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The Quarterly Journal of Mathematics
, Volume Advance Article (2) – Jan 3, 2018

20 pages

/lp/ou_press/inflection-points-on-hyperbolic-tori-of-s-3-QmcXTm1Leb

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hax058
- Publisher site
- See Article on Publisher Site

Abstract Families of hyperbolic tori in S3 (the asymptotic lines are globally defined) without double inflection points is provided. More precisely, a small deformation of the Clifford torus para-metrized by asymptotic lines is analyzed, and it is described the set of inflections of the two families of asymptotic lines 𝒜1 and 𝒜2. Denote by ℐi the set of inflections of the asymptotic lines of the associated asymptotic foliation 𝒜i, also called flecnodal set. The intersection ℐ1∩ℐ2 is called the set of double inflections. It is shown that by an appropriated deformation of the Clifford torus the set ℐ1∩ℐ2 is empty for the deformed surface. This gives a negative answer to a problem formulated by S. Tabachnikov and V. Ovsienko [Hyperbolic Carathéodory Conjecture, Proc. of the Steklo Inst. of Math. 258 (2007), pp. 178–193] in the context of spherical surfaces. Also, a family of immersed tori in S3, without double inflection set, is obtained. 1. Introduction More than one hundred years ago were discovered two global theorems on smooth closed strictly convex plane curves, see [12]. The first one is known as the four-vertex theorem: the curvature of a plane oval has at least four critical points. These critical points are the points at which the osculating circles are hyper osculating, that is, are third-order tangent to the curve. The second theorem concerns osculating conics, and states that a smooth convex closed curve has at least six distinct points at which the osculating conics are hyper osculating. Naturally, it is to be expected that there are multi-dimensional versions of these theorems, see [15]. In order to analyze the bidimensional version of these results, Ovsienko and Tabachnikov, see [15], considered a hyperbolic surface M of P3R, this is, M is orien-table and the second quadratic form is non-degenerate and indefinite everywhere. It is known that a smooth surface M in P3R can be approximated by a quadric at every point up to order 2. A point p0∈M is called quadratic if M can be approximated by a quadric at p0 up to order 3. In this direction, Ovsienko and Tabachnikov (see [15]) conjectured that: “every closed hyperbolic surface in P3Rhas no less than eight distinct quadratic points”. This conjecture is supported by the evidences that small deformations of the Clifford torus of the form (α+εhN)/∣α+εhN∣ the set of quadratic points is, at first order in ε, defined by the system of equations huuu+hu=0,hvvv+hv=0,and in various situations analyzed in [15] the authors confirmed this conjecture. Clifford torus was considered because it is a hyperbolic surface and every point is quadratic. In fact, the Clifford torus in S3 is the intersection of the quadratic cone xy−zw=0 with unitary sphere x2+y2+z2+w2=1. At each point of M, one has two orthogonal asymptotic directions. The projective space P3R can be seen as the quotient space of unit sphere S3 by the equivalence relation A(p)=−p, p∈S3 and the Clifford torus is also a surface of P3R. In this work, we will consider hyperbolic surface in S3. There is a considerable difference between the cases of surfaces in the Euclidean and in the Spherical spaces. In R3, for example, the asymptotic lines are never globally defined for immersions of compact and oriented surfaces. This is due to the fact that in these surfaces there are always elliptic points, at which Kext>0 (see [18, Chapter 2, p. 64]). Let α:M→S3 be an immersion or embedding of class Cr,r≥3, of a smooth, oriented and compact two-dimensional manifold M into the three-dimensional sphere S3 endowed with the canonical inner product ⟨.,.⟩ of R4. The Fundamental Forms of α at a point p of M are the symmetric bilinear forms on TpM defined as follows (see [18]): Iα(p;v,w)=⟨Dα(p;v),Dα(p;w)⟩,IIα(p;v,w)=⟨−DNα(p;v),Dα(p;w)⟩.Here, Nα is the positive unit normal of the immersion α and ⟨Nα,α⟩=0. Through every point p of the hyperbolic region Hα of the immersion α, characterized by the condition that the extrinsic Gaussian Curvature Kext=det(DNα) is negative, pass two transverse asymptotic lines of α, tangent to the two asymptotic directions through p. Assuming r≥3 this follows from the usual existence and uniqueness theorems on Ordinary Differential Equations. In fact, on Hα, the local line fields are defined by the kernels Lα,1, Lα,2 of the smooth one-forms ωα,1, ωα,2 which locally split IIα as the product of ωα,1 and ωα,2. The forms ωα,i are locally defined up to a non-vanishing factor and a permutation of their indexes. Therefore, their kernels and integral foliations are locally well defined only up to a permutation of their indexes. Under the orientability hypothesis imposed on M, it is possible to globalize, to the whole Hα, the definition of the line fields Lα,1, Lα,2 and of the choice of an ordering between them, as established in [7–9]. These two line fields, called the asymptotic line fields of α, are of class Cr−2 on Hα; they are distinctly defined together with the ordering between them given by the sub-indexes {1,2} which define their orientation ordering: “1” for the first asymptotic line field Lα,1, “2” for the second asymptotic line field Lα,2. The asymptotic foliations of α are the integral foliations Aα,1 of Lα,1 and Aα,2 of Lα,2; they fill out the hyperbolic region Hα. In a local chart (u,v), the asymptotic directions of an immersion α are defined by the implicit differential equation II=edu2+2fdudv+gdv2=0. In S3, with the second fundamental form relative to the normal vector N=α∧αu∧αv, it follows that e=det[α,αu,αv,αuu]EG−F2,f=det[α,αu,αv,αuv]EG−F2,g=det[α,αu,αv,αvv]EG−F2. The study of asymptotic lines on surfaces M of R3 and S3 is a classical subject of Differential Geometry. See ([1, Chapter 3]), ([4, Chapter II]), ([18, Vol. IV]) and ([19, Chapter 2]). In S3, the asymptotic lines can be globally defined, an example is the Clifford torus, C=S1(r)×S1(r)⊂S3, where S1(r)={(x,y)∈R2:x2+y2=r2} and r=2/2. In C, all asymptotic lines are closed curves. In fact, they are Villarceau circles illustrated in Fig. 1. Figure 1. View largeDownload slide Villarceau circles are the leaves of the asymptotic foliations Aα,i of the Clifford torus. Figure 1. View largeDownload slide Villarceau circles are the leaves of the asymptotic foliations Aα,i of the Clifford torus. In [10], a deformation of the Clifford torus is considered to obtain an embedded torus such that all asymptotic lines are dense. In this paper, two families of tori are constructed in order to obtain a negative answer to the conjecture formulated by Ovsienko and Tabachnikov for hyperbolic surfaces of S3. In Section 3, the first example is constructed performing a small deformation (see Eq. (3.3)) of the Clifford torus. In order to do the analysis, the quadratic points are characterized up to order three in ε. We observe that in [15], it was considered only first order approximation in ε. In Section 4, the second example is a family of immersed torus, obtained by a small deformation of minimal immersed torus, where the set of quadratic points or inflection points is empty. Also, in Section 5, it will be analyzed the inflection sets of a small deformation of the torus of revolution in R3. 2. Inflections and quadratic points A smooth surface M in S3 can be approximated by a quadric at every point up to order 2. A point p0∈M is called a quadratic point if M can be approximated by a quadric at p0 up to order 3. In what follows, quadratic and inflection points will be characterized, see [15]. Lemma 1. Let α:M→S3be a smooth surface given by α(u,v)=(u,v,p(u,v),q(u,v)), where p(u,v)=a11uv+16a03v3+12a12uv2+12a21u2v+16a30u3+h.o.t.q(u,v)=1+12b02v2+b11uv+12b20u2+16b03v3+12b12uv2+12b21u2v+16b30u3+h.o.t. (2.1) A point p0is a quadratic point if, and only if, the coefficients a30and a03vanish at p0. Proof For completeness, a proof will be sketched, see also [15]. Without loss of generality, we consider p0=(0,0,0,1). Consider the quadric in S3 tangent to M through the point p0 q02v2+2q11uv+q20u2+r20z2+s13uz+s23vz+z=0. (2.2)Solving (2.2) in z and writing the terms up to order 3, we obtain z=−q02v2−2q11uv−q20u2+s13q20u3+(2q11s13+q20s23)u2v+(q02s13+2q11s23)uv2+s23q02v3+O(4). (2.3)As the quadric is in S3, we have to u2+v2+z2+w2=1. Solving this last equation in w and writing the terms up to order 3, we obtain w=1−12u2−12v2+O(4). (2.4) To get cubic contact, we must solve the system {z=p(u,v),w=q(u,v)}. This system is equivalent to −q02v2−q20u2+(q20s13−16a30)u3+(q02s23−16a03)v3−(2q11+a11)uv+(2q11s13+q20s23−12a21)u2v+(q02s13+2q11s23−12a12)uv2=0,12(1+b02)v2+12(1+b20)u2+16b30u3+16b03v3b11uv+12b21u2v+12b12uv2=0. We obtain the solution {b20=b02=−1,b11=b03=b30=b12=b21=0,q11=−a112,q20=q02=0,s13=−a212a11,s23=−a122a11,a30=a03=0}. Therefore, M can be approximated by a quadric at p0 up to order 3 if, and only if, a30=a03=0.□ Definition 1. A point p0 is called an inflection point if an asymptotic line passing through p0 has zero geodesic curvature at p0. If both asymptotic lines passing through p0 has inflection, the point p0 is called a point of double inflection. The inflections of the asymptotic lines of the asymptotic foliation Ai will be denoted by Ii, (i=1,2). So the set of double inflection is the set I1∩I2. For generic hyperbolic surfaces, it is expected that the set of inflections Ii is regular arcs of curves, see Fig. 2. In classical and recent literature, they are called flecnodal curves, see [2, 20]. Figure 2. View largeDownload slide Inflection points of asymptotic foliation Ai=Aα,i (left and center) and double inflection (right). Figure 2. View largeDownload slide Inflection points of asymptotic foliation Ai=Aα,i (left and center) and double inflection (right). We recall that Gauss equations of structure of a smooth immersion X:U⊂R2→S3 are given by Xuu=Γ111Xu+Γ112Xv+eN−EXXuv=Γ121Xu+Γ122Xv+fN−FXXvv=Γ221Xu+Γ222Xv+gN−GX. (2.5)Here Γijk are the Christoffel symbols and expressed in terms of the coefficients of first fundamental forms and its first derivatives. The Christoffel symbols are given by Γ111=GEu−2FFu+FEv2(EG−F2),Γ112=2EFu−EEv−FEu2(EG−F2)Γ121=GEv−FGu2(EG−F2),Γ122=EGu−FEv2(EG−F2)Γ221=2GFv−GGu−FGv2(EG−F2),Γ222=EGv−2FFv+FGu2(EG−F2). (2.6)In Proposition 1, we characterize the set of inflections of the asymptotic lines. Proposition 1. Let X:U⊂R2⟶Mbe a local parametrization of a surface M of S3. Then the inflection points are given by: I1={(u,v):I1(u,v,p)=H(u,v,p)=0}or I2={(u,v):I2(u,v,q)=L(u,v,p)=0}where I1(u,v,p)=2(gp+f)[Γ221p3+(2Γ121−Γ222)p2+(Γ111−2Γ122)p−Γ112]+gvp3+(gu+2fv)p2+(ev+2fu)p+eu=0,H(u,v,p)=gp2+2fp+e=0. (2.7)Or equivalently, in the projective coordinate q=du/dv, by I2(u,v,q)=2(eq+f)[Γ112q3+(2Γ122−Γ111)q2+(Γ222−2Γ121)q−Γ221]+euq3+(2fu+ev)q2+(gu+2fv)q+gv=0,L(u,v,q)=eq2+2fq+g=0. (2.8) Proof Let γ(s)=X(u(s),v(s)), where u and v are the solutions of e(u′)2+2fu′v′+g(v′)2=0. Differentiating with respect to s it follows that: γ=X(u,v)⟹γ′=Xuu′+Xvv′⟹γ″=Xuu(u′)2+2Xuvu′v′+Xvv(v′)2+Xuu″+Xvv″. From Eq. (2.5), it follows that: γ″=(u′)2(Γ111Xu+Γ112Xv+eN−EX)+2u′v′(Γ121Xu+Γ122Xv+fN−FX)+(v′)2(Γ221Xu+Γ222Xv+gN−GX)+Xuu′′+Xvv′′. ⟹γ′′=((u′)2Γ111+2u′v′Γ121+(v′)2Γ221+u′′)Xu+((u′)2Γ112+2u′v′Γ122+(v′)2Γ222+v′′)Xv−((u′)2E+2u′v′F+(v′)2G)X. The inflection points are the points where the tangent component of γ″ is zero. That is, the points where the geodesic curvature is zero. Then it follows that {(u′)2Γ111+2u′v′Γ121+(v′)2Γ221+u″=0,(u′)2Γ112+2u′v′Γ122+(v′)2Γ222+v″=0. From the equations above it follows that d2vdu2=Γ221(dvdu)3+(2Γ121−Γ222)(dvdu)2+(Γ111−2Γ122)dvdu−Γ112,d2udv2=Γ112(dudv)3+(2Γ122−Γ111)(dudv)2+(Γ222−2Γ121)dudv−Γ221. (2.9) From equation e(u′)2+2fu′v′+g(v′)2=0 write p=dvdu and so H(u,v,p)=gp2+2fp+e=0. From Eq. (2.9), it follows that d2vdu2=p′=Γ221p3+(2Γ121−Γ222)p2+(Γ111−2Γ122)p−Γ112. (2.10)From the equation H(u,v,p)=0, the Lie–Cartan vector field is given by {u′=1v′=pp′=−(Hu+pHv)Hp.From the construction above, the asymptotic lines are the projections of integral curves of the Lie–Cartan vector field. Therefore, the inflection points I1 given by Eq. (2.10) are the projections of the solutions of the system of equations below: I1(u,v,p)=2(gp+f)[Γ221p3+(2Γ121−Γ222)p2+(Γ111−2Γ122)p−Γ112]+gvp3+(gu+2fv)p2+(ev+2fu)p+eu=0,H(u,v,p)=gp2+2fp+e=0. In the chart (u,v,q) the analysis is similar.□ Proposition 2. In a hyperbolic surface of S3, a point is a quadratic point, if and only if, it is a double inflection point. Proof See [15]. For convenience to the reader, we give a proof here. Consider α(u,v)=(u,v,p(u,v),q(u,v)), where p(u,v) and q(u,v) are given by (2.1). Using the Proposition 1, the geodesic curvature at p0=(0,0,0,1) for the first asymptotic line Aα,1 is kg=−a30 and for the second asymptotic line Aα,2 is kg=−a03. If p0 is a quadratic point, by Lemma 1 a30=a03=0 and thus p0 is a double inflection point. Reciprocally, if p0 is a double inflection point, we obtain a30=a03=0 and by Lemma 1, p0 is a quadratic point.