Some remarks on the generalized Tanaka-Webster connection of a contact metric manifold

TL;DR

This paper investigates conditions under which the bi-Legendrian connection on a contact metric manifold aligns with the generalized Tanaka-Webster connection, providing new insights into their relationships and interpretations within Sasakian geometry.

Contribution

It establishes necessary and sufficient conditions for the bi-Legendrian connection to be metric and to coincide with the Tanaka-Webster connection, enhancing understanding of their interplay.

Findings

Conditions for bi-Legendrian connection to be metric

Criteria for coincidence with Tanaka-Webster connection

Interactions between Tanaka-Webster, bi-Legendrian, and Levi-Civita connections in Sasakian manifolds

Abstract

We find necessary and sufficient conditions for the bi-Legendrian connection $\nabla$ associated to a bi-Legendrian structure $(\cal F,\cal G)$ on a contact metric manifold $(M,\phi,\xi,\eta,g)$ being a metric connection and then we give conditions ensuring that $\nabla$ coincides with the (generalized) Tanaka-Webster connection of $(M,\phi,\xi,\eta,g)$. Using these results, we give some interpretations of the Tanaka-Webster connection and we study the interactions between the Tanaka-Webster, the bi-Legendrian and the Levi Civita connection in a Sasakian manifold.