Normal forms and classification problems in affine differential geometry
Normal forms are useful in differential geometry. The
Moser normal form of a CR hypersurface is a good example. Affine differential
geometry in its simplest guise is the study of those geometric properties
of submanifolds in Rn invariant under affine transformations.
For hypersurfaces, such a study may be undertaken by choosing affine coördinates
so that the hypersurface is defined by a function whose Taylor series is
in a preferred normal form. A hypersurface which is the orbit of a Lie
subgroup of the group of affine motions is said to be affine homogeneous.
The Archimedean screw in R3 is a good example as are
ellipsoids and the parabolic dish. This talk will discuss normal forms
and how they may be used to define invariants and classify homogeneous
surfaces in R3 and more besides. It is joint work with
Vladimir Ezhov.
Spring 1999 meeting of the PNGS