= Clifford module bundle
{c}
{title2=$c:T^*M\to\operatorname{End}(E)$}
A Clifford module bundle is a <vector bundle> over a <Riemannian manifold> with a fibrewise <Clifford multiplication>. A compatible <connection on a vector bundle> satisfies $[\nabla_X,c(\alpha)]=c(\nabla_X\alpha)$ for the <Levi-Civita connection> on covectors. On a graded Clifford module bundle, <Clifford multiplication> exchanges the two grading summands. The associated <Dirac operator> is $D=\sum_i c(e^i)\nabla_{e_i}$. Its fibres are modules over the fibres of the <Clifford algebra bundle>.
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