bundle out of the cotangent bundle T*M. The metric induces a natural isomorphism TM = T*M and therefore an isomorphism Cℓ(TM) = Cℓ(T*M). There is a natural vector bundle isomorphism between the Clifford bundle of M and the exterior bundle of M: This is an isomorphism of vector bundles not algebra bundles. The isomorphism is induced from the corresponding isomorphism on each fiber. In this way one can think of sections of the Clifford bundle as differential forms on M equipped with Clifford multiplication rather than the wedge product (which is independent of the metric). The above isomorphism respects the grading in the sense that Local description For a vector at , and a form the Clifford multiplication is defined as , where the metric duality to change vector to the one form is used in the first term. Then the exterior derivative and coderivative can be related to the metric connection using the choice of an orthonormal base by . Using these definitions, the Dirac-Kähler operator is defined by . On a star domain the operator can be inverted using Poincare lemma for exterior derivative and its Hodge star dual for coderivative. Practical way of doing this is by homotopy and cohomotopy operators. See also Orthonormal frame bundle Spinor Spin manifold Spinor representation Spin geometry Spin structure Clifford module bundle Notes References Clifford algebras Vector bundles Riemannian geometry