Math @ Duke

Publications [#361358] of Curtis W. Porter
Papers Published
 Porter, C, Unit Tangent Bundles, CR Leaf Spaces, and Hypercomplex Structures
(February, 2021)
(last updated on 2022/08/06)
Abstract: Unit tangent bundles $UM$ of semiRiemannian manifolds $M$ are shown to be
examples of dynamical Legendrian contact structures, which were defined in
recent work [25] of SykesZelenko to generalize leaf spaces of 2nondegenerate
CR manifolds. In doing so, SykesZelenko extended the classification in
PorterZelenko [20] of regular, 2nondegenerate CR structures to those that can
be recovered from their leaf space. The present paper treats dynamical
Legendrian contact structures associated with 2nondegenerate CR structures
which were called "strongly regular" in PorterZelenko, named "Lcontact
structures." Closely related to Liecontact structures, Lcontact manifolds
have homogeneous models given by isotropic Grassmannians of complex 2planes
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 2plane in the homogeneous
model is a splitquaternionic or quaternionic line, respectively, and more
general Lcontact 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 "Riccishifted" Lcontact structure on
$UM$, whose Nijenhuis tensor vanishes when $M$ is conformally flat. In the
language of SykesZelenko (for $M$ analytic), such $UM$ is the leaf space of a
2nondegenerate CR manifold which is "recoverable" from $UM$, providing a new
source of examples of 2nondegenerate CR structures.


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