Clifford Bundles and Compatible Connections

Abstract

We show how the natural operations for vector spaces with quadratic forms carry over to vector bundles with metrics. For a Riemannian manifold X (with or without boundary), we obtain the Clifford bundle Cℓ(X) ≔ Cℓ(TX, g). We show that there exists a connection D for any bundle S of complex left modules over Cℓ(X) which is compatible with Clifford multiplication and extends the Riemannian connection on X to S.