Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

Details Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2 Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

2009-10-19

arxiv.org/abs/0910.3609v1

SIGMA 5 (2009), 097, 22 pages

Mathematics

Differential Geometry

Scientific paper

10.3842/SIGMA.2009.097

An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of ${\rm Aut}(T_p M)$ for all $p\in M$, which preserves (pointwise) the affine metric $h$, the difference tensor $K$ and the affine shape operator $S$. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. $S= HId$ (and thus $S$ is trivially preserved). In Part 1 we found the possible symmetry groups $G$ and gave for each $G$ a canonical form of $K$. We started a classification by showing that hyperspheres admitting a pointwise ${\mathbb Z}_2\times {\mathbb Z}_2$ resp. ${\mathbb R}$-symmetry are well-known, they have constant sectional curvature and Pick invariant $J<0$ resp. J=0. Here, we continue with affine hyperspheres admitting a pointwise ${\mathbb Z}_3$- or SO(2)-symmetry. They turn out to be warped products of affine spheres (${\mathbb Z}_3$) or quadrics (SO(2)) with a curve.

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