# Isometric Embeddings into Heisenberg Groups

## TL;DR

This paper investigates conditions under which isometric embeddings of Euclidean or Heisenberg spaces into higher-dimensional Heisenberg groups are necessarily homogeneous homomorphisms, focusing on geodesic properties and distance invariance.

## Contribution

It establishes that if all infinite geodesics in the target are straight lines, then the embedding must be a homogeneous homomorphism, and explores conditions for geodesic linearity.

## Key findings

- Embeddings are homogeneous homomorphisms under geodesic linearity conditions.
- Identifies necessary and sufficient conditions for the geodesic linearity property.
- Provides examples illustrating the geodesic linearity property in Heisenberg groups.

## Abstract

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this `geodesic linearity property', and we provide various examples.