# Morse index theorem

From Diffgeom

## Statement

### Setup

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 .

Let be the unique -map from such that .

Define:

- The multiplicity of is the nullity of
- is said to be a
**focal point**if its multiplicity is positive

Define the index form corresponding to as the following bilinear form on :