□ 3. Clifford torus and its deformations Consider the Clifford torus C=S1(12)×S1(12)⊂S3 parametrized by α(u,v)=22(cos(u+v),sin(u+v),cos(−u+v),sin(−u+v)), (3.1)where (u,v)∈[0,2π]×[0,2π]. Proposition 3. The asymptotic lines on the Clifford torus in the coordinates given by Eq. (3.1) are given by dudv=0, that is, the asymptotic lines are the coordinate curves (Villarceau circles). Also the asymptotic lines are geodesics. Proof Direct calculations shows that E(u,v)=1,F(u,v)=0,G(u,v)=1,e(u,v)=0,f(u,v)=−1,g(u,v)=0.Therefore, the asymptotic lines are the coordinates curves, since e(u,v)=g(u,v)=0 and f(u,v)≠0. The affirmation about geodesics is immediate since the metric is flat.□ Theorem 1. The set of inflections of the embedding β=α+εhN∣α+εhN∣ is given, up to order 3, given by I1=−[4hu+huuu]ε+3[−4huhuv+huuvhuu]ε2+[72h2hu−36(hu)3−36huhuuh+18h2huuu+18huvhuuhv+18(huu)2hu+18huuhuhvv−9huvhuuhuuv+18hvhuhuuv−92huvv(huu)2]ε3+O(ε4),I2=−[4hv+hvvv]ε+3[−4hvhuv+huvvhvv]ε2+[72h2hv−36(hv)3+18huvhuhvv+18hvvhvhuu+18(hvv)2hv−36hvhhvv+18h2hvvv−92huuv(hvv)2−9hvvhuvhuvv+18hvhuhuvv]ε3+O(ε4). (3.2) Proof The coefficients of the first and second fundamental forms of β are given by E=1+ε2(hu)2F=ε(2ε2h3+εhvhu−2h)G=1+ε2(hv)2e=−ε(−3ε2h2huu+2ε2h(hu)2+2εhvhu+huu)f=−1−εhuv+(4h2−(hu)2−(hv)2)ε2+(3h2huv−2huhhv)ε3g=−εhvv−2ε2hvhu+(3hvvh2−2(hv)2h)ε3. By Proposition 1, the inflection points of β are defined by Eqs. (2.7) and (2.8). In order to obtain I1, we write p=p1ε+p2ε2+p3ε3 in H(u,v,p)=gp2+2fp+e=0, where the coefficients p1,p2 and p3 are obtained solving, respectively, the equations Hε∣ε=0=0,Hεε∣ε=0=0 and Hεεε∣ε=0=0. Replacing this solution p=p1ε+p2ε2+p3ε3 in the equation below I1(u,v,p)=2(gp+f)[Γ221p3+(2Γ121−Γ222)p2+(Γ111−2Γ122)p−Γ112]+gvp3+(gu+2fv)p2+(ev+2fu)p+eu=0,we obtain I1=I1(u,v,p(u,v,ε))=0. The set of inflections defined by I2(u,v,ε)=0 is obtained analogously working with Eq. (2.8).□ Remark 1. The set of inflections given by Theorem 1 appears in [15] with approximation of first order in ε. In order to obtain the main result of this work, it is necessary to obtain the set of inflections with approximation of order ε3 and deformations as considered below, see Proposition 4 and Theorem 2 below. Consider the small deformation of the Clifford torus parametrized by α as defined by A=α+ε[ψ(u,v)αu+ξ(u,v)αv+h(u,v)α∧αu∧αv+φ(u,v)α]β=A∣A∣. (3.3) Proposition 4. The double inflection set of βdefined by Eq. (3.3) is given by I1(u,v,ε)=I2(u,v,ε)=0where, I1(u,v,ε)=−(4hu+huuu)ε+J12ε2+J13ε3+O(ε4),I2(u,v,ε)=−(4hv+hvvv)ε+J22ε2+J23ε3+O(ε4). (3.4)The terms J12,J13,J22and J23are presented in AppendixA. Proof By the Proposition 1, the inflection points of β are defined by Eqs. (2.7) and (2.8). We will work first with (2.7). In H(u,v,p)=gp2+2fp+e=0, we write p=p1ε+p2ε2+p3ε3. The coefficients p1,p2 and p3 are obtained solving respectively the equations Hε∣ε=0=0,Hεε∣ε=0=0 and Hεεε∣ε=0=0. Replacing this p=p1ε+p2ε2+p3ε3 in the factor 2(gp+f)[Γ221p3+(2Γ121−Γ222)p2+(Γ111−2Γ122)p−Γ112]+gvp3+(gu+2fv)p2+(ev+2fu)p+eu=0,we obtain I1(u,v,ε) given by (3.4). The factor I2(u,v,ε) is obtained analogously working with (2.8), writing q=q1ε+q2ε2+q3ε3 in L(u,v,q)=eq2+2fq+g=0. These long expressions involved in the functions J12, J13, J22 and J23 were obtained using symbolic computation systems, but can be calculated by hand.□ In order to obtain a family of torus without double inflection points, we consider initially families of tori such that the set of inflections are regular curves (may be coincident or sufficiently close). In other words, I1(u,v,ε)=0 and I2(u,v,ε)=0 given by (3.4) are regular curves. The idea is to deform these families in such a way that the curves I1(u,v,ε)=0 and I2(u,v,ε)=0 have empty intersection. In Example 1, we present a case such that the regular curves I1(u,v,ε)=0 and I2(u,v,ε)=0 are equal and that will be used to obtain the main result of Theorem 2. See also section 5. Example 1. In (3.3) consider ψ(u,v)=φ(u,v)=ξ(u,v)=0 and h(u,v)=cos(u+v)+q0cos(3u+3v)+q1cos(4u+4v) with q0 and q1 small. By (3.4), we obtain I1(u,v,ε)=I2(u,v,ε)=[3sin(v+u)−15q0sin(3u+3v)−48q1sin(4u+4v)]ε+O(ε2). Therefore, for q0 and q1 small, I1(u,v,ε)=0 and I2(u,v,ε)=0 are close to, u+v=π+O1(ε), and u+v=2π+O2(ε). In order to conclude that I1(u,v,ε)=I2(u,v,ε) if, and only if, u+v=π, u+v=2π, we observe that h(u,v)=h(v,u)=h(u+π,v)=h(u,v+π) for all (u,v). Theorem 2. There exists a real analytic embedded torus, close to the Clifford torus, such that the set of double inflection is empty. Proof Considerer the space of first harmonics H1={1,sinu,cosu,cosv,sinv} and let H=(1,sinu,cosu,sinv,cosv). Also, consider the the vectors A=(a0,a1,a2,a3,a4), B=(b0,b1,b2,b3,b4), C=(c0,c1,c2,c3,c4),D=(d1,d2). Let h(u,v)=cos(u+v)+q0cos(3u+3v)+q1cos(4u+4v)+a0(1−ε)+ε⟨H,A⟩ψ(u,v)=⟨C,H⟩+ε(d1sin2u+d2cos2u),φ(u,v)=⟨B,H⟩,ξ(u,v)=0. (3.5) The introduction of Q=(q0,q1)≠0 in the first-order deformation cos(u+v)+q0cos(3u+3v)+q1cos(4u+4v) is essential to break the symmetry. For small ε≠0 and appropriated non-zero vectors A, B, C, D=(d1,d2) and Q=(q0,q1), the set of inflections is regular curves and the set of double inflection is empty. In fact, consider I˜1(u,v,ε)=I1(u,v,ε)ε,I˜2(u,v,ε)=I2(u,v,ε)ε,where I1(u,v,ε) and I2(u,v,ε) are given by Proposition 4. Note that I˜1(u,v,ε)=3sin(v+u)−15q0sin(3u+3v)−48q1sin(4u+4v)+J12(u,v)ε+J13(u,v)ε2+O(ε3),I˜2(u,v,ε)=3sin(v+u)−15q0sin(3u+3v)−48q1sin(4u+4v)+J22(u,v)ε+J23(u,v)ε2+O(ε3). The solutions of I˜1(u,v,ε)=0 and I˜2(u,v,ε)=0 are obtained applying the Implicit Function Theorem. Let v=r−u+π, where r=r1ε+r2ε2+⋯. The coefficient r1 is obtained solving the equation dI˜1dε∣ε=0=0 in r1, and the coefficient r2 is obtained solving the equation d2I˜1dε2∣ε=0=0 in r2. Replacing this r in I˜1 we obtain I˜1(R1(u,ε),u,ε)=0.Analogously, we obtain I˜2(R2(u,ε),u,ε)=0. So to obtain that the set of double inflections, near the set u+v=π, is empty, we need to show that Δ12(u,ε)=(R1−R2)(u,ε) is different from zero for all u∈[0,2π] and ε≠0 small. Also, near the set defined by u+v=2π the set of inflections are regular and defined implicitly by I˜1(R3(u,ε),u,ε)=0,I˜2(R4(u,ε),u,ε)=0. Again, we need to show that Δ34(u,ε)=(R3−R4)(u,ε) is different from zero for all u∈[0,2π] and ε≠0 small. The main idea is to construct a deformation such that in the first order the set of double inflections are two closed regular curves, parametrized by u+v=π and u+v=2π. In the sequence with an appropriate deformation in the space of first harmonics, we obtain a torus, close to the Clifford torus, such that the set of double inflections is empty. The crucial point is to obtain deformations with non-zero means for the functions Δ12(u,ε) and Δ34(u,ε). At this point, we will consider the equations Δ12(u,0)=0 and Δ34(u,0)=0. From equation (3.4), it follows, after a long calculation corroborated with symbolic manipulators, that Δ12(u,0)=C1sinu+C2cosu=0Δ34(u,0)=C3sinu+C4cosu=0C1=(8(−b2+b4)+9(−c1+c3))q0+(15(b2−b4)+16(c1−c3))q115q0−64q1−1+−a0b2+a0b4+a2−a4−c1+c315q0−64q1−1C2=(8(b1+b3)−9(c2+c4))q0+(−15(b1+b3)+16(c2+c4))q115q0−64q1−1+a0b1+a0b3−a1−a3−c2−c415q0−64q1−1C3=(16b4+18c3)q0+(30b2+32c1)q1−2a0b2+2a2+2c315q0+64q1−1C4=(16b3−18c4)q0+(−30b1+32c2)q1+2a0b1−2a1−2c415q0+64q1−1.The equation above is solved explicitly and we found that ai=ai(Q,B,C,a0), i=1,…,4. In fact, a1=a0b1−15b1q1+8b3q0+16c2q1−9c4q0−c4,a2=a0b2−15b2q1−8b4q0−16c1q1−9c3q0−c3,a3=a0b3+8b1q0−15b3q1−9c2q0+16c4q1−c2,a4=a0b4−8b2q0−15b4q1−9c1q0−16c3q1−c1. (3.6) With the above relations, the functions δ12=Δ12/ε and δ34=Δ34/ε can be extended to ε equal to zero. From the structure of equations, it follows that δ12(u,ε)=δ0(a0,B,C,D,Q)+p10sinu+p01cos(u)+p20sin2u+p02cos2u+O(ε),δ34(u,ε)=δ1(a0,B,C,D,Q)+q10sinu+q01cos(u)+q20sin2u+q02cos2u+O(ε),where pij=pij(a0,B,C,D,Q) and qij=qij(a0,B,C,D,Q). Also, pij and qij are polynomials of degree 3 in the variables and they are quadratic if a0 is fixed. These long expressions are in Appendix B fixing some variables. The system of equations p10=p01=p20=p02=q10=q01=q20=q02=0 has non-empty solution. In fact, we have a map R:R15→R8, with R−1(0) being a variety of codimension 7, is the solution of the mentioned system. Taking the numerical values, see Appendix B, A0=(1.5,8.5951,−5.0918,−2.2717,−9.8657),B0=(−2,4.4373,0.8442,2.0814,−4.2707),C0=(−1,2.9243,4.7664,5.6263,−1.5845),D0=(10,30),Q0=(0.0083,0.01),it follows that pij(1.5,B0,C0,D0,Q0)=qij(1.5,B0,C0,D0,Q0)=0, δ0(1.35,B0,C0,D0,Q0)=−0.1352 and δ1(1.35,B0,C0,Q0)=−9.9554. Here A0 is obtained from equation (3.6) with the numerical values a0=1.5, B0=(b0,b1,b2,b3,b4) and C0=(c0,c1,c2,c3,c4) obtained solving the system of equations pij=qij=0. So it follows that Δ12(u,ε)=R1(u,ε)−R2(u,ε)=−0.1352ε+O(ε2)Δ34(u,ε)=R3(u,ε)−R4(u,ε)=−9.9554ε+O(ε2). This shows that the set of double inflections is empty for this deformation of the Clifford torus. We need to choose the vectors Q=(q0,q1), A, B and C non-zeros, in order to guarantee that ∫02πδ12(u,0)du=2πδ0(a0,B,C,D,Q)≠0 and ∫02πδ34(u,0)du=2πδ1(a0,B,C,D,Q)≠0. We remark that if A=0, B=0 or C=0, then δ0δ1=0, and the set of double inflections will be not empty. So we need to deform the Clifford torus in at least three directions in order to construct an example with empty double inflection set using only first- and second-order harmonics. Otherwise the set of double inflections will be non-empty and generically will have at least four points.