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Developments of curves with respect to symmetric tensors and existence of isometric immersions with prescribed second fundamental form

Chengjie Yu

Abstract

In this paper, we introduce the notion of developments of curves with respect to symmetric tensors and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. Our method provides a geometric construction of such an isometric immersion.

Developments of curves with respect to symmetric tensors and existence of isometric immersions with prescribed second fundamental form

Abstract

In this paper, we introduce the notion of developments of curves with respect to symmetric tensors and use it to prove the existence of isometric immersions into a general ambient space with prescribed second fundamental form. Our method provides a geometric construction of such an isometric immersion.

Paper Structure

This paper contains 3 sections, 5 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Developments of curves with respect to symmetric tensors
  3. Existence of isometric immersions with prescribed second fundamental form

Key Result

Theorem 1.1

Let $(M^n,g)$ and $(\widetilde{M}^n,\widetilde{g})$ be two Riemannian manifolds with $M$ simply connected and $\widetilde{M}$ complete. Let $p\in M$, $\widetilde{p}\in \widetilde{M}$ and $\varphi:T_pM\to T_{\widetilde{p}}\widetilde{M}$ be a linear isometry. For any smooth curve $\gamma:[0,1]\to M$ w Then, the map $f(\gamma(1))=\widetilde{\gamma}(1)$ from $M$ to $\widetilde{M}$ is well defined and

Theorems & Definitions (12)

  • Definition 1.1
  • Theorem 1.1: An alternative form of Cartan-Ambrose-Hicks theorem in Yu
  • Definition 1.2: Development of curve w.r.t. a symmetric tensor
  • Theorem 1.2
  • Theorem 1.3: Local version of Theorem \ref{['thm-main']}
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm-Cartan']}
  • ...and 2 more