On the geometry of Riemannian isometric embeddings

TL;DR

This paper investigates the geometric properties of isometric embeddings of certain Riemannian manifolds, providing new bounds on embedding dimensions and exploring equivariant embeddings related to Bieberbach groups.

Contribution

It introduces novel bounds for isometric embeddings of manifolds with Bieberbach group actions, including bounded embeddings and equivariant embeddings, using a new approach based on a known trick.

Findings

Embedding dimension bounds: D1 = N+2n, D2 = N+n

Bounded embeddings into Euclidean space are possible

Nash dimension relates to embedding complexity

Abstract

This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $\Gamma$ acting by isometries. The first problem concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space $\RR^{D_1}$. The second problem seeks a $\Gamma$-equivariant isometric embdding of $M$ into $\RR^{D_2}$. By using a known trick in a novel way, our idea yields results with $D_1 = N+2n$ and $D_2 = N+n$, where $N$ is the Nash dimension of $ M/\Gamma $. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of $\RR^{2N}$.