□ Proposition 5 below is an indicative that, considering only normal deformations of the Clifford torus, the double inflection set consists of at least four points. Proposition 5. Consider a smooth double periodic function h:R2→R, h(u,v)=h(u+2π,v)=h(u,v+2π)=h(u+2π,v+2π). Let H1={(u,v)∈[0,2π)×[0,2π):huuu+4hu=0}and H2={(u,v)∈[0,2π)×[0,2π):hvvv+4hv=0}. Let h(u,v,ε)=cos(u+v)+εsinu. For ε≠0we have that #(H1∩H2)=4. Proof We have that h1=huuu+4hu=3sin(u+v)h2=hvvv+4hv=3sin(u+v)−3εcosu. The intersection set is given by the four points p1=(π2,π),p2=(3π2,π),p3=(3π2,π2),p4=(3π2,3π2).□ 4. A family of immersed tori of S3 without double inflections In this section, will be considered a family of immersed tori such that the set of double inflections (quadratic points) is empty. Consider a family of minimal immersed tori, see [11], defined by α(u,v)=(cos(mv)sinu,sin(mv)sinu,cos(nv)cosu,sin(nv)cosu). (4.1) Lemma 2. The asymptotic lines of the immersion αgiven by Eq. (4.1) are the coordinate curves. Moreover, the geodesic curvature of the coordinate curves v=v0is zero and the geodesic curvature of the coordinates curves u=u0are given by kg∣u=u0=∣cosu0sinu0(m2−n2)∣n2cos2u0+m2sin2u0. (4.2) Proof Let v=v1n2cos2u+m2sin2u. Consider the parametrization α(u,v1). Direct calculations shows that ⟨αu,αu⟩=1,⟨αu,αv1⟩=0,⟨αv1,αv1⟩=1. The coefficients of the second fundamental form are given by e(u,v)=g(u,u)=0 and f(u,v)=−mn≠0. So the coordinate curves are asymptotic lines. Also k=∣αv1v1∣=m4sin2u+n4cos2un2cos2u+m2sin2u.As kn=−⟨αv1v1,α⟩=1 it follows that kg2=k2−kn2 is as stated in Eq. (4.2). For the coordinates curves v=v0 we obtain k=∣αuu∣=1 and kn=−⟨αuu,α⟩=1 and it follows that kg=0.□ The unit normal vector N to the immersion α is proportional to α∧αu∧αv. Consider the deformation β(u,v)=α(u,v)+εh(u,v)N∣α+εh(u,v)N∣. (4.3) Proposition 6. The set of inflections of the two families of asymptotic lines of the immersion βdefined by Eq. (4.3) is given by I1(u,v,ε)=I2(u,v,ε)=0where I1(u,v,ε)=[(6m2−2n2)cos2u+(−2m2+6n2)sin2u]huε+6huusinucosu(m2−n2)ε+(n2cos2u+m2sin2u)huuuε+O(ε2),I2(u,v,ε)=−2mn(m2−n2)sinucosu+3εhuv((m2−n2)2cos3u−m2(m2−n2)cosu)sinu−ε(hvn4+3hvm2n2+hvvvn2)cos2u−ε(hvm4+3hvm2n2+hvvvm2)sin2u+O(ε2). (4.4) Proof Direct calculations show that e=−12ε[(m2−n2)(2husin(2u)−huucos(2u))+(m2+n2)huu]+O(ε2)f=−mn+ε((m2−n2)cos2u−m2)huv+O(ε2)g=ε(m2−n2)((m2+n2)(cos2u−m2))h+ε((m2−n2)cos2u−m2)[(m2−n2)cosusinuhu+hvv]+O(ε2)E=1+O(ε2)F=−2εmnh+O(ε2)G=m2sin2u+n2cos2u+O(ε2).The sequence of proof is analogous to that of Proposition 4.□ Theorem 3. For h(u,v)=cosu+sinuand m=2and n=1the immersion βfor ε≠0small has no points of double inflection. Proof With h(u,v)=cosu+sinu, m=2 and n=1, by Proposition 6, the inflections are given by I1(u,v,ε)=0 and I2(u,v,ε)=0 where I1(u,v,ε)=14(21sinu−39cosu+45sin3u−45cos3u)ε+O(ε2),I2(u,v,ε)=−6sin2u+O(ε2). Using the Implicit Function Theorem, solving the equation I2(u,v,ε)=0 in the variable u we obtain that I2(u,v,ε)=0 is the union of four regular curves near the lines u=0,u=π2,u=π and u=3π2. The inflection set I1(u,v,ε)=0 is the union of six regular curves near the lines u1=0.410124,u2=1.378403,u3=2.138464,u4=3.551716,u5=4.519995 and u6=5.280056. As I1(0,v,0)=−21,I1(π2,v,0)=−6,I1(π,v,0)=21 and I1(3π2,v,0)=6 it follows, for ε≠0 small, that the set of double inflection is empty.