Geometric connections and geometric Dirac operators on contact manifolds

We construct several natural connections and Dirac type operators on a general metric contact manifold which are more sensitive to the geometric background. In the special case of CR manifolds these connections are also compatible with the CR structure and include among them the Webster connection. We also describe several Weitzenbock type formulae. Our method is based on work of P. Gauduchon applied to two almost complex manifolds naturally associated to a given metric contact manifold.