi as follows: ϕ=12δij5δij05δijδij000ψ,\phi =\frac{1}{2}\left[\begin{array}{ccc}{\delta }_{ij}& \sqrt{5}{\delta }_{ij}& 0\\ \sqrt{5}{\delta }_{ij}& {\delta }_{ij}& 0\\ 0& 0& \psi \end{array}\right],where ψ\psi is the Golden mean. Then ϕ\phi is golden structure on Rn2n+m{{\mathbb{R}}}_{n}^{2n+m}. Moreover, if the conjugate tensor field ϕ∗{\phi }^{\ast }is defined as ϕ∗=12δij−5δij05δij−δij000ψ.{\phi }^{\ast }=\frac{1}{2}\left[\begin{array}{ccc}{\delta }_{ij}& -\hspace{-0.25em}\sqrt{5}{\delta }_{ij}& 0\\ \sqrt{5}{\delta }_{ij}& -\hspace{-0.25em}{\delta }_{ij}& 0\\ 0& 0& \psi \end{array}\right].Then we can easily see that (Rn2n+m,g,ϕ)\left({{\mathbb{R}}}_{n}^{2n+m},g,\phi )and (Rn2n+m,g,ϕ∗)\left({{\mathbb{R}}}_{n}^{2n+m},g,{\phi }^{\ast })are Golden-like statistical manifolds. Also, this verifies Proposition 1.Let (M=Mp(cp)×Mq(cq),g,ϕ)\left(M={M}_{p}\left({c}_{p})\times {M}_{q}\left({c}_{q}),g,\phi )be a Golden product space form. Then the Riemannian curvature tensor RRof MMis given by [32]: (2.7)R(X,Y)Z=−(1−ψ)cp−ψcq25{g(Y,Z)X−g(X,Z)Y+g(ϕY,Z)ϕX−g(ϕX,Z)ϕY}+−(1−ψ)cp+ψcq4{g(ϕY,Z)X−g(ϕX,Z)Y+g(Y,Z)ϕX−g(X,Z)ϕY},\begin{array}{rcl}R\left(X,Y)Z& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left\{g\left(Y,Z)X-g\left(X,Z)Y+g\left(\phi Y,Z)\phi X-g\left(\phi X,Z)\phi Y\right\}\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)\left\{g\left(\phi Y,Z)X-g\left(\phi X,Z)Y+g\left(Y,Z)\phi X-g\left(X,Z)\phi Y\right\},\end{array}where Mp{M}_{p}and Mq{M}_{q}are space forms with constant sectional curvatures cp{c}_{p}and cq{c}_{q}, respectively. We can obtain the curvature tensor R∗{R}^{\ast }with respect to dual connection just by replacing ϕ\phi by ϕ∗.{\phi }^{\ast }.Let Mn{M}^{n}be statistical submanifold of (Nm,g,ϕ)\left({N}^{m},g,\phi ). The Gauss and Weingarten formulae are ∇XY=∇XY+σ(X,Y),∇Xξ=−AξX+∇X⊥ξ∇X∗Y=∇X∗Y+σ∗(X,Y),∇X∗ξ=−Aξ∗X+∇X∗⊥ξ\begin{array}{lcl}{\nabla }_{X}Y& =& {\nabla }_{X}Y+\sigma \left(X,Y),\hspace{1.51em}{\nabla }_{X}\xi =-{A}_{\xi }X+{\nabla }_{X}^{\perp }\xi \\ {\nabla }_{X}^{\ast }Y& =& {\nabla }_{X}^{\ast }Y+{\sigma }^{\ast }\left(X,Y),\hspace{1.0em}{\nabla }_{X}^{\ast }\xi =-{A}_{\xi }^{\ast }X+{\nabla }_{X}^{\ast \perp }\xi \end{array}for all X,Y∈TMX,Y\in TMand ξ∈T⊥M\xi \in {T}^{\perp }M, respectively. Moreover, we have the following equations: Xg(Y,Z)=g(∇XY,Z)+g(Y,∇X∗Z)g(σ(X,Y),ξ)=g(Aξ∗X,Y),g(σ∗(X,Y),ξ)=g(AξX,Y)Xg(ξ,η)=g(∇X⊥ξ,η)+g(ξ,∇X∗⊥η).\begin{array}{l}Xg\left(Y,Z)=g\left({\nabla }_{X}Y,Z)+g\left(Y,{\nabla }_{X}^{\ast }Z)\\ g\left(\sigma \left(X,Y),\xi )=g\left({A}_{\xi }^{\ast }X,Y),\hspace{1.0em}g\left({\sigma }^{\ast }\left(X,Y),\xi )=g\left({A}_{\xi }X,Y)\\ Xg\left(\xi ,\eta )=g\left({\nabla }_{X}^{\perp }\xi ,\eta )+g\left(\xi ,{\nabla }_{X}^{\ast \perp }\eta ).\end{array}The mean curvature vector fields for an orthonormal tangent frame {e1,e2,…,en}\left\{{e}_{1},{e}_{2},\ldots ,{e}_{n}\right\}and a normal frame {en+1,…,em}\left\{{e}_{n+1},\ldots ,{e}_{m}\right\}, respectively, are defined as H=1n∑i=1nσ(ei,ei)=1n∑γ=n+1m∑i=1nσiiγξγ,σijγ=g(σ(ei,ej),eγ)\hspace{1em}H=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}\sigma \left({e}_{i},{e}_{i})=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\gamma }\right){\xi }_{\gamma },\hspace{2.0em}{\sigma }_{ij}^{\gamma }=g\left(\sigma \left({e}_{i},{e}_{j}),{e}_{\gamma })\hspace{1.65em}and H∗=1n∑i=1nσ∗(ei,ei)=1n∑γ=n+1m∑i=1nσii∗γξγ,σij∗γ=g(σ∗(ei,ej),eγ){H}^{\ast }=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{\sigma }^{\ast }\left({e}_{i},{e}_{i})=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\ast \gamma }\right){\xi }_{\gamma },\hspace{1.0em}{\sigma }_{ij}^{\ast \gamma }=g\left({\sigma }^{\ast }\left({e}_{i},{e}_{j}),{e}_{\gamma })for 1≤i,j≤n1\le i,j\le n, and 1≤l≤m1\le l\le m. Moreover, we have 2h0=h+h∗2{h}^{0}=h+{h}^{\ast }and 2H0=H+H∗2{H}^{0}=H+{H}^{\ast }, where the second fundamental form h0{h}^{0}and the mean curvature H0{H}^{0}are calculated with respect to Levi-Civita connection ∇0{\nabla }^{0}on MM.The squared mean curvatures are defined as ‖H‖2=1n2∑γ=n+1m∑i=1nσiiγ2,‖H∗‖2=1n2∑γ=n+1m∑i=1nσii∗γ2.\Vert H{\Vert }^{2}=\frac{1}{{n}^{2}}\mathop{\sum }\limits_{\gamma =n+1}^{m}{\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\gamma }\right)}^{2},\hspace{1.0em}\Vert {H}^{\ast }{\Vert }^{2}=\frac{1}{{n}^{2}}\mathop{\sum }\limits_{\gamma =n+1}^{m}{\left(\mathop{\sum }\limits_{i=1}^{n}{\sigma }_{ii}^{\ast \gamma }\right)}^{2}.The Casorati curvatures are defined as C=1n∑γ=n+1m∑i,j=1n(σijγ)2,C∗=1n∑γ=n+1m∑i,j=1n(σij∗γ)2.