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.
[1] It is an instance of a Dirac-type operator.
be a compact Riemannian manifold of even dimension
Let be the exterior derivative on
-th order differential forms on
The Riemannian metric on
allows us to define the Hodge star operator
and with it the inner product on forms.
Denote by the adjoint operator of the exterior differential
This operator can be expressed purely in terms of the Hodge star operator as follows: Now consider
acting on the space of all forms
One way to consider this as a graded operator is the following: Let
be an involution on the space of all forms defined by: It is verified that
Consequently, Definition: The operator
with the above grading respectively the above operator
is called the signature operator of
[2] In the odd-dimensional case one defines the signature operator to be
acting on the even-dimensional forms of
is a multiple of four, then Hodge theory implies that: where the right hand side is the topological signature (i.e. the signature of a quadratic form on
defined by the cup product).
The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that: where
[4] Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.