Given a spin manifold, the analog of the metric connection is the spin connection ; this is effectively "the same thing" as the normal connection, just with spin indexes attached to it in a consistent fashion. The covariant derivative can be defined in terms of the connection in an entirely conventional way. It acts naturally on the Clifford bundle ; the Clifford bundle is the space in which the spinors live. The general exploration of such structures and their relationships is termed spin geometry.