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. General construction 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. One is most often interested in the case where g is positive-definite or at least nondegenerate; that is, when (E, g) is a Riemannian or pseudo-Riemannian vector bundle. For concreteness, suppose that (E, g) is a Riemannian vector bundle. The Clifford bundle of E can be constructed as follows. Let CℓnR be the Clifford algebra generated by Rn with the Euclidean metric. The standard action of the orthogonal group O(n) on Rn induces a graded automorphism of CℓnR. The homomorphism is determined by where vi are all vectors in Rn. The Clifford bundle of E is then given by where F(E) is the orthonormal frame bundle of E. It is clear from this construction that the structure group of Cℓ(E) is O(n). Since O(n) acts by graded automorphisms on CℓnR it follows that Cℓ(E) is a bundle of Z2-graded algebras over M. The Clifford bundle Cℓ(E) can then be decomposed into even and odd subbundles: If the vector bundle E is orientable then one can reduce the structure group of Cℓ(E) from O(n) to SO(n) in the natural manner. Clifford bundle of a Riemannian manifold If M is a Riemannian manifold with metric g, then the Clifford bundle of M is the Clifford bundle generated by the tangent bundle TM. One can also build a Clifford