{\mathcal{C}}=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\mathop{\sum }\limits_{i,j=1}^{n}{({\sigma }_{ij}^{\gamma })}^{2},\hspace{1.0em}{{\mathcal{C}}}^{\ast }=\frac{1}{n}\mathop{\sum }\limits_{\gamma =n+1}^{m}\mathop{\sum }\limits_{i,j=1}^{n}{({\sigma }_{ij}^{\ast \gamma })}^{2}.If we suppose that W{\mathcal{W}}is a dd-dimensional subspace of TMTM, d≥2d\ge 2, and {e1,e2,…,ed}\left\{{e}_{1},{e}_{2},\ldots ,{e}_{d}\right\}is an orthonormal basis of W{\mathcal{W}}, then the scalar curvature of the dd-plane section is given as τ(W)=∑1≤u<v≤dK(eu∧ev),\tau \left({\mathcal{W}})=\sum _{1\le u\lt v\le d}K\left({e}_{u}\wedge {e}_{v}),and the normalized scalar curvature ρ\rho is defined as ρ=2τs(s−1).\rho =\frac{2\tau }{s\left(s-1)}.Also, the Casorati curvature of the subspace W{\mathcal{W}}is given by C(W)=1d∑γ=r+1m∑i,j=1d(σijγ)2,C∗(W)=1d∑γ=r+1m∑i,j=1d(σij∗γ)2.{\mathcal{C}}\left({\mathcal{W}})=\frac{1}{d}\mathop{\sum }\limits_{\gamma =r+1}^{m}\mathop{\sum }\limits_{i,j=1}^{d}{({\sigma }_{ij}^{\gamma })}^{2},\hspace{1.0em}{{\mathcal{C}}}^{\ast }\left({\mathcal{W}})=\frac{1}{d}\mathop{\sum }\limits_{\gamma =r+1}^{m}\mathop{\sum }\limits_{i,j=1}^{d}{({\sigma }_{ij}^{\ast \gamma })}^{2}.A point x∈Mx\in Mis called as quasi-umbilical point, if at xxthere exist m−nm-nmutually orthogonal unit normal vectors ei{e}_{i}, i∈{n+1,…,m}i\in \left\{n+1,\ldots ,m\right\}in a way the shape operators with respect to all vectors ei{e}_{i}have an eigenvalue with multiplicity n−1n-1and for each ei{e}_{i}the distinguished eigen vector is the same.The normalized δ\delta -Casorati curvatures δc(n−1){\delta }_{c}\left(n-1)and δ^c(n−1){\widehat{\delta }}_{c}\left(n-1)of the submanifold Ms{M}^{s}are, respectively, given by [δc(n−1)]x=12Cx+n+12sinf{C(W)∣Wa hyperplane ofTxM}{{[}{\delta }_{c}\left(n-1)]}_{x}=\frac{1}{2}{{\mathcal{C}}}_{x}+\frac{n+1}{2s}{\rm{\inf }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\}and [δ^c(n−1)]x=2Cx−2n−12nsup{C(W)∣Wa hyperplane ofTxM}.{{[}{\widehat{\delta }}_{c}\left(n-1)]}_{x}=2{{\mathcal{C}}}_{x}-\frac{2n-1}{2n}{\rm{\sup }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\}.In [5], Decu et al. generalized the notion of normalized δ\delta -Casorati curvature to the generalized normalized δ\delta -Casorati curvatures δC(k;n−1){\delta }_{C}\left(k;\hspace{0.33em}n-1)and δ^C(k;n−1).{\widehat{\delta }}_{C}\left(k;\hspace{0.33em}n-1).For a submanifold Mn{M}^{n}and for any positive real number k≠n(n−1)k\ne n\left(n-1), the generalized normalized δ\delta -Casorati curvature is given by: [δC(k;n−1)]x=kCx+(n−1)(n+k)(n2−n−k)kninf{C(W)∣Wa hyperplane ofTxM},{{[}{\delta }_{C}\left(k;n-1)]}_{x}=k{{\mathcal{C}}}_{x}+\frac{\left(n-1)\left(n+k)\left({n}^{2}-n-k)}{kn}{\rm{\inf }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\},if 0<k<n2−n0\lt k\lt {n}^{2}-n, and [δ^C(k;n−1)]x=kCx−(n−1)(n+k)(k−n2+n)knsup{C(W)∣Wa hyperplane ofTxM},{{[}{\widehat{\delta }}_{C}\left(k;n-1)]}_{x}=k{{\mathcal{C}}}_{x}-\frac{\left(n-1)\left(n+k)\left(k-{n}^{2}+n)}{kn}{\rm{\sup }}\left\{{\mathcal{C}}\left({\mathcal{W}})| {\mathcal{W}}\hspace{0.33em}\hspace{0.1em}\text{a hyperplane of}\hspace{0.1em}\hspace{0.33em}{T}_{x}M\right\},if k>n2−nk\gt {n}^{2}-n.The generalized normalized δ\delta -Casorati curvatures δC(k:n−1){\delta }_{C}\left(k:n-1)and δˆC(k:n−1){\hat{\delta }}_{C}\left(k:n-1)are generalizations of normalized δ\delta -Casorati curvatures δC(n−1){\delta }_{C}\left(n-1)and δˆC(n−1){\hat{\delta }}_{C}\left(n-1). In fact, we have the following relations (see [5]): (2.8)δCn(n−1)2;n−1x=n(n−1)[δC(n−1)]x,{\left[{\delta }_{C}\left(\frac{n\left(n-1)}{2};n-1\right)\right]}_{x}=n\left(n-1){\left[{\delta }_{C}\left(n-1)]}_{x},(2.9)[δˆC(2n(n−1);n−1)]x=n(n−1)[δˆC(n−1)]x.\hspace{0.5em}{{[}{\hat{\delta }}_{C}(2n\left(n-1);n-1)]}_{x}=n\left(n-1){\left[{\hat{\delta }}_{C}\left(n-1)]}_{x}.In the same way, the dual Casorati curvatures are obtained just by replacing δ\delta and δ∗{\delta }^{\ast }and C{\mathcal{C}}by C∗{{\mathcal{C}}}^{\ast }.Now, we state the following fundamental results on statistical manifolds.Proposition 2[20] Let MMbe statistical submanifold of (M,g,ϕ)\left(M,g,\phi ). Let RRand R∗{R}^{\ast }be the Riemannian curvature tensors on MMfor ∇\nabla and ∇∗{\nabla }^{\ast }, respectively. Then we have the following.g(R(X,Y)Z,W)=g(R(X,Y)Z,W)+g(σ(X,Z),σ∗(Y,W))−g(σ∗(X,W),σ(Y,Z)),g(R∗(X,Y)Z,W)=g(R∗(X,Y)Z,W)+g(σ∗(X,Z),σ(Y,W))−g(σ(X,W),σ∗(Y,Z)),g(R⊥(X,Y)ξ,η)=g(R(X,Y)ξ,η)+g([Aξ∗,Aη]X,Y),g(R∗⊥(X,Y)ξ,η)=g(R∗(X,Y)ξ,η)+g([Aξ,Aη∗]X,Y),\begin{array}{l}g\left(R\left(X,Y)Z,W)=g\left(R\left(X,Y)Z,W)+g\left(\sigma \left(X,Z),{\sigma }^{\ast }\left(Y,W))-g\left({\sigma }^{\ast }\left(X,W),\sigma \left(Y,Z)),\\ g\left({R}^{\ast }\left(X,Y)Z,W)=g\left({R}^{\ast }\left(X,Y)Z,W)+g\left({\sigma }^{\ast }\left(X,Z),\sigma \left(Y,W))-g\left(\sigma \left(X,W),{\sigma }^{\ast }\left(Y,Z)),\\ g\left({R}^{\perp }\left(X,Y)\xi ,\eta )=g\left(R\left(X,Y)\xi ,\eta )+g\left(\left[{A}_{\xi }^{\ast },{A}_{\eta }]X,Y),\\ g\left({{R}^{\ast }}^{\perp }\left(X,Y)\xi ,\eta )=g\left({R}^{\ast }\left(X,Y)\xi ,\eta )+g\left(\left[{A}_{\xi },{A}_{\eta }^{\ast }]X,Y),\end{array}where [Aξ,Aη∗]=AξAη∗−Aη∗Aξ\left[{A}_{\xi },{A}_{\eta }^{\ast }]={A}_{\xi }{A}_{\eta }^{\ast }-{A}_{\eta }^{\ast }{A}_{\xi }and [Aξ∗,Aη]=Aξ∗Aη−AηAξ∗\left[{A}_{\xi }^{\ast },{A}_{\eta }]={A}_{\xi }^{\ast }{A}_{\eta }-{A}_{\eta }{A}_{\xi }^{\ast }, for X,Y,Z,W∈TMX,Y,Z,W\in TMand ξ,η∈T⊥M\xi ,\eta \in {T}^{\perp }M.Now, we state two important lemmas which we use to prove the main results in the upcoming sections.Lemma 1Let n≥3n\ge 3be an integer and a1,a2,…,an{a}_{1},{a}_{2},\ldots ,{a}_{n}are nnreal numbers. Then, we have∑1≤i<j≤nnaiaj−a1a2≤n−22(n−2)∑i=1nai2.