On equivariant isometric embeddings of Riemannian manifolds with symmetries

TL;DR

This paper proves the existence of smooth, equivariant isometric embeddings of certain Riemannian manifolds with symmetries into Euclidean space, extending previous results to manifolds with group actions.

Contribution

It establishes the existence of equivariant isometric embeddings for manifolds with symmetries, generalizing G"unther's results to manifolds with group actions.

Findings

Existence of smooth equivariant isometric embeddings proven.

Embedding dimension matches G"unther's bounds.

Applicable to manifolds diffeomorphic to ^n with group actions.

Abstract

Let $(M,g)$ be a $C^\infty$-smooth, $n$-dimensional Riemannian manifold which is diffeomorphic to $\RR^n$ and admit an action of a properly discontinuous and cocompact group. This work proves the existence of a $C^\infty$ equivariant isometric embedding of $M$ in some Euclidean space $\RR^q$ where $q = \max\{s_n+2n, s_n+n+5\}$ is the same as the dimension of Matthias G\"unther's results.