Symmetry in the Geometry of Metric Contact Pairs

TL;DR

This paper characterizes the geometric structure of complete symmetric normal metric contact pair manifolds, showing they are closely related to Calabi-Eckmann manifolds and products of symmetric spaces.

Contribution

It proves that universal coverings of such manifolds are Calabi-Eckmann manifolds and describes conditions under which these manifolds decompose into products of symmetric spaces.

Findings

Universal cover of complete symmetric normal metric contact pair manifolds is Calabi-Eckmann.

Certain conditions imply the manifold is a product of symmetric spaces and fibers over a symplectic pair.

Reflections in integral submanifolds are isometries under specified conditions.

Abstract

We prove that the universal covering of a complete locally symmetric normal metric contact pair manifold is a Calabi-Eckmann manifold. Moreover we show that a complete, simply connected, normal metric contact pair manifold such that the foliation induced by the vertical subbundle is regular and reflections in the integral submanifolds of the vertical subbundle are isometries, then the manifold is the product of globally $\phi$-symmetric spaces and fibers over a locally symmetric space endowed with a symplectic pair.