Abstract
An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T p M) for all p ∈ M, which preserves (pointwise) the affine metric h, the difference tensor K (resp. the cubic form) and the affine shape operator S. In this paper, we deal with locally strongly convex affine hypersurfaces of dimension three. First we solve an algebraic problem. We determine the non-trivial stabilizers G of the pair (K, S) under the action of SO(3) on a Euclidean vector space (V, h) and find a representative (canonical form of K and S) of each SO(3)/G-orbit. Then, we classify hypersurfaces admitting a pointwise G-symmetry for all non-trivial stabilizers G (apart of Z 2). Besides well-known hypersurfaces (for Z 2 × Z 2 we get the locally homogeneous hypersurface (x 1 −) 1/2x 32 (x 2 −) 1/2x 42) = 1) we obtain e.g. warped products of two-dimensional affine spheres (resp. quadrics) and curves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. L. Bryant, Second order families of special Lagrangian 3-folds, Perspectives in Comparison, Generalized and Special Geometry (V. Apostolov, A. Dancer, N. Hitchin, and M. Wang, eds.), CRM Proceedings and Lecture Notes, AMS, 2005 (to appear).
F. Dillen, Equivalence theorems in affine differential geometry, Geometriae Dedicata 32 (1989), 81–92.
F. Dillen and L. Vrancken, Homogeneous affine hypersurfaces with rank one shape operators, Math. Z. 212 (1993), 61–72.
F. Dillen and L. Vrancken, Calabi type composition of affine spheres, Differential Geometry and its applications 4 (1994), 303–328.
S. Hiepko, Eine innere Kennzeichung der verzerrten Produkte, Math. Ann. 241 (1979), 209–215.
A. M. Li, U. Simon and G. S. Zhao, Global affine differential geometry of hypersurfaces, de Gruyter Expositions in Mathematics, vol. 11, Walter De Gruyter & Co., Berlin, 1993.
Y. Lu and F. Dillen, Affine hypersurfaces with symmetric shape operator, Soochow J. Math. 30 (2004), 351–354.
S. Nölker, Isometric immersions of warped products, Differential Geometry and its applications 6 (1996), 1–30.
K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge University Press, 1994.
C. Scharlach, www.math.tu-berlin.de/∼schar/symmSO2.ps resp. symmZ3.ps.
L. Vrancken, Special classes of three dimensional affine hyperpsheres characterized by properties of their cubic form, Contemporary geometry and related topics, World Sci. Publishing, River Edge, NJ, 2004, pp. 431–459.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lu, Y., Scharlach, C. Affine hypersurfaces admitting a pointwise symmetry. Results. Math. 48, 275–300 (2005). https://doi.org/10.1007/BF03323369
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03323369