= Clifford algebra bundle
{c}
{title2=$\operatorname{Cl}(E,Q)\longrightarrow M$}
= Clifford bundle
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{synonym}
A <vector bundle> $E$ with a fibrewise <quadratic form> $Q$ defines a bundle whose fibre over $x$ is the <Clifford algebra> $\operatorname{Cl}(E_x,Q_x)$. When the polar form is a <nondegenerate bilinear form>, local orthonormal frames identify its fibres with a fixed <Clifford algebra>, and orthogonal transition maps induce the algebra transition maps. A <Clifford module bundle> is a bundle of modules over these algebras, rather than the algebra bundle itself. For a <Riemannian manifold>, use its <cotangent bundle> and the metric <quadratic form>, with the sign convention specified.
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