Unit Tangent Bundles, CR Leaf Spaces, and Hypercomplex Structures

TL;DR

This paper explores the geometric structures of unit tangent bundles of semi-Riemannian manifolds, showing they serve as examples of dynamical Legendrian contact structures and relate to hypercomplex and CR geometries.

Contribution

It introduces new connections between unit tangent bundles, L-contact structures, and 2-nondegenerate CR structures, expanding the classification and understanding of these geometric entities.

Findings

Unit tangent bundles are examples of dynamical Legendrian contact structures.

L-contact structures relate to hypercomplex structures and homogeneous models.

The Ricci curvature defines Ricci-shifted structures with vanishing Nijenhuis tensor in conformally flat cases.

Abstract

Unit tangent bundles $UM$ of semi-Riemannian manifolds $M$ are shown to be examples of dynamical Legendrian contact structures, which were defined in recent work [25] of Sykes-Zelenko to generalize leaf spaces of 2-nondegenerate CR manifolds. In doing so, Sykes-Zelenko extended the classification in Porter-Zelenko [20] of regular, 2-nondegenerate CR structures to those that can be recovered from their leaf space. The present paper treats dynamical Legendrian contact structures associated with 2-nondegenerate CR structures which were called "strongly regular" in Porter-Zelenko, named "L-contact structures." Closely related to Lie-contact structures, L-contact manifolds have homogeneous models given by isotropic Grassmannians of complex 2-planes whose algebra of infinitesimal symmetries is one of $\mathfrak{so}(p+2,q+2)$ or $\mathfrak{so}^*(2p+4)$ for $p\geq1$, $q\geq0$. Each 2-plane in the homogeneous model is a split-quaternionic or quaternionic line, respectively, and more general L-contact structures arise on contact manifolds with hypercomplex structures, unit tangent bundles being a prime example. The Ricci curvature tensor of $M$ is used to define the "Ricci-shifted" L-contact structure on $UM$, whose Nijenhuis tensor vanishes when $M$ is conformally flat. In the language of Sykes-Zelenko (for $M$ analytic), such $UM$ is the leaf space of a 2-nondegenerate CR manifold which is "recoverable" from $UM$, providing a new source of examples of 2-nondegenerate CR structures.