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On equivariant isometric embeddings of Riemannian manifolds with symmetries

Hongda Qiu

TL;DR

This work addresses the problem of equivariant isometric embeddings for Riemannian manifolds with symmetry under a Bieberbach group action. It adapts Günther's embedding machine to preserve group actions and proves the existence of a smooth equivariant isometric embedding into $\mathbb{R}^q$ with $q = \max\{ s_n+2n, s_n+n+5\}$ and $s_n = \tfrac{1}{2}n(n+1)$. The construction starts with a free equivariant initial embedding on $M$ and proceeds by decomposing the metric defect into a finite sum of $\Gamma$-periodic tensors of Property (E), followed by localized perturbations that converge to an isometry while maintaining equivariance. The result shows that symmetry preservation does not incur an extra dimension cost relative to Günther's bound and highlights open questions for non co-compact actions and potential generic dimension reductions.

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 Günther's results.

On equivariant isometric embeddings of Riemannian manifolds with symmetries

TL;DR

This work addresses the problem of equivariant isometric embeddings for Riemannian manifolds with symmetry under a Bieberbach group action. It adapts Günther's embedding machine to preserve group actions and proves the existence of a smooth equivariant isometric embedding into with and . The construction starts with a free equivariant initial embedding on and proceeds by decomposing the metric defect into a finite sum of -periodic tensors of Property (E), followed by localized perturbations that converge to an isometry while maintaining equivariance. The result shows that symmetry preservation does not incur an extra dimension cost relative to Günther's bound and highlights open questions for non co-compact actions and potential generic dimension reductions.

Abstract

Let be a -smooth, -dimensional Riemannian manifold which is diffeomorphic to and admit an action of a properly discontinuous and cocompact group. This work proves the existence of a equivariant isometric embedding of in some Euclidean space where is the same as the dimension of Matthias Günther's results.

Paper Structure

This paper contains 5 sections, 8 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Initial embedding
  4. Proof of Theorem \ref{['theorem main']}
  5. Günther's Proof of Proposition \ref{['proposition gunther']}

Key Result

Theorem 1

Let $(M,g)$ be a Riemannian manifold which is diffeomorphic to $\mathbb{R}^n$ and admits an action of a Bieberbach group $\Gamma$. Let $u_0:M\to\mathbb{R}^q, q\geq s_n+n+5$ be a smooth free equivariant embedding such that $g-du_0\cdot du_0$ is positive-definite. Then there exists a smooth free equiv

Theorems & Definitions (16)

  • Definition 1.1: Free map
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Remark 2.1
  • Proof
  • Claim 1
  • Definition 2.1: "Property (E)", Günther
  • Proposition 1: Günther
  • Lemma 2
  • ...and 6 more