Existence of isometric immersions into nilpotent Lie groups

TL;DR

This paper provides a complete characterization of when a simply connected Riemannian manifold can be isometrically immersed into two-step nilpotent Lie groups, including H-type groups, by establishing necessary and sufficient conditions.

Contribution

It introduces the first comprehensive criteria for the existence of isometric immersions into two-step nilpotent Lie groups, expanding understanding in differential geometry.

Findings

Derived necessary and sufficient conditions for such immersions.

Extended results to include H-type groups.

Clarified geometric constraints for isometric embeddings.

Abstract

We establish necessary and sufficient conditions for existence of isometric immersions of a simply connected Riemannian manifold into a two-step nilpotent Lie group. This comprises the case of immersions into $H$-type groups.