Indefinite affine hyperspheres admitting a pointwise symmetry; part 1

Christine Scharlach

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 $p\in M$, 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. $S= H\Id$ (and thus $S$ is trivially preserved). First we solve an algebraic problem. We determine the non-trivial stabilizers $G$ of a traceless cubic form on a Lorentz-Minkowski 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 well-known, 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: 3-dimensional 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

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