On the Yamabe problem on contact Riemannian manifolds

Abstract

A contact Riemannian manifold, whose complex structure is not necessarily integrable, is the generalization of the notion of a pseudohermitian manifold in CR geometry. The Tanaka–Webster–Tanno connection plays the role of the Tanaka–Webster connection for a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By using special frames and normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, the Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. So the Yamabe problem on a contact Riemannian manifold is always solvable.