Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2

Christine Scharlach

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of $\operatorname{Aut}(T_p M)$ 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. $S= H\operatorname{Id}$ (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 $J0$ 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.

Détail

Keywords: affine hyperspheres; indefinite affine metric; pointwise symmetry; affine differential geometry; affine spheres; warped products.

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