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Regularity of the geodesic flow on submanifolds

Christian Lange

TL;DR

This work addresses the regularity of the geodesic flow and exponential map on submanifolds with finite smoothness. It combines a high-regularity analysis via Jacobi fields and the Gauss equation with a low-regularity approximation scheme to establish that, for a $C^k$ submanifold with $k\ge 2$, the geodesic flow and the exponential map achieve $C^{k-1}$ regularity, with a detailed treatment distinguishing $k\ge 3$ from $k=2$. The approach leverages ODE theory, curvature arguments, and, in the low-regularity regime, smooth approximations and compactness to obtain a robust flow construction and uniqueness. The results clarify the sharp regularity threshold and have implications for Taylor expansions and geometric analysis in spaces with limited smoothness, consistent with Nash embedding and classical examples.

Abstract

We show that the geodesic flow and the exponential map of a $C^k$ submanifold of $\mathbb{R}^n$ with $k\geq 2$ are of class $C^{k-1}$.

Regularity of the geodesic flow on submanifolds

TL;DR

This work addresses the regularity of the geodesic flow and exponential map on submanifolds with finite smoothness. It combines a high-regularity analysis via Jacobi fields and the Gauss equation with a low-regularity approximation scheme to establish that, for a submanifold with , the geodesic flow and the exponential map achieve regularity, with a detailed treatment distinguishing from . The approach leverages ODE theory, curvature arguments, and, in the low-regularity regime, smooth approximations and compactness to obtain a robust flow construction and uniqueness. The results clarify the sharp regularity threshold and have implications for Taylor expansions and geometric analysis in spaces with limited smoothness, consistent with Nash embedding and classical examples.

Abstract

We show that the geodesic flow and the exponential map of a submanifold of with are of class .
Paper Structure (7 sections, 5 theorems, 22 equations)

This paper contains 7 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and proof in high regularity
  3. Geodesic flow and exponential map
  4. Differentiability
  5. Jacobi fields and Gauß equation
  6. Interchanging partial derivatives
  7. The low regularity case

Key Result

Theorem 1.1

Let $M$ be a $\mathcal{C}^{k}$ submanifold of $\mathbb{R}^n$ for some $k\geq 2$. Then for any $v \in TM$ there exists a unique maximal locally length minimizing solution $\gamma_v$ of the geodesic equation with $\gamma'(0)=v$. Let $\phi: V \rightarrow TM$, $V \subset \mathbb{R} \times TM$, $\phi(t,v

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 2.2: Peano
  • Lemma 3.1
  • proof
  • Lemma 3.2: Osgood's lemma
  • Lemma 3.3
  • proof
  • Remark 3.4