\mathop{\sum }\limits_{1\le i\lt j\le n}^{n}{a}_{i}{a}_{j}-{a}_{1}{a}_{2}\le \frac{n-2}{2\left(n-2)}{\left(\mathop{\sum }\limits_{i=1}^{n}{a}_{i}\right)}^{2}.Moreover, the equality holds if and only if a1+a2=a3=⋯=an{a}_{1}+{a}_{2}={a}_{3}=\cdots ={a}_{n}.The optimization techniques have a pivotal role in improving inequalities involving Chen invariants. Oprea [35] applied the constrained extremum problem to prove Chen-Ricci inequalities for Lagrangian submanifolds of complex space forms. In the characterization of our main result, we will use the following lemma.Let MMbe a Riemannian submanifold in a Riemannian manifold (Mˆ,g)\left(\hat{M},g)and y:Mˆ→Ry:\hat{M}\to {\mathbb{R}}be a differentiable function. If we have the constrained extremum problem (2.10)minx∈M[y(x)].{{\rm{\min }}}_{x\in M}[y\left(x)].Then we have the following lemma.Lemma 2[35] If x0∈M{x}_{0}\in Mis a solution of the problem (2.10), then(1)(grady)(x0)∈Tx0⊥M\left({\rm{grad}}\hspace{0.33em}y)\left({x}_{0})\in {T}_{{x}_{0}}^{\perp }M;(2)The bilinear form Ω:Tx0M×Tx0M→R\Omega :{T}_{{x}_{0}}M\times {T}_{{x}_{0}}M\to {\mathbb{R}}defined byΩ(X,Y)=Hessy(X,Y)+g(σ(X,Y),(grady)(x0))\Omega \left(X,Y)={{\rm{Hess}}}_{y}\left(X,Y)+g\left(\sigma \left(X,Y),\left({\rm{grad}}\hspace{0.33em}y)\left({x}_{0}))is positive semi-definite, where σ\sigma is the second fundamental form of MMin Mˆ\hat{M}and grady{\rm{grad}}\hspace{0.33em}yis the gradient of yy.In principle, the bilinear form Ω\Omega is Hessy∣M(x0){}_{y| M}\left({x}_{0}). Therefore, if Ω\Omega is positive semi-definite on MM, then the critical points of y∣My| M, which coincide with the points where grad yyis normal to MM, are global optimal solutions of the problem (2.6) (for instance see [36, Remark 3.2]).3Main inequalitiesLet π\pi be a two plane spanned by {e1,e2}\left\{{e}_{1},{e}_{2}\right\}and denote g(ϕe1,e1)g(ϕ2,e2)=Ψ(π)g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})=\Psi \left(\pi ). Also, as in [37], for an orthonormal basis {e1,e2}\left\{{e}_{1},{e}_{2}\right\}of two-plane section, we denote Θ(π)=g(ϕe1,e2)g(ϕ∗e1,e2),\Theta \left(\pi )=g\left(\phi {e}_{1},{e}_{2})g\left({\phi }^{\ast }{e}_{1},{e}_{2}),where Θ(π)\Theta \left(\pi )is a real number in [0,1]\left[0,1].Theorem 1Let (N,g,ϕ)\left(N,g,\phi )be a Golden-like statistical manifold of dimension mmand MMbe its statistical submanifold of dimension nn. Then, we have the following: (τ−K(π))−(τ0−K0(π))≥−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]−n2(n−2)4(n−1)[‖H‖2+‖H∗‖2]+2Kˆ0(π)−2τˆ0.\begin{array}{l}\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))\\ \hspace{1.0em}\ge \left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )\\ \hspace{2.0em}+\left(\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]-\frac{{n}^{2}\left(n-2)}{4\left(n-1)}{[}\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2}]+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.\end{array}\hspace{2.55em}ProofLet {e1,e2,…,en}\left\{{e}_{1},{e}_{2},\ldots ,{e}_{n}\right\}and {en+1,…,em}\left\{{e}_{n+1},\ldots ,{e}_{m}\right\}be the orthonormal frames of TMTMand T⊥M{T}^{\perp }M, respectively.The scalar curvature corresponding to the sectional KK-curvature isτ=12∑1≤i<j≤n[g(R(ei,ej)ej,ei)+g(R∗(ei,ej)ej,ei)−2g(R0(ei,ej)ej,ei)].\tau =\frac{1}{2}\sum _{1\le i\lt j\le n}{[}g(R\left({e}_{i},{e}_{j}){e}_{j},{e}_{i})+g({R}^{\ast }\left({e}_{i},{e}_{j}){e}_{j},{e}_{i})-2g({R}^{0}\left({e}_{i},{e}_{j}){e}_{j},{e}_{i})].Using (2.7) and Gauss equation for RRand R∗{R}^{\ast }and doing some simple calculations, we obtain τ=−(1−ψ)cp−ψcq25[n(n−1)+tr2(ϕ)−tr(ϕ∗2)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−τ0+12∑γ=n+1m∑1≤i<j≤n[σii∗γσjj∗γ+σiiγσjj∗γ−2σij∗γσijγ].\tau =\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-1)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({{\phi }^{\ast }}^{2})]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )-{\tau }_{0}+\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}\hspace{0.25em}\sum _{1\le i\lt j\le n}{[}{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }+{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\ast \gamma }-2{\sigma }_{ij}^{\ast \gamma }{\sigma }_{ij}^{\gamma }].