Umbilic points and Real hyperquadrics

TL;DR

This paper refines the existence and uniqueness of Chern-Moser normal form for real hypersurfaces, linking umbilic points to hyperquadrics and exploring their symmetry properties.

Contribution

It provides a refined theorem for normal forms and characterizes umbilic points via this form, connecting local biholomorphic equivalence to hyperquadrics.

Findings

Refined existence and uniqueness theorem for Chern-Moser normal form.

Characterization of umbilic points through normal form.

Nondegenerate hypersurfaces are locally biholomorphic to hyperquadrics if all points are umbilic.

Abstract

We show a refined version of the existence and uniqueness theorem to Chern-Moser normal form. The class of nondegenerate real hypersurfaces in normal form has a natural group action. Umbilic point is defined via normal form. Nondegenerate analytic real hypersurfaces are locally biholomorphic to a real hyperquadric whenever every point is umbilic in this sense.