n-space, and Rn is considered to be embedded in a Clifford algebra. This is a special case of the Atiyah–Singer–Dirac operator acting on sections of a spinor bundle. Example 5 For a spin manifold, M, the Atiyah–Singer–Dirac operator is locally defined as follows: For and e1(x), ..., ej(x) a local orthonormal basis for the tangent space of M at x, the Atiyah–Singer–Dirac operator is where is the spin connection, a lifting of the Levi-Civita connection on M to the spinor bundle over M. The square in this case is not the Laplacian, but instead where is the scalar curvature of the connection. Example 6 On Riemannian manifold of dimension with Levi-Civita connection and an orthonormal basis , we can define exterior derivative and coderivative as . Then we can define a Dirac-Kähler operator , as follows . The operator acts on sections of Clifford bundle in general, and it can be restricted to spinor bundle, an ideal of Clifford bundle, only if the projection operator on the ideal is parallel. Generalisations In Clifford analysis, the operator acting on spinor valued functions defined by is sometimes called Dirac operator in k Clifford variables. In the notation, S is the space of spinors, are n-dimensional variables and is the Dirac operator in the i-th variable. This is a common generalization of the Dirac operator () and the Dolbeault operator (, k arbitrary). It is an invariant differential operator, invariant under the action of the group . The resolution of D is known only in some special cases. See also AKNS hierarchy Dirac equation Clifford algebra Clifford analysis Connection Dolbeault operator Heat kernel Spinor bundle References Differential operators Quantum mechanics Mathematical physics