The CR immersion into a sphere with the degenerate CR Gauss map

TL;DR

This paper explores an analogue of the classical algebraic geometry problem in CR geometry, showing that under certain conditions, a CR map between spheres is totally geodesic if its CR Gauss map is degenerate.

Contribution

It establishes a new characterization of totally geodesic CR maps between spheres via the degeneracy of the CR Gauss map, extending classical algebraic geometry concepts.

Findings

CR map between spheres is totally geodesic iff the CR Gauss map is degenerate

Develops an analogue theory of Gauss maps in CR geometry

Provides conditions under which the CR Gauss map degeneracy characterizes geometric properties

Abstract

It is a classical problem in algebraic geometry to characterize the algebraic subvariety by using the Gauss map. In this note, we try to develop the analogue theory in CR geometry. In particular, under some assumptions, we show that a CR map between spheres is totally geodesic if and only if the CR Gauss map of the image is degenerate.