\hspace{20.4em}In view of (2.5), we obtain τ=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−τ0+12∑γ=n+1m∑1≤i<j≤n[σii∗γσjj∗γ+σiiγσjj∗γ−2σij∗γσijγ],\tau =\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )-{\tau }_{0}+\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}\hspace{0.25em}\sum _{1\le i\lt j\le n}{[}{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }+{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\ast \gamma }-2{\sigma }_{ij}^{\ast \gamma }{\sigma }_{ij}^{\gamma }],\hspace{20.35em}which can be written as τ=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−τ0+2∑γ=n+1m∑1≤i<j≤n[σii0γσjj0γ−(σij0γ)2]−12∑γ=n+1m∑1≤i<j≤n[{σiiγσjjγ+(σijγ)2}+{σii∗γσjj∗γ−(σij∗γ)2}].\hspace{-46em}\begin{array}{rcl}\tau & =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )\\ & & -{\tau }_{0}+2\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\hspace{0.16em}\displaystyle \sum _{1\le i\lt j\le n}\left[{\sigma }_{ii}^{0\gamma }{\sigma }_{jj}^{0\gamma }-{\left({\sigma }_{ij}^{0\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\hspace{0.25em}\displaystyle \sum _{1\le i\lt j\le n}\left[\left\{{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\gamma }+{\left({\sigma }_{ij}^{\gamma })}^{2}\right\}+\left\{{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\ast \gamma })}^{2}\right\}].\end{array}By using Gauss equation for the Levi-Civita connection, we have (3.1)τ=τ0+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)−2τˆ0−12∑γ=n+1m∑1≤i<j≤n[{σiiγσjj∗γ−(σijγ)2}+{σii∗γσjj∗γ−(σij∗γ)2}].\begin{array}{rcl}\tau & =& {\tau }_{0}+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )-2{\hat{\tau }}_{0}\\ & & -\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\hspace{0.16em}\displaystyle \sum _{1\le i\lt j\le n}{[}\left\{{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\gamma })}^{2}\right\}+\left\{{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\ast \gamma })}^{2}\right\}].\end{array}Now, the sectional KK-curvature K(π)K\left(\pi )of the plane section π\pi is(3.2)K(π)=12[g(R(e1,e2)e2,e1)+g(R∗(e1,e2)e2,e1)−2g(R0(e1,e2)e2,e1)].K\left(\pi )=\frac{1}{2}{[}g(R\left({e}_{1},{e}_{2}){e}_{2},{e}_{1})+g({R}^{\ast }\left({e}_{1},{e}_{2}){e}_{2},{e}_{1})-2g({R}^{0}\left({e}_{1},{e}_{2}){e}_{2},{e}_{1})].Using (2.7) and Gauss equation for RRand R∗{R}^{\ast }and putting the values in (3.2), we obtain K(π)=−(1−ψ)cp−ψcq25[1+g(ϕe1,e1)g(ϕ2,e2)−g(ϕe1,e2)g(ϕe2,e1)]−K0(π)+12∑γ=n+1m{[σ11γσ22∗γ+σ11∗γσ22γ−2σ12∗γσ12γ]}.\begin{array}{rcl}K\left(\pi )& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})-g\left(\phi {e}_{1},{e}_{2})g\left(\phi {e}_{2},{e}_{1})]-{K}_{0}\left(\pi )\\ & & +\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\{{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\ast \gamma }+{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\gamma }-2{\sigma }_{12}^{\ast \gamma }{\sigma }_{12}^{\gamma }]\}.\end{array}\hspace{2.75em}Using σ+σ∗=2σ0\sigma +{\sigma }^{\ast }=2{\sigma }^{0}, we obtain K(π)=−(1−ψ)cp−ψcq25[1+g(ϕe1,e1)g(ϕ2,e2)−g(ϕe1,e2)g(ϕ2,e1)]−K0(π)+2∑γ=n+1m[σ110γσ220γ−(σ120γ)2]−12∑γ=n+1m{[σ11γσ22γ−(σ12γ)2]+[σ11∗γσ22∗γ−(σ12∗γ)2]}.\begin{array}{rcl}K\left(\pi )& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})-g\left(\phi {e}_{1},{e}_{2})g\left({\phi }_{2},{e}_{1})]\\ & & -{K}_{0}\left(\pi )+2\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{0\gamma }{\sigma }_{22}^{0\gamma }-{\left({\sigma }_{12}^{0\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\{{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]+{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}]\}.\end{array}\hspace{0.3em}Using Gauss equation with respect to Levi-Civita connection, we have K(π)=K0(π)+−(1−ψ)cp−ψcq25[1+g(ϕe1,e1)g(ϕ2,e2)+g(ϕe1,e2)g(ϕ∗e1,e2)]−2Kˆ0(π)−12∑γ=n+1m[σ11γσ22γ−(σ12γ)2]−12∑γ=n+1m[σ11∗γσ22∗γ−(σ12∗γ)2].\begin{array}{rcl}K\left(\pi )& =& {K}_{0}\left(\pi )+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+g\left(\phi {e}_{1},{e}_{1})g\left({\phi }_{2},{e}_{2})+g\left(\phi {e}_{1},{e}_{2})g\left({\phi }^{\ast }{e}_{1},{e}_{2})]\\ & & -2{\hat{K}}_{0}\left(\pi )-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}].\end{array}\hspace{2.45em}The above equation can be written in the form(3.3)K(π)=K0(π)+−(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]−2Kˆ0(π)−12∑γ=n+1m[σ11γσ22γ−(σ12γ)2]−12∑γ=n+1m[σ11∗γσ22∗γ−(σ12∗γ)2].K\left(\pi )={K}_{0}\left(\pi )+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]-2{\hat{K}}_{0}\left(\pi )-\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]-\frac{1}{2}\mathop{\sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}].From (3.1) and (3.3), we have (τ−K(π))−(τ0−K0(π))=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]\begin{array}{rcl}\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))& =& \left(-\hspace{-0.25em}\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )+\left(\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]\end{array}−12∑γ=n+1m[σiiγσjjγ−(σijγ)2]−12∑γ=n+1m[σii∗γσjj∗γ−(σij∗γ)2]+12∑γ=n+1m∑α=13{[σ11γσ22γ−(σ12γ)2]+[σ11∗γσ22∗γ−(σ12∗γ)2]}+2Kˆ0(π)−2τˆ0.