Calculus of Variations and Geometric Measure Theory

C. Mantegazza

On $C^2$ umbilical hypersurfaces

created by root on 22 Apr 2021
modified on 29 Aug 2021


Short Note

Inserted: 22 apr 2021
Last Updated: 29 aug 2021

Year: 2021


We show by an elementary argument that the second fundamental form of a connected, totally umbilical hypersurface of class $C^2$ is a constant multiple of the metric tensor. It follows that the hypersurface is smooth and it is either a piece of a hyperplane or of a sphere.