The Symplectic-to-Contact Dictionary

TL;DR

This paper reviews the Symplectic-to-Contact Dictionary, a formal framework connecting Symplectic and Contact Geometry, and explores its applications to various geometric structures, revealing new insights and relationships.

Contribution

It provides a comprehensive overview of the Symplectic-to-Contact Dictionary and demonstrates its utility in relating different geometric structures and uncovering new geometrical insights.

Findings

The dictionary formalizes the relationship between Symplectic and Contact Geometry.

Applications extend to complex and G-structures, linking diverse geometries.

The framework offers new perspectives on classical and modern geometric theories.

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.