Geometry of the contactomorphism group

TL;DR

This paper explores the Riemannian geometry of the contactomorphism group on compact contact manifolds, focusing on curvature, geodesics, conjugate points, and the exponential map.

Contribution

It provides explicit curvature calculations, solutions to the Jacobi equation, and analyzes the properties of the exponential map in the context of contactomorphism groups.

Findings

Sectional curvature is non-negative in certain sections.

Explicit solutions to the Jacobi equation are obtained.

The exponential map is a non-linear Fredholm map of index zero.

Abstract

In this paper we examine the Riemannian geometry of the group of contactomorphisms of a compact contact manifold. We compute the sectional curvature of $\mathcal{D}_\theta(M)$ in the sections containing the Reeb field and show that it is non-negative. We also solve explicitly the Jacobi equation along the geodesic corresponding to the flow of the Reeb field and determine the conjugate points. Finally, we show that the Riemannian exponential map is a non-linear Fredholm map of index zero.