Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 48, No. 2, pp. 469491 (2007) 

Indefinite affine hyperspheres admitting a pointwise symmetry; part 1Christine ScharlachTechnische Universität Berlin, Fakultät II, Institut für Mathematik, MA 83, 10623 Berlin, Germany, email: schar@math.tuberlin.deAbstract: An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of $\Aut(T_p M)$ for all $p\in M$, which preserves (pointwise) the affine metric $h$, the difference tensor $K$ and the affine shape operator $S$. Here, we consider 3dimensional indefinite affine hyperspheres, i. e. $S= H\Id$ (and thus $S$ is trivially preserved). First we solve an algebraic problem. We determine the nontrivial stabilizers $G$ of a traceless cubic form on a LorentzMinkowski space $\Min$ under the action of the isometry group $\textit{SO}(1,2)$ and find a representative of each $\textit{SO}(1,2)/G$orbit. Since the affine cubic form is defined by $h$ and $K$, this gives us the possible symmetry groups $G$ and for each $G$ a canonical form of $K$. In this first part, we show that hyperspheres admitting a pointwise $\Z_2\times \Z_2$ resp. $\R$symmetry are wellknown, they have constant sectional curvature and Pick invariant $J<0$ resp. $J=0$. The classification of affine hyperspheres admitting a pointwise $G$symmetry will be continued elsewhere. Keywords: 3dimensional affine hyperspheres, indefinite affine metric, pointwise symmetry, $\textit{SO}(1,2)$action, stabilizers of a cubic form, affine differential geometry, affine spheres Classification (MSC2000): 53A15; 15A21, 53B30 Full text of the article:
Electronic version published on: 7 Sep 2007. This page was last modified: 28 Jun 2010.
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