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Geodesic vector fields, induced contact structures and tightness in dimension three

Tilman Becker

Abstract

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the 'mixed' sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on $\mathbb{R}^3$ whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact $3$-manifolds with isometric Reeb flows (also known as $R$-contact manifolds) up to diffeomorphism.

Geodesic vector fields, induced contact structures and tightness in dimension three

Abstract

In this paper, we provide new and simpler proofs of two theorems of Gluck and Harrison on contact structures induced by great circle or line fibrations. Furthermore, we prove that a geodesic vector field whose Jacobi tensor is parallel along flow lines (e.g. if the underlying manifold is locally symmetric) induces a contact structure if the 'mixed' sectional curvatures are nonnegative, and if a certain nondegeneracy condition holds. Additionally, we prove that in dimension three, contact structures admitting a Reeb flow which is either periodic, isometric, or free and proper, must be universally tight. In particular, we generalise an earlier result of Geiges and the author, by showing that every contact form on whose Reeb vector field spans a line fibration is necessarily tight. Furthermore, we provide a characterisation of isometric Reeb vector fields. As an application, we recover a result of Kegel and Lange on Seifert fibrations spanned by Reeb vector fields, and we classify closed contact -manifolds with isometric Reeb flows (also known as -contact manifolds) up to diffeomorphism.
Paper Structure (7 sections, 21 theorems, 72 equations)

This paper contains 7 sections, 21 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonnegative Ricci curvature
  4. Adapted Jacobi fields
  5. Space forms
  6. Geodesic fields with parallel Jacobi tensor
  7. Reeb flows and tightness

Key Result

Lemma 2.1

Let $X$ be a unit vector field on a Riemannian manifold $(M,g)$ with dual 1-form $\alpha := i_X g$. Then, the following are equivalent.

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Example 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 3.1
  • Corollary 3.2
  • proof : Proof of Theorem \ref{['thm:inducedcontact']}
  • Definition 4.1
  • ...and 34 more