Clifford bundle

In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold M which is called the Clifford bundle of M.

Let V be a (real or complex) vector space together with a symmetric bilinear form <·,·>. The Clifford algebra Cℓ(V) is a natural (unital associative) algebra generated by V subject only to the relation

for all v in V. One can construct Cℓ(V) as a quotient of the tensor algebra of V by the ideal generated by the above relation.

Like other tensor operations, this construction can be carried out fiberwise on a smooth vector bundle. Let E be a smooth vector bundle over a smooth manifold M, and let g be a smooth symmetric bilinear form on E. The Clifford bundle of E is the fiber bundle whose fibers are the Clifford algebras generated by the fibers of E:

The topology of Cℓ(E) is determined by that of E via an associated bundle construction.

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