# Seminar Archives

## : Affine Cross-Sectional Curvature of Hypersurfaces from the Point of View of Singularity Theory

8 May 2008 13:15 in

We consider the affine differential geometry of smoothly embedded hypersurfaces using tools from singularity theory. Affine differential geometry studies the differential invariants of manifolds under the action of the affine special linear group SL(n+1,R) x R^{n+1}, i.e. volume preserving transformations and translations.

We seek an affine analogue of the Euclidean cross-section curvature of a surface in three-space. We see that applying an exact analogue fails, and to do it properly we need to use standard ideas from singularity theory. Once we arrive at an idea of affine cross-sectional curvature we generalise it to hypersurfaces in R^{n+1}.

From the singularity theory definition of affine cross-sectional curvature we derive an expression for it using connexions and metrics. We also prove some of the nice properties of affine cross-sectional curvature. We link it to affine principal directions, affine principal curvatures, affine umbilics, and affine mean curvature.