Morse index theorem
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 .
- 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 :