A spin structure on a contact Riemannian manifold carries a spin structure on a circle bundle over the manifold. We have interest in the Dirac operators associated with those structures. In terms of a modified Tanno connection, relations between them are studied and some kinds of their explicit expressions are offered.
"Dirac operators on the Fefferman spin spaces in almost CR-geometry." Osaka J. Math. 56 (3) 507 - 524, July 2019.