□ 5. Inflection points in the torus of revolution in R3 In this section, we consider perturbations of the torus of revolution to obtain examples in R3 such that the set of double inflection points (also quadratic points) is empty. The characterization of double inflection points and quadratic points for surfaces of R3 can be established as in Propositions 1 and 2. Proposition 7. Let T2be the torus of revolution, obtained by the rotation of the circle (x−R)2+z2=r2, r<R, around the z-axis. Then there is a surface near T2such that the set of double inflection points is empty. Proof Consider the following parametrization of the torus of revolution X(u,v)=((R+rcosu)cosv,(R+rcosu)sinv,rsinu),and consider the perturbation β(u,v)=X(u,v)+εh(u,v)N, (5.1)where N is the unit normal vector of X and h(u,v) is a smooth double periodic function. The coefficients of the first and second fundamental forms of β are given by E=−h2ε2+hu2ε2−2hrε+r2F=hvhuε2G=(hε−r)2(cosu)2−2R(hε−r)cosu+hv2ε2+R2e=T2ε(3εhu2+2rhuu)+2r(−rhv2ε22+T2(−εh+r))2rT2f=−(rhv(Rεh−rT)sinu−T(2rhuhvεcosu+Rhuhvε+rhuvT))εrT2g=ε2cosu2(T2hu2−3r2hv2)−ε((Rεh+rT)hu−hvv)−Tr2(cosu(εhcosu−T))Tr2,where T=R+rcosu>0. Note that for ε=0, the Gaussian curvature is given by K=cosur(R+rcosu)and thus, K<0 for π2<u<3π2. Proceeding analogously as in the proof of Theorem 1 (writing p=p0+p1ε+h.o.t,.) we obtain the set of inflections. For h(u,v)=sinu, the set of inflection points are given by I1=3Tr3Rsinucosu−3r4εcos3u−3r4εcosusin2u+3r4εcosu−4r2εT−Trcosu−9r3εRsin2u+6r3εR−3r3εRcos2u−9r2εR2sin2ucosu+3r2εR2cosu+O(ε2),I2=3Tr3Rsinucosu−3r4εcos3u−3r4εcosusin2u+3r4εcosu+4r2εT−Trcosu−9r3εRsin2u+6r3εR−3r3εRcos2u−9r2εR2sin2ucosu+3r2εR2cosu+O(ε2). (5.2)Note that for ε=0, I1=I2=0 if, and only if, u=π. For ε≠0, I1−I2=−8r2εT−Trcosu+O(ε2)≠0,for π2<u<3π2. Therefore, the intersection I1=0 and I2=0 is empty.□ 6. Conclusions In this work, we obtained an embedded torus (close to the Clifford torus) such that each asymptotic foliation has four regular curves of inflection points (points where the geodesic curvature of the corresponding asymptotic line is zero), but with no point of double inflection. The central idea is to obtain, first an example of a torus with coincident set of inflections for both asymptotic foliations and in the sequence to perform a deformation to separate the sets of inflections, see Section 3. Also, we obtained a family of immersed tori in S3 such that the set of double inflection is empty, see Section 4. It would be challenge to prove (or disprove) the following bidimensional version of the four vertex theorem, see [14, 15]. Consider a smooth, or at least of class C3, double periodic function h:R2→R, h(u,v)=h(u+2π,v)=h(u,v+2π)=h(u+2π,v+2π). Let H1={(u,v)∈[0,2π)×[0,2π):huuu+4hu=0} and H2={(u,v)∈[0,2π)×[0,2π):hvvv+4hv=0}. Is it possible that H1∩H2=∅ ? Generically, #(H1∩H2)≥4? Finally we mention that in [20] is shown that, for a hyperbolic disc of a generic smooth surface, the set of double inflections has an odd number of transversal intersection points. Also in [13] is analyzed the inflections of graphs of polynomial surfaces in R3. For classical and recent works about inflections, quadratic points, flecnodal curves and bitangencies of lines or planes on surfaces see [2, 3, 5, 6, 16, 17, 20]. Funding This work was partially supported by Pronex FAPEG/CNPq. Acknowledgments The second author is fellow of CNPq. The authors are very grateful to the anonymous referee for useful remarks that improved the final version. References 1 M. P. do Carmo, Differential Geometry of Curves and Surfaces , Prentice-Hall, Inc, Englewood Cliffs, NJ, 1976. 2 A. Cayley, On the singularities of surfaces, Camb. Dublin Math. J. 7 ( 1852), 166– 171. Collected Math. Papers II, 28–32. 3 M. Craizer and R. Garcia, Quadratic points of surfaces in projective 3-space, preprint 2017. 4 G. Darboux, Leçons sur la Théorie des Surfaces Vols. I, II, III, IV, Gauthier Villars, Paris, 1896. 5 D. Dreibelbis, Bitangencies on surfaces in four dimension, Quart. J. Math. 52 ( 2001), 137– 160. Google Scholar CrossRef Search ADS 6 W. L. 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Struik, Lectures on Classical Differential Geometry, Addison Wesley Pub. Co., Reprinted by Dover Publications, Inc., New York, 1988. 