Signature operator

In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator. In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator. Let M {displaystyle M} be a compact Riemannian manifold of even dimension 2 l {displaystyle 2l} . Let be the exterior derivative on i {displaystyle i} -th order differential forms on M {displaystyle M} . The Riemannian metric on M {displaystyle M} allows us to define the Hodge star operator ⋆ {displaystyle star } and with it the inner product