\begin{array}{rcl}& & -\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{ii}^{\gamma }{\sigma }_{jj}^{\gamma }-{\left({\sigma }_{ij}^{\gamma })}^{2}]-\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}{[}{\sigma }_{ii}^{\ast \gamma }{\sigma }_{jj}^{\ast \gamma }-{\left({\sigma }_{ij}^{\ast \gamma })}^{2}]\\ & & +\frac{1}{2}\mathop{\displaystyle \sum }\limits_{\gamma =n+1}^{m}\mathop{\displaystyle \sum }\limits_{\alpha =1}^{3}\{{[}{\sigma }_{11}^{\gamma }{\sigma }_{22}^{\gamma }-{\left({\sigma }_{12}^{\gamma })}^{2}]+{[}{\sigma }_{11}^{\ast \gamma }{\sigma }_{22}^{\ast \gamma }-{\left({\sigma }_{12}^{\ast \gamma })}^{2}]\}+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.\end{array}Using Lemma 1, we can obtain the above equation in simplified form as (τ−K(π))−(τ0−K0(π))≥−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+(1−ψ)cp−ψcq25[1+Ψ(π)+Θ(π)]−n2(n−2)4(n−1)[‖H‖2+‖H∗‖2]+2Kˆ0(π)−2τˆ0.\begin{array}{rcl}\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))& \ge & \left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )+\left(\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[1+\Psi \left(\pi )+\Theta \left(\pi )]\\ & & -\frac{{n}^{2}\left(n-2)}{4\left(n-1)}{[}\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2}]+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.\end{array}This proves our claims.□Corollary 1Let (N,g,ϕ)\left(N,g,\phi )be a Golden-like statistical manifold of dimension mmand MMbe its totally real statistical submanifold of dimension nn. Then, we have the following(τ−K(π))−(τ0−K0(π))≥−(1−ψ)cp−ψcq25[n(n−2)−1]−n2(n−2)4(n−1)[‖H‖2+‖H∗‖2]+2Kˆ0(π)−2τˆ0.\left(\tau -K\left(\pi ))-\left({\tau }_{0}-{K}_{0}\left(\pi ))\ge \left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)-1]-\frac{{n}^{2}\left(n-2)}{4\left(n-1)}{[}\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2}]+2{\hat{K}}_{0}\left(\pi )-2{\hat{\tau }}_{0}.Theorem 2Let Mn{M}^{n}be a statistical submanifold of a Golden-like statistical manifold Nm{N}^{m}. Then for the generalized normalized δ\delta -Casorati curvature, we have the following optimal relationships: (i)For any real number kk, such that 0<k<n(n−1)0\lt k\lt n\left(n-1), (3.4)ρ≤δC0(k;n−1)n(n−1)+1(n−1)C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+1n(n−1)−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+2n−(1−ψ)cp+ψcq4tr(ϕ),\rho \le \frac{{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{\left(n-1)}{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\frac{1}{n\left(n-1)}\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right){[}n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ),where δC0(k;n−1)=12[δC(k;n−1)+δC∗(k;n−1)]{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\delta }_{C}\left(k;\hspace{0.33em}n-1)+{\delta }_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].(ii)For any real number k>n(n−1)k\gt n\left(n-1), (3.5)ρ≤δ^C0(k;n−1)n(n−1)+1(n−1)C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+1n(n−1)−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+2n−(1−ψ)cp+ψcq4tr(ϕ),\rho \le \frac{{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{\left(n-1)}{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\frac{1}{n\left(n-1)}\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right){[}n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ),where δ^C0(k;n−1)=12[δ^C(k;n−1)+δ^C∗(k;n−1)]{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\widehat{\delta }}_{C}\left(k;\hspace{0.33em}n-1)+{\widehat{\delta }}_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].ProofLet p∈Mp\in Mand {e1,…,en}\left\{{e}_{1},\ldots ,{e}_{n}\right\}, {en+1,…,em}\left\{{e}_{n+1},\ldots ,{e}_{m}\right\}be the orthonormal basis of TpM{T}_{p}Mand Tp⊥M{T}_{p}^{\perp }M, respectively. From Gauss equation, we obtain 2τ=n2g(H,H∗)−n∑1≤i,j≤ng(σ∗(ei,ej),σ(ei,ej))+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ).2\tau ={n}^{2}g\left(H,{H}^{\ast })-n\sum _{1\le i,j\le n}g\left({\sigma }^{\ast }\left({e}_{i},{e}_{j}),\sigma \left({e}_{i},{e}_{j}))+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi ).\hspace{24.85em}Denote H+H∗=2H0H+{H}^{\ast }=2{H}^{0}and C+C∗=2C0{\mathcal{C}}+{{\mathcal{C}}}^{\ast }=2{{\mathcal{C}}}^{0}. Then the above equation becomes 2τ=−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ)+2n2‖H0‖2−n22(‖H‖2+‖H∗‖2)−2nC0+n2(C+C∗).2\tau =\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi )+2{n}^{2}\Vert {H}^{0}{\Vert }^{2}-\frac{{n}^{2}}{2}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})-2n{{\mathcal{C}}}^{0}+\frac{n}{2}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast }).We define a polynomial P{\mathcal{P}}in the components of second fundamental form as: (3.6)P=kC0+a(k)C0(W)+n2(C+C∗)−n22(‖H‖2+‖H∗‖2)−2τ(p)+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ),{\mathcal{P}}=k{{\mathcal{C}}}^{0}+a\left(k){{\mathcal{C}}}^{0}\left({\mathcal{W}})+\frac{n}{2}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast })-\frac{{n}^{2}}{2}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})-2\tau \left(p)+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi ),where W{\mathcal{W}}is a hyperplane in TpM{T}_{p}M. Assuming W{\mathcal{W}}is spanned by {e1,…,en−1}\left\{{e}_{1},\ldots ,{e}_{n-1}\right\}, we have (3.7)P=∑α=n+1m2n+kn∑i,j=1n(σij0α)2+a(k)1n−1∑i,j=1n−1(σij0α)2−2∑i,j=1nσij0α2.{\mathcal{P}}=\mathop{\sum }\limits_{\alpha =n+1}^{m}\left[\frac{2n+k}{n}\mathop{\sum }\limits_{i,j=1}^{n}{\left({\sigma }_{ij}^{0\alpha })}^{2}+a\left(k)\frac{1}{n-1}\mathop{\sum }\limits_{i,j=1}^{n-1}{\left({\sigma }_{ij}^{0\alpha })}^{2}-2{\left(\mathop{\sum }\limits_{i,j=1}^{n}{\sigma }_{ij}^{0\alpha }\right)}^{2}\right].\hspace{31.75em}Equation (3.7) yields P=∑α=n+1n2(2n+k)n+2a(k)n−1∑1≤i<j≤n−1(σij0α)2+2(2n+k)n+2a(k)n−1∑i=1n−1(σin0α)22n+kn+a(k)n−1−2∑i=1n−1(σii0α)2−4∑1≤i<j≤nσii0ασjj0α+2n+kn−2(σnn0α)2≥∑α=n+1mk(n−1)+a(k)nn(n−1)∑i=1n−1(σii0α)2+km(σnn0α)2−4∑1≤i<j≤nσii0ασjj0α.\begin{array}{rcl}{\mathcal{P}}& =& \mathop{\displaystyle \sum }\limits_{\alpha =n+1}^{n}\left[\left(\frac{2\left(2n+k)}{n}+\frac{2a\left(k)}{n-1}\right)\displaystyle \sum _{1\le i\lt j\le n-1}{\left({\sigma }_{ij}^{0\alpha })}^{2}+\left(\frac{2\left(2n+k)}{n}+\frac{2a\left(k)}{n-1}\right)\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{\left({\sigma }_{in}^{0\alpha })}^{2}\left(\frac{2n+k}{n}+\frac{a\left(k)}{n-1}-2\right)\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{\left({\sigma }_{ii}^{0\alpha })}^{2}-4\displaystyle \sum _{1\le i\lt j\le n}{\sigma }_{ii}^{0\alpha }{\sigma }_{jj}^{0\alpha }+\left(\frac{2n+k}{n}-2\right){\left({\sigma }_{nn}^{0\alpha })}^{2}\right]\\ & \ge & \mathop{\displaystyle \sum }\limits_{\alpha =n+1}^{m}\left[\frac{k\left(n-1)+a\left(k)n}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{\left({\sigma }_{ii}^{0\alpha })}^{2}+\frac{k}{m}{\left({\sigma }_{nn}^{0\alpha })}^{2}-4\displaystyle \sum _{1\le i\lt j\le n}{\sigma }_{ii}^{0\alpha }{\sigma }_{jj}^{0\alpha }\right].\end{array}Let yα{y}_{\alpha }be a quadratic form defined by yα=Rn→R{y}_{\alpha }={{\mathbb{R}}}^{n}\to {\mathbb{R}}for any α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}, yα(σ110α,σ220α,…,σnn0α)=∑i=1n−1k(n−1)+a(k)nn(n−1)(σii0α)2+kn(σnn0α)2−4∑1≤i<j≤nσii0ασjj0α,{y}_{\alpha }\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })=\mathop{\sum }\limits_{i=1}^{n-1}\frac{k\left(n-1)+a\left(k)n}{n\left(n-1)}{\left({\sigma }_{ii}^{0\alpha })}^{2}+\frac{k}{n}{\left({\sigma }_{nn}^{0\alpha })}^{2}-4\sum _{1\le i\lt j\le n}{\sigma }_{ii}^{0\alpha }{\sigma }_{jj}^{0\alpha },and the optimization problem min{yα}{\rm{\min }}\{{y}_{\alpha }\}subject to G:σ110α+σ220α+…+σnn0α=pα{\mathcal{G}}:{\sigma }_{11}^{0\alpha }+{\sigma }_{22}^{0\alpha }+\ldots +{\sigma }_{nn}^{0\alpha }={p}^{\alpha }, where pα{p}^{\alpha }is a real constant.The partial derivatives of yα{y}_{\alpha }are (3.8)∂yα∂σii0α=2k(n−1)+a(k)nn(n−1)σii0α−4∑l=1nσll0α−σii0α=0∂yα∂σnn0α=2knσnn0α−4∑l=1n−1σll0α=0,\left\{\begin{array}{l}\frac{\partial {y}_{\alpha }}{\partial {\sigma }_{ii}^{0\alpha }}=2\frac{k\left(n-1)+a\left(k)n}{n\left(n-1)}{\sigma }_{ii}^{0\alpha }-4\left(\mathop{\displaystyle \sum }\limits_{l=1}^{n}{\sigma }_{ll}^{0\alpha }-{\sigma }_{ii}^{0\alpha }\right)=0\\ \frac{\partial {y}_{\alpha }}{\partial {\sigma }_{nn}^{0\alpha }}=\frac{2k}{n}{\sigma }_{nn}^{0\alpha }-4\mathop{\displaystyle \sum }\limits_{l=1}^{n-1}{\sigma }_{ll}^{0\alpha }=0,\end{array}\right.for every i∈{1,…,n−1}i\in \left\{1,\ldots ,n-1\right\}, α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}.For an optimal solution (σ110α,σ220α,…,σnn0α)\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })of the problem, the vector grad yα{y}_{\alpha }is normal at G{\mathcal{G}}. It is collinear with the vector (1,…,1)\left(1,\ldots ,1). From (3.8) and ∑i=1nσii0α=pα{\sum }_{i=1}^{n}{\sigma }_{ii}^{0\alpha }={p}^{\alpha }it follows that a critical point of the corresponding problem has the form σii0α=2n(n−1)(n−1)(2n+k)+na(k)pασnn0α=2n2n+kpα,\left\{\begin{array}{l}{\sigma }_{ii}^{0\alpha }=\frac{2n\left(n-1)}{\left(n-1)\left(2n+k)+na\left(k)}{p}^{\alpha }\\ {\sigma }_{nn}^{0\alpha }=\frac{2n}{2n+k}{p}^{\alpha },\end{array}\right.\hspace{8.25em}for any i∈{1,…,n−1}i\in \left\{1,\ldots ,n-1\right\}, α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}.For p∈Gp\in {\mathcal{G}}, let A{\mathcal{A}}be a 2-form, A:TpG×G→R{\mathcal{A}}:{T}_{p}{\mathcal{G}}\times {\mathcal{G}}\to {\mathbb{R}}defined by A(X,Y)=Hess(yα)(X,Y)+⟨σ′(X,Y),(grad(y)(p))⟩,{\mathcal{A}}\left(X,Y)={\rm{Hess}}({y}_{\alpha })\left(X,Y)+\langle {\sigma }^{^{\prime} }\left(X,Y),\left({\rm{grad}}(y)\left(p))\rangle ,where σ′{\sigma }^{^{\prime} }is the second fundamental form of G{\mathcal{G}}in Rn{{\mathbb{R}}}^{n}and ⟨⋅,⋅⟩\langle \cdot ,\hspace{0.33em}\cdot \rangle is the standard inner product on Rn{{\mathbb{R}}}^{n}.Moreover, it is easy to see that the Hessian matrix of yα{y}_{\alpha }has the form Hess(yα)=2(n−1)(k+2n)+na(k)n(n−1)−4⋯−4−4−42(n−1)(k+2n)+na(k)n(n−1)⋯−4−4⋮⋮⋱⋮⋮−4−4⋯2(n−1)(k+2n)+na(k)n(n−1)−4−4−4⋯−42kn.{\rm{Hess}}({y}_{\alpha })=\left(\begin{array}{ccccc}2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}& -\hspace{-0.25em}4& \cdots \hspace{0.33em}& -4& -\hspace{-0.25em}4\\ -\hspace{-0.25em}4& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}& \cdots \hspace{0.33em}& -4& -\hspace{-0.25em}4\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ -\hspace{-0.25em}4& -\hspace{-0.25em}4& \cdots \hspace{0.33em}& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}& -\hspace{-0.25em}4\\ -\hspace{-0.25em}4& -\hspace{-0.25em}4& \cdots \hspace{0.33em}& -4& \frac{2k}{n}\end{array}\right).As G{\mathcal{G}}is a totally geodesic hyperplane in Rn,{{\mathbb{R}}}^{n},considering a vector X=(X1,…,Xn)X=\left({X}_{1},\ldots ,{X}_{n})tangent to G{\mathcal{G}}at an arbitrary point xxon G{\mathcal{G}}, that is, verifying the relation ∑i=1n=0{\sum }_{i=1}^{n}=0, we obtain A(X,X)=2(n−1)(k+2n)+na(k)n(n−1)∑i=1n−1Xi2+2knXn2−8∑i,j=1nXiXj,i≠j=2(n−1)(k+2n)+na(k)n(n−1)∑i=1n−1Xi2+2knXn2+4∑i=1nXi2−8∑i,j=1nXiXj,i≠j=2(n−1)(k+2n)+na(k)n(n−1)∑i=1n−1Xi2+2knXn2+4∑i=1nXi2≥0.\begin{array}{rcl}{\mathcal{A}}\left(X,X)& =& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{X}_{i}^{2}+\frac{2k}{n}{X}_{n}^{2}-8\mathop{\displaystyle \sum }\limits_{i,j=1}^{n}{X}_{i}{X}_{j},\hspace{1.0em}i\ne j\\ & =& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{X}_{i}^{2}+\frac{2k}{n}{X}_{n}^{2}+4{\left(\mathop{\displaystyle \sum }\limits_{i=1}^{n}{X}_{i}\right)}^{2}-8\mathop{\displaystyle \sum }\limits_{i,j=1}^{n}{X}_{i}{X}_{j},\hspace{1.0em}i\ne j\\ & =& 2\frac{\left(n-1)\left(k+2n)+na\left(k)}{n\left(n-1)}\mathop{\displaystyle \sum }\limits_{i=1}^{n-1}{X}_{i}^{2}+\frac{2k}{n}{X}_{n}^{2}+4\mathop{\displaystyle \sum }\limits_{i=1}^{n}{X}_{i}^{2}\ge 0.\end{array}Hence, by Lemma 2, the critical point (σ110α,σ220α,…,σnn0α)\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })of yα{y}_{\alpha }is the global minimum point of the problem. Moreover, since yα(σ110α,σ220α,…,σnn0α)=0,{y}_{\alpha }\left({\sigma }_{11}^{0\alpha },{\sigma }_{22}^{0\alpha },\ldots ,{\sigma }_{nn}^{0\alpha })=0,we obtain P≥0.{\mathcal{P}}\ge 0.This implies that 2τ≤kC0+a(k)C0(W)+n2(C+C0)−n22(‖H‖2+‖H∗‖2)+−(1−ψ)cp−ψcq25[n(n−2)+tr2(ϕ)−tr(ϕ∗)]+−(1−ψ)cp+ψcq42(n−1)tr(ϕ).2\tau \le k{{\mathcal{C}}}^{0}+a\left(k){{\mathcal{C}}}^{0}\left({\mathcal{W}})+\frac{n}{2}\left({\mathcal{C}}+{{\mathcal{C}}}^{0})-\frac{{n}^{2}}{2}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[n\left(n-2)+{{\rm{tr}}}^{2}\left(\phi )-{\rm{tr}}\left({\phi }^{\ast })]+\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right)2\left(n-1){\rm{tr}}\left(\phi ).Thus, we have (3.9)ρ≤kn(n−1)C0+(n+k)(n2−n−k)n2kC0(W)+12(n−1)(C+C∗)−n2(n−1)(‖H‖2+‖H∗‖2)+−(1−ψ)cp−ψcq25(n−2)(n−1)+tr2(ϕ)n(n−1)−tr(ϕ∗)n(n−1)+2n−(1−ψ)cp+ψcq4tr(ϕ).\begin{array}{rcl}\rho & \le & \frac{k}{n\left(n-1)}{{\mathcal{C}}}^{0}+\frac{\left(n+k)\left({n}^{2}-n-k)}{{n}^{2}k}{{\mathcal{C}}}^{0}\left({\mathcal{W}})+\frac{1}{2\left(n-1)}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast })-\frac{n}{2\left(n-1)}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[\frac{\left(n-2)}{\left(n-1)}+\frac{{{\rm{tr}}}^{2}\left(\phi )}{n\left(n-1)}-\frac{{\rm{tr}}\left({\phi }^{\ast })}{n\left(n-1)}\right]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ).\end{array}Now taking the infimum over all tangent hyperplanes W{\mathcal{W}}of TpM{T}_{p}M, we obtain ρ≤δC(k;n−1)n(n−1)+12(n−1)(C+C∗)−n2(n−1)(‖H‖2+‖H∗‖2)+−(1−ψ)cp−ψcq25(n−2)(n−1)+tr2(ϕ)n(n−1)−tr(ϕ∗)n(n−1)+2n−(1−ψ)cp+ψcq4tr(ϕ).\begin{array}{rcl}\rho & \le & \frac{{\delta }_{C}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{2\left(n-1)}\left({\mathcal{C}}+{{\mathcal{C}}}^{\ast })-\frac{n}{2\left(n-1)}\left(\Vert H{\Vert }^{2}+\Vert {H}^{\ast }{\Vert }^{2})\\ & & +\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left[\frac{\left(n-2)}{\left(n-1)}+\frac{{{\rm{tr}}}^{2}\left(\phi )}{n\left(n-1)}-\frac{{\rm{tr}}\left({\phi }^{\ast })}{n\left(n-1)}\right]+\frac{2}{n}\left(-\frac{\left(1-\psi ){c}_{p}+\psi {c}_{q}}{4}\right){\rm{tr}}\left(\phi ).\end{array}This gives us the inequality (3.4). Similarly on taking the supremum over all tangent hyperplanes W{\mathcal{W}}of TpM{T}_{p}Min (3.9), we obtain the inequality (3.5).□Corollary 2Let Mn{M}^{n}be a totally real statistical submanifold of a Golden-like statistical manifold Nm{N}^{m}. Then for the generalized normalized δ\delta -Casorati curvature, we have the following optimal relationships: (i)For any real number kk, such that 0<k<n(n−1)0\lt k\lt n\left(n-1), ρ≤δC0(k;n−1)n(n−1)+C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+−(1−ψ)cp−ψcq25n−2n−1,\rho \le \frac{{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left(\frac{n-2}{n-1}\right),where δC0(k;n−1)=12[δC(k;n−1)+δC∗(k;n−1)]{\delta }_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\delta }_{C}\left(k;\hspace{0.33em}n-1)+{\delta }_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].(ii)For any real number k>n(n−1)k\gt n\left(n-1), ρ≤δ^C0(k;n−1)n(n−1)+1(n−1)C0−n(n−1)g(H,H∗)−2nn(n−1)‖H0‖2+−(1−ψ)cp−ψcq25n−2n−1,\rho \le \frac{{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)}{n\left(n-1)}+\frac{1}{\left(n-1)}{{\mathcal{C}}}^{0}-\frac{n}{\left(n-1)}g\left(H,{H}^{\ast })-\frac{2n}{n\left(n-1)}\Vert {H}^{0}{\Vert }^{2}+\left(-\frac{\left(1-\psi ){c}_{p}-\psi {c}_{q}}{2\sqrt{5}}\right)\left(\frac{n-2}{n-1}\right),where δ^C0(k;n−1)=12[δ^C(k;n−1)+δ^C∗(k;n−1)]{\widehat{\delta }}_{C}^{0}\left(k;\hspace{0.33em}n-1)=\frac{1}{2}\left[{\widehat{\delta }}_{C}\left(k;\hspace{0.33em}n-1)+{\widehat{\delta }}_{C}^{\ast }\left(k;\hspace{0.33em}n-1)].3.1Equality caseThe submanifolds enjoying the equality for the Casorati curvature at every point are called Casorati ideal submanifolds (for instance see [38]). In this subsection, we investigate the Casorati ideal submanifolds for (3.4) and (3.5) in terms of their second fundamental form.Theorem 3The Casorati ideal Lagrangian submanifolds for (3.4) and (3.5) are totally geodesic submanifolds with respect to Levi-Civita connection.ProofFirst, we find out the critical points of P{\mathcal{P}}σc=(σ110n+1,σ120n+1,…,σnn0n+1,…,σ110m,…,σnn0m){\sigma }^{c}=\left({\sigma }_{11}^{0n+1},{\sigma }_{12}^{0n+1},\ldots ,{\sigma }_{nn}^{0n+1},\ldots ,{\sigma }_{11}^{0m},\ldots ,{\sigma }_{nn}^{0m})as the solutions of the following system of linear homogeneous equations ∂P∂σii0α=22n+kk+(k+n)(n2−n−k)nk−2σii0α−4∑l=1,l≠inσll0α=0,∂P∂σnn0α=2knσnn0α−4∑l=1n−1σll0α=0,∂P∂σij0α=42n+kk+(k+n)(n2−n−k)nkσij0α=0,i≠j,∂P∂σin0α=42n+kk+(k+n)(n2−n−k)nkσin0α=0.\left\{\begin{array}{l}\frac{\partial {\mathcal{P}}}{\partial {\sigma }_{ii}^{0\alpha }}=2\left[\frac{2n+k}{k}+\frac{\left(k+n)\left({n}^{2}-n-k)}{nk}-2\right]{\sigma }_{ii}^{0\alpha }-4\mathop{\displaystyle \sum }\limits_{l=1,l\ne i}^{n}{\sigma }_{ll}^{0\alpha }=0,\\ \frac{\partial {\mathcal{P}}}{\partial {\sigma }_{nn}^{0\alpha }}=2\frac{k}{n}{\sigma }_{nn}^{0\alpha }-4\mathop{\displaystyle \sum }\limits_{l=1}^{n-1}{\sigma }_{ll}^{0\alpha }=0,\\ \frac{\partial {\mathcal{P}}}{\partial {\sigma }_{ij}^{0\alpha }}=4\left[\frac{2n+k}{k}+\frac{\left(k+n)\left({n}^{2}-n-k)}{nk}\right]{\sigma }_{ij}^{0\alpha }=0,\hspace{1.0em}i\ne j,\\ \frac{\partial {\mathcal{P}}}{\partial {\sigma }_{in}^{0\alpha }}=4\left[\frac{2n+k}{k}+\frac{\left(k+n)\left({n}^{2}-n-k)}{nk}\right]{\sigma }_{in}^{0\alpha }=0.\end{array}\right.The critical points satisfy σij0α=0{\sigma }_{ij}^{0\alpha }=0, i,j=∈{1,…,n}i,j=\in \left\{1,\ldots ,n\right\}and α={n+1,…,m}\alpha =\left\{n+1,\ldots ,m\right\}. Moreover, we know that P≥0{\mathcal{P}}\ge 0and P(σc)=0,{\mathcal{P}}\left({\sigma }^{c})=0,then the critical point σc{\sigma }^{c}is a minimum point of P.{\mathcal{P}}.Consequently, the equality holds in (3.4) and (3.5) if and only if σijα+σij∗α=0,{\sigma }_{ij}^{\alpha }+{\sigma }_{ij}^{\ast \alpha }=0,for i,j∈{1,…,n}i,j\in \left\{1,\ldots ,n\right\}and α∈{n+1,…,m}\alpha \in \left\{n+1,\ldots ,m\right\}. In other words, the equalities hold identically at all points p∈Mp\in Mif and only if σ+σ∗=0,\sigma +{\sigma }^{\ast }=0,where σ\sigma and σ∗{\sigma }^{\ast }are the imbedding curvature tensors of the submanifold associated with the dual connection ∇\nabla and ∇∗{\nabla }^{\ast }, respectively. Hence, the equality in (3.4) and (3.5) holds at ppif and only if ppis totally geodesic point with respect to Levi-Civita connection.□Remark 1The results for normalized Casorati curvature can be easily obtained by using (2.8) and (2.9) in the inequalities (3.4) and (3.5).4ConclusionIn this paper, we introduced and studied Golden-like statistical manifolds. We obtained some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provided a couple of examples.

### Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: Chen invariant; Casorati curvature; Golden manifold; statistical manifold; 53C15; 53C25; 53C40; 53B25

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