Submanifolds with the Harmonic Gauss Map in Lie Groups

TL;DR

This paper establishes criteria for when the Gauss map of submanifolds in Lie groups is harmonic, providing conditions related to the submanifold's geometry and the Lie group's metric structure.

Contribution

It introduces a new criterion for harmonic Gauss maps in Lie groups and characterizes harmonicity for totally geodesic submanifolds and geodesics in specific Lie groups.

Findings

Harmonic Gauss map criterion derived for submanifolds in Lie groups.

Necessary and sufficient conditions for harmonicity in totally geodesic submanifolds.

Geodesic Gauss map harmonicity characterized in 2-step nilpotent groups.

Abstract

In this paper we find a criterion for the Gauss map of an immersed smooth submanifold in some Lie group with left invariant metric to be harmonic. Using the obtained expression we prove some necessary and sufficient conditions for the harmonicity of this map in the case of totally geodesic submanifolds in Lie groups admitting biinvariant metrics. We show that, depending on the structure of the tangent space of a submanifold, the Gauss map can be harmonic in all biinvariant metrics or non-harmonic in some metric. For 2-step nilpotent groups we prove that the Gauss map of a geodesic is harmonic if and only if it is constant.