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F. Milán (2013)
Singularities of improper affine maps and their Hessian equationJournal of Mathematical Analysis and Applications, 405
Buqing Su (1983)
Affine differential geometry
A. Pogorelov (1972)
On the improper convex affine hyperspheresGeometriae Dedicata, 1
K. Jörgens (1955)
Harmonische Abbildungen und die Differentialgleichungrt−s2=1Mathematische Annalen, 129
F Dillen, Y Lu (2004)
Affine surfaces with symmetric shape operatorSoochow J. Math., 30
K Jörgens (1954)
Über die Lösungen der Differentialgleichung $$rt-s^2=1$$ r t - s 2 = 1Math. Ann., 127
T. Binder (2004)
On relative hypersurfaces with parallel shape operatorResults in Mathematics, 46
K. Jörgens (1954)
Über die Lösungen der Differentialgleichungrt−s2=1Mathematische Annalen, 127
Ying Lu (2011)
Blaschke hypersurfaces with symmetric shape operatorActa Mathematica Sinica, English Series, 27
J. Gálvez, Antonio Martínez, Pablo Mira (2004)
The space of solutions to the hessian one equation in the finitely punctured planeJournal de Mathématiques Pures et Appliquées, 84
F. Dillen, L. Verstraelen, L. Vrancken, I. Woestijne (1993)
Geometry and Topology of Submanifolds, V
E. Calabi (1958)
Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens.Michigan Mathematical Journal, 5
(2016)
Pergamon, Oxford (1969) Ying Lü School of Mathematical Science Xiamen University Xiamen 361005 Fujian People’s Republic of China e-mail: lueying@xmu.edu.cn Received: June 9, 2016
W. Jelonek (1992)
Affine surfaces with parallel shape operatorsAnnales Polonici Mathematici, 56
AZ Petrov (1969)
Einstein Spaces
F. Dillen (1988)
Locally symmetric complex affine hypersurfacesJournal of Geometry, 33
We prove the following: a relative hypersurface with parallel shape operator is either a relative hypersphere, or it is affinely equivalent to an example constructed by Th. Binder. Furthermore, based on Binder’s example, we give another simple and more explicit example; this way we improve the classification and show that it is completely determined by functions $$\kappa (t)$$ κ ( t ) and $$C(v_i;t)$$ C ( v i ; t ) , the latter being solutions of certain Monge–Ampère equations. Our example geometrically is constructed from a plane curve and a family of relative hyperspheres. In case of an affine sphere with Blaschke geometry we show that our classification can be considered as a construction coming from a plane curve together with a family of improper affine hyperspheres. Especially in $$\mathbb R^4$$ R 4 , this construction is determined by only three functions of a single variable.
Results in Mathematics – Springer Journals
Published: Dec 23, 2016
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