Flatness of CR Submanifolds in a Sphere

TL;DR

This paper proves that a CR submanifold in a sphere with vanishing CR second fundamental form must be totally geodesic, highlighting a geometric rigidity property of such embeddings.

Contribution

It establishes a rigidity result for CR submanifolds in spheres, showing that vanishing CR second fundamental form implies total geodesicity.

Findings

CR submanifold with zero CR second fundamental form is totally geodesic

Vanishing CR second fundamental form implies rigidity of the embedding

Provides geometric characterization of CR submanifolds in spheres

Abstract

Let $M$ be the image of a smooth CR embedding of a strictly pseudoconvex CR real hypersurface into a sphere. If the CR second fundamental form of $M$ vanishes, we show that $M$ is a totally geodesic submanifold.