Adapted connections on metric contact manifolds

Abstract

In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to completely describe the space of adapted connections. We use this description to obtain characterizations of the generalized Tanaka–Webster connection and to describe adapted connections whose Dirac operators are essentially self-adjoint. Finally, Schrödinger–Lichnerowicz-type formulae are provided for these operators.