On geometry of sub-Riemannian η-Einstein manifolds

Abstract

On a sub-Riemannian manifold of contact type a connection with torsion is considered, called in the work a Ψ-connection. A Ψ-connection is a particular case of an N-connection. On a sub-Riemannian manifold, a Ψ-connection is defined up to an endomorphism of a distribu­tion D, this endomorphism is called in the work the structure endomor­phism. The endomorphism ψ is uniquely defined by the following rela­tions: If the distribution of a sub-Riemannian manifold is integrable, then the Ψ-connection is of the class of the quarter-symmetric connections. It is proved that the Ψ-connection is a metric connection if and only if the structure vector field of the sub-Riemannian structure is integrable. A formula expressing the Ψ-connections in terms of the Levi-Civita connection of the sub-Riemannian manifold is obtained. The components of the curvature ten­sors and the Ricci-tensors of the Ψ-connection and of the Levi-Civita connection are computed. It is proved that if a sub-Riemannian manifold is an η-Einstein manifold, then it is also an η-Einstein manifold with re­spect to the Ψ-connection. The converse holds true only under the condi­tion that the trace of the structure endomorphism Ψ is a constant not de­pending on a point of the manifold. The paper is completed by the theo­rem stating that a Sasaki manifold is an η-Einstein manifold if and only if M is an η-Einstein manifold with respect to the Ψ-connection.

Keywords

• sub-riemannian manifold
• interior connection
• ψ-connec­tion
• η-einstein manifold