Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the geometry of Riemannian isometric embeddings

Dmitri Burago, Hongda Qiu

TL;DR

Let $(M,g)$ be an $n$-manifold diffeomorphic to $\mathbb{R}^n$ carrying a Bieberbach group $\Gamma$ acting by isometries. The authors address two embedding problems: embedding $(M,g)$ into a bounded subset of some $\nR^{D_1}$ and obtaining a $\Gamma$-equivariant embedding into $\nR^{D_2}$, with bounds $D_1=N+2n$ and $D_2=N+n$ where $N$ is the Nash dimension of $M/ Gamma$. They also prove that a general $n$-manifold with Nash dimension $N$ can be isometrically embedded into a bounded subset of $\nR^{2N}$. The method splits the metric into $g=g_1+g_2$ with $g_2$ flat, uses the Nash embedding of the quotient for $(M,g_1)$, and a bounded embedding of the flat part, then glues via a diagonal map to produce the desired embeddings and their equivariant versions.

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 $Γ$ 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 $Γ$-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/Γ$. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of $\RR^{2N}$.

On the geometry of Riemannian isometric embeddings

TL;DR

Let be an -manifold diffeomorphic to carrying a Bieberbach group acting by isometries. The authors address two embedding problems: embedding into a bounded subset of some and obtaining a -equivariant embedding into , with bounds and where is the Nash dimension of . They also prove that a general -manifold with Nash dimension can be isometrically embedded into a bounded subset of . The method splits the metric into with flat, uses the Nash embedding of the quotient for , and a bounded embedding of the flat part, then glues via a diagonal map to produce the desired embeddings and their equivariant versions.

Abstract

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

Paper Structure

This paper contains 2 sections, 5 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 1

Let $M$ be a Riemannian manifold that is diffeomorphic to $\mathbb{R}^n$ and admits an action of a Bieberbach group $\Gamma$. Then there exists an isometric embedding ${u}: (M,g)\to\mathbb{R}^{N+2n}$ such that ${u}(M)$ is a bounded subset of $\mathbb{R}^{N+2n}$.

Figures (2)

  • Figure 1: A diagram illustrating a $\Gamma$-equivariant embedding $u:M\to \mathbb{R}^D$. Here we use the same notation $g$ for an element of $\Gamma$ and its corresponding actions.
  • Figure 2: A spiral curve winding in an annulus

Theorems & Definitions (10)

  • Theorem 1: Isometric embedding into a bounded region
  • Theorem 2
  • Theorem 3: Equivariant isometric embedding
  • Proposition 1
  • Proof
  • Proposition 2
  • Proof
  • Proof : Theorem \ref{['theorem general bounded']}
  • Proof : Theorem \ref{['theorem bounded']}
  • Proof : Theorem \ref{['theorem equivariant']}