20 R. Uribe-Vargas, A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve, Mosc. Math. J. 6 ( 2006), 731– 768. The terms J12,J13,J22 and J23 described in Proposition 4 are given by J12=(ψuuu+4φ+6huv−4ξv−12ψu+3φuu)hu−(4ξ+huuu+3ξuu)ψu+(3(huuv−ψuu)+4(hv−ψ))ξu−(ξv−φ)huuu+32[2ψuu−huuv+2φu]huu+4hφu−ψuuuξ+3ξuuhuv+(−ψ+hv)ξuuu+hφuuu,J13=12(ξuvhuvhuu−2ξuvhuvξu)+6(huvψuuhuu−hvξuuφu)+12ξξuuψuv+6huuξ−9huuψuv+3huuhuvv−3huuφv+32hξ+21hξuu+4hhuuu−16hφv−212hφuuv+16ξψv+12ψuξuv+12ψuψuu+36ψuφu−24ψuψ+36ψuhv+12ψhuv+6ψξuuv+32ψφ+12ψφuu−12φhuuv−32φhv−12hvhuv−6hvξuuv−18hvφuu−6ξuuhvv+6ξuuψv−12(ψu)2−(3ξuv+3ψuu−32huuv)hvξuu−3ξuu(ψu)2+3(2ψuv−huvv)(ξu)2+[12(huu)2+12huuhvv−48huuψv−9ξuhuu−24huuh+6huuψuuv+12huuφuv−12hvψuu−24hvξuv+12hvhuuv−24φuhv+24ψuhuv−48φhuv−6ξuhvv−24(ψu)2−24ψuξv+96ψuφ−12(ψuu)2+24ψψuu+6ψuuhuuv−36φuψuu+18ξuψv−4ξuuuψv+16hψv−12ξuuψuv+24ξψuv+6(ξu)2+12ξuh−3ξuψuuv−6ξuφuv+24ξξuu−12ξuuφv−4hξuuu+4ψuuuξv+32φξv+12ξvφuu+24ψξuv+48h2−24hφuv+16ψ2−12ψhuuv+48ψφu−4ψφuuu+16(ξ2+ξφv−φ2)+12φuhuuv−8ψuuuφ−24(φφuu+(φu)2)]hu−36huvφu+6ξψuuv+34[−4ξ+4ψuv−huvv+2φv](huu)2−3(−4ξ+ξuu+2ψuv+2φv)(hu)2−24ξuhuvψuu−12φuuξhuv−4ξφuuuhv+12huvhuuvξu−6huvhuuvhuu−6ξψuuhuuv−12ξhuuvφu+4huuuψvξu+4ξhuuuφv+8ξφψuuu+4ξξuuuψv−2(ξuuuhv+3ξuuhuv−huuuξv)φ+12[6ξuuψuu+6ξuuξuv−3ξuuhuuv+12φuξuu+4ξuuuφ+2φuhuuu]ψ+[ξuuuhv+3ξuuhuv+6φξuuψξuuu−huuuξv+2φhuuu]ψu+[−4ψuhv−4ψuuuhv−12huvψuu−24huvφu+6ψuψuu+8ψuψ−3ψuhuuv+12ψuφu+ψuφuuu−12ψuuφuu+4huuuψv−12ξuuφuv+2ξξuuu−4ξuuuφv+16φuξv+4φuuuξv+8ψuuuψ+6huuvφuu−4φuψuuu−32φuφ−8φφuuu−24φuφuu]h−24φuφ+12φuξv−6(hu)3+6ψuφuu−16ξhvφu+24ξψuuφu−4ψuuuξξv+6huuξvψuu+3h2huuu−φ2huuu+ξ2huuu+12ξ(ψuu)2+ψ2huuu+6ψuhuv+8ψuφ−4ψuξv+32ψφu+8ψφuuu+4ψuuuψ−4ψuuuhv, J22=(6(huv−2ξv)+ξvvv+4(φ−ψu)+3φvv)hv+(3(huvv−ξvv)+4(hu−ξ))ψv−(3ψvv+hvvv+4ψ)ξv−(−φ+ψu)hvvv+32(−huvv+2φv+2ξvv)hvv+4hφv+(−ξ+hu)ψvvv+3ψvvhuv−ψξvvv+hφvvv,J23=34(4ξuv−huuv+2φu−4ψ)(hvv)2+[ψvvvhu+3ψvvhuv−hvvvψu+6φψvv−ξψvvv+2φhvvv]ξv−24ψuvhuvψv+12ψuvhuvhvv−2(ψvvvhu+3ψvvhuv−hvvvψu)φ−3ξvhuvv+ξvφvvv+(12ψvvψuv+12ψvvξvv−6ψvvhuvv+6φvψvv+φvhvvv+8ψvvvφ)ξ+6huvξvvhvv−6φvψvvhu−8hφφvvv−24hφvφvv−12hξvvφvv−12hξvvhuv−24hhuvφv+6ψuhvvξvv+4hhvvvξu+6hhuvvφvv+12huvψvhuvv−3ψvv(ξvv+ψuv−12huvv)hu+6ξuvvψ+2ψvvvψ+12ξξvvv−4ξvvvhu−6huvhuvvhvv−12hψvvφuv−32hφφv+16hψuφv−4hξvvvφv+4hφvvvψu−24ψvξvvhvu−4hψvvvφu+4hvvvψvξu−4hξvvvhu−9hvvξuv+3hvvhuuv−3hvvφu+6hvvψ+21hψvv+4hhvvv−212hφuvv+32hψ−16ξvξ+18ξvξvv+12ξvψuv+48ξvφv+12ξhuv+6ξψuvv+32ξφ+12ξφvv−12φhuvv−6ψvvhuu+6ψvvξu−6huψuvv−18huφvv+[6(ψv)2+14[−24huu−36hvv−12ξuvv+48h−24φuv+72ξu]ψv+3(hvv)2+32(2huu+ξuvv−4h+2φuv−8ξu)hvv+12h2+(−ψvvv−6φuv+4ξu)h+4ψ2+2(3ξuv+3ψvv+2φu)ψ−6(ξv)2+6(huv+4φ−ψu)ξv+4ξ2+(6ξvv+6ψuv−3huvv+12φv−φvvv)ξ−6(φv)2−3(3ξvv−huvv+2hu)φv−4φ2−2(6huv+ξvvv+3φvv−4ψu)φ−3(ξuv+φu)ψvv−3(ξvv+2ψuv−huvv)hu−3(ξvv)2+32ξvvhuvv+(ξvvv+3φvv)ψu−ψvvvξu]hv−36huvφv−6(hv)3+3(2ξuv−huuv)(ψv)2−24φvφ+12φvψu+[−16φvhu−4φvvvhu+6ξvhuv−12huvφvv−4ξvψu−4ξvvvψu+12ξuvψvv+4ψvvvξu−12(ξv)2+8ξvφ+6ξvφvv+12(ξvv)2−6ξvvhuvv+24φvξvv+8ξvvvφ+32ξφv+8ξφvvv−12φvhuvv+4hvvvφu]ψ−3(2ξuv+ψvv+2φu−4ψ)(hv)2−16hφu−32φhu−12huhuv+32huξv+16ψξu−3(ξv)2ψvv+ψ2hvvv+ξ2hvvv+3h2hvvv−φ2hvvv. Let a0=3/2,b0=−2,c0=−1,d1=10,d2=30,q0=1/120,q1=1/100. The functions pij and qij, i,j=0,1,2 with these values are given by p10=−1.8702c2+3.2517b2−3.2517b4−2.2865c3−0.6040b3+1.8702c4+2.2865c1+0.6040b1,p01=1.8702c3+1.8702c1+0.6040b2+0.6040b4−3.2517b3+2.2865c2+2.2865c4−3.2517b1,q10=−10.9220c2+27.1478b2+27.1478b4+18.9498c3−5.2553b3−10.9220c4+18.9498c1−5.2553b1,q01=−10.9220c3+10.9220c1−5.2553b2+5.2553b4+27.1478b3+18.9498c2−18.9498c4−27.1478b1,p20=24.1584+0.5742b12−2.2056b2c1−1.6457b4c3+0.1662b2c3−0.9267c2c4−1.6457b3c4−2.2056b1c2−0.9267c1c3−0.1662b1c4−0.2102b3c2+0.595b1b3+0.595b2b4+0.2102b4c1+0.5742b32−1.0909c32+1.0909c42−0.5742b42+1.5893c22−0.5742b22−1.5893c12,p02=72.4752+1.1484b2b1+1.6457b4c4−1.1484b3b4−0.1662b2c4−2.2056b2c2+0.0595b2b3−1.6457b3c3+2.1819c4c3−3.1786c1c2+0.2102c1b3−0.9267c2c3+0.9267c1c4+2.2056b1c1−0.1662b1c3+0.2102b4c2−0.595b1b4,q20=−210.2128−5.7573b12+34.0613b2c1+29.0896b4c3−13.3237b2c3−63.1459c2c4+29.0896b3c4+34.0613b1c2−63.1459c1c3+13.3237b1c4+13.0400b3c2−1.0040b1b3−1.0040b2b4−13.0400b4c1−5.7573b32+49.8318c32−49.8318c42+5.7573b42−55.7680c22+5.7573b22+55.7680c12,q02=−630.6383−11.5146b2b1−29.0896b4c4+11.5146b3b4+13.3237b2c4+34.0613b2c2−1.0040b2b3+29.0896b3c3−99.6636c4c3+111.5360c1c2−13.0400c1b3−63.1459c2c3+63.1459c1c4−34.0613b1c1+13.3237b1c3−13.0400b4c2+1.0040b1b4. (B.1)Also, δ0=0.1993(a2b1−a1b2−a1b4−a2b3)+0.0621(a1c1+a1c3+a2c2−a2c4)+0.2824(b2b3+b1b4)+0.0316(b1c1+b2c2)+0.2140(b1c3−b2c4)+0.1196(b4c2−c1b3)−0.3020(b4c4+b3c3)+0.0568(c2c3+c1c4),δ1=7.4340(b1c1b1c3+b2c2+b2c4+c1b3−b3c3−b4c2−b4c4).A numeric solution of the system of equations p10=p01=p20=p02=q10=q01=q20=q02=0 given by Eq. (B.1) is given by {b1=4.4373,b2=0.8442,b3=2.0814,b4=−4.2707,c1=2.9243,c2=4.7664,c3=5.6263,c4=−1.5845}. Therefore, A0=(1.5,8.5951,−5.0918,−2.2717,−9.8657),B0=(−2,4.4373,0.8442,2.0814,−4.2707),C0=(−1,2.9243,4.7664,5.6263,−1.5845),D0=(10,30),Q0=(0.0083,0.01). © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** Jan 3, 2018

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