# Index form

From Diffgeom

## Definition

### Given data

Let be Euclidean space, and let be the linear space of piecewise -maps from to . Let denote the subspace of comprising maps which are zero at the endpoints (viz maps satisfying ).

Let be a -map from into the collection of self-adjoint linear transformations on .

### Definition part

Then the **index form** associated with is the bilinear form on defined as follows: where is the inner product on .