  Mathai-Quillen forms are used to give an integral formula for the Lefschetz
number of a smooth map of a closed manifold. Applied to the identity map, this
formula reduces to the Chern-Gauss-Bonnet theorem. The formula is computed
explicitly for constant curvature metrics. