Contact Metric Spaces and pseudo-Hermitian Symmetry

Jong Taek Cho

doi:10.3390/math8091583

Abstract

We prove that a contact strongly pseudo-convex CR (Cauchy–Riemann) manifold M2n+1, n≥2, is locally pseudo-Hermitian symmetric and satisfies ∇ξh=μhϕ, μ∈R, if and only if M is either a Sasakian locally ϕ-symmetric space or a non-Sasakian (k,μ)-space. When n=1, we prove a classification theorem of contact strongly pseudo-convex CR manifolds with pseudo-Hermitian symmetry.

Highlights

For a contact manifold ( M2n+1 ; η ), we have the two fundamental structures associated with the contact form η: the Riemannian metric g and the Levi form related with an endomorphism J
In previous works [6,7,8,9,10,11], the generalized Tanaka-Webster connection has been playing an important part when we studied the interplay between the contact Riemannian structure and the contact strongly pseudo-convex almost CR structure
A manifold M equipped with structure tensors (η, ξ, φ, g) satisfying (1) is said to be a contact Riemannian manifold or contact metric manifold and is denoted by M = ( M; η, g)

Summary

Introduction

Making a generalization of this definition, in Reference [5], the authors call a contact metric manifold locally φ-symmetric if it satisfies the same curvature condition (∗) as in the Sasakian case. In previous works [6,7,8,9,10,11], the generalized Tanaka-Webster connection has been playing an important part when we studied the interplay between the contact Riemannian structure and the contact strongly pseudo-convex almost CR structure. We give non-homogeneous examples of locally pseudo-Hermitian symmetric contact strongly pseudo-convex CR manifolds

Contact Riemannian Structures

Strongly Pseudo-Convex almost CR Structures

The pseudo-Hermitian Symmetric Spaces

The Contact Strongly Pseudo-Convex 3-Manifolds