# On $C^2$ umbilical hypersurfaces

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

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Inserted: 22 apr 2021
Last Updated: 29 aug 2021

Year: 2021

Abstract:

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.

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