Symmetries for Cubic Forms and Adapted Frames
Thursday May 8, 2008
02:00PM - 03:00PM
Speakers:
Christine Scharlach
Abstract:
Symmetries for Cubic Forms and Adapted Frames
Christine Scharlach
Abstract
In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the cubic form C (resp. the difference tensor K). A hypersurface is said to admit a pointwise group symmetry if at every point h, S and K are preserved under the group action. Necessarily the possible groups must be subgroups of the isometry group and the symmetry condition helps to restrict the frame bundle.
The study of submanifolds which admit a pointwise group symmetry was initiated by Bryant; he studied 3-dimensional Lagrangian submanifolds of C3 . The concept was introduced in affine hypersurface theory by Vrancken. He gave a classification of 3dimensional positive definite affine hyperspheres admitting pointwise isometries, which was extended by Lu and myself to hypersurfaces. In all cases the isometry group is SO(3), only in the latter the invariance of the shape operator is an extra condition.
We will report on the classification of 3-dimensional indefinite affine hyperspheres
(i. e. the isometry group is SO(1, 3)). Some of the classes are well known (constant curvature, homogeneous), but also we obtain many interesting classes which are warped products of 2-dimensional affine spheres with curves.
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