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A remark on Euler-like vector fields

Haoyuan Gao

TL;DR

The paper proves that every germ of a tubular neighborhood embedding along an embedded submanifold $N$ can be realized as the normal exponential image of a neighborhood in the Riemannian normal bundle, i.e., via a suitable metric near $N$. It establishes this by first handling the point-case and then extending to general $N$ through an isometry with an ambient metric and the canonical normal-bundle isomorphism, yielding a commutative diagram with the normal exponential map. This result strengthens the link between Euler-like vector fields (and their germ bijection with tubular embeddings) and geometric realizations through Riemannian geometry, enabling a metric-based realization of deformation-to-normal-cone-type structures. The appendix provides a vector-bundle extension tool to facilitate germ-level constructions.

Abstract

In this note, we show that (the germ of) each Euler-like vector field comes from a tubular neighborhood embedding given by the normal exponential map of some Riemannian metric.

A remark on Euler-like vector fields

TL;DR

The paper proves that every germ of a tubular neighborhood embedding along an embedded submanifold can be realized as the normal exponential image of a neighborhood in the Riemannian normal bundle, i.e., via a suitable metric near . It establishes this by first handling the point-case and then extending to general through an isometry with an ambient metric and the canonical normal-bundle isomorphism, yielding a commutative diagram with the normal exponential map. This result strengthens the link between Euler-like vector fields (and their germ bijection with tubular embeddings) and geometric realizations through Riemannian geometry, enabling a metric-based realization of deformation-to-normal-cone-type structures. The appendix provides a vector-bundle extension tool to facilitate germ-level constructions.

Abstract

In this note, we show that (the germ of) each Euler-like vector field comes from a tubular neighborhood embedding given by the normal exponential map of some Riemannian metric.

Paper Structure

This paper contains 3 sections, 8 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Normal exponential map
  3. Proof of the main theorem

Key Result

Theorem 1.4

BLM The correspondence that associates to a tubular neighborhood embedding its associated (pushforward) Euler-like vector field determines a bijection from germs of tubular neighborhood embeddings to germs of Euler-like vector fields. Here a germ means a germ near $N$.

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 3.1
  • ...and 10 more