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The Symplectic-to-Contact Dictionary

Fabrizio Pugliese, Giovanni Sparano, Luca Vitagliano

TL;DR

This work formalizes a precise bridge between Symplectic and Contact Geometry via two interrelated dictionaries. It shows that a Contact structure corresponds to a symplectic Atiyah form on the normal line bundle, with the Jacobi bracket on sections encoded by a Jacobi tensor, and that the same data can be packaged in a homogeneous, line-bundle framework via symplectization. The approach extends to Jacobi structures, Poissonization, and odd geometries, including complex and G-structures, revealing multiple consistent translation schemes and suggesting broader applicability beyond traditional symplectic contexts. The resulting framework unifies several odd-dimensional analogues and provides a systematic route to derive new structures such as cosymplectic, almost complex Atiyah, and various homogeneous G-structures from known symplectic data.

Abstract

Contact Geometry is an odd dimensional analogue of Symplectic Geometry. This vague idea can actually be formalized in a rather precise way by means of a Symplectic-to-Contact Dictionary. The aim of this review paper is discussing the basic entries in this dictionary. Surprisingly, the dictionary can also be applied to apparently far away situations like complex and $G$-structures, to get old and new interesting geometries.

The Symplectic-to-Contact Dictionary

TL;DR

This work formalizes a precise bridge between Symplectic and Contact Geometry via two interrelated dictionaries. It shows that a Contact structure corresponds to a symplectic Atiyah form on the normal line bundle, with the Jacobi bracket on sections encoded by a Jacobi tensor, and that the same data can be packaged in a homogeneous, line-bundle framework via symplectization. The approach extends to Jacobi structures, Poissonization, and odd geometries, including complex and G-structures, revealing multiple consistent translation schemes and suggesting broader applicability beyond traditional symplectic contexts. The resulting framework unifies several odd-dimensional analogues and provides a systematic route to derive new structures such as cosymplectic, almost complex Atiyah, and various homogeneous G-structures from known symplectic data.

Abstract

Contact Geometry is an odd dimensional analogue of Symplectic Geometry. This vague idea can actually be formalized in a rather precise way by means of a Symplectic-to-Contact Dictionary. The aim of this review paper is discussing the basic entries in this dictionary. Surprisingly, the dictionary can also be applied to apparently far away situations like complex and -structures, to get old and new interesting geometries.
Paper Structure (8 sections, 10 theorems, 65 equations)

This paper contains 8 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Contact Structures
  3. Jacobi Brackets and the Gauge Algebroid
  4. Atiyah Forms and the Dictionary
  5. Homogeneous Symplectic Structures
  6. Jacobi bundles
  7. Odd Complex Structures
  8. Homogeneous $G$-Structures

Key Result

Theorem 2.4

Every contact manifold of dimension $2n+1$ is locally contactomorphic to $(\mathbb{R}^{2n+1}, H_{can})$.

Theorems & Definitions (38)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4: Darboux Lemma
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • Definition 4.1
  • Theorem 4.2
  • ...and 28 more