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This article is cited in 1 scientific paper (total in 1 paper)
Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2
Christine Scharlach Technische Universität Berlin, Fak. II, Inst. f. Mathematik, MA 8-3, 10623 Berlin, Germany
Abstract:
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 $J<0$ 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.
Keywords:
affine hyperspheres; indefinite affine metric; pointwise symmetry; affine differential geometry; affine spheres; warped products.
Received: May 8, 2009; in final form October 6, 2009; Published online October 19, 2009
Citation:
Christine Scharlach, “Indefinite Affine Hyperspheres Admitting a Pointwise Symmetry. Part 2”, SIGMA, 5 (2009), 097, 22 pp.
Linking options:
https://www.mathnet.ru/eng/sigma443 https://www.mathnet.ru/eng/sigma/v5/p97
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Abstract page: | 211 | Full-text PDF : | 32 | References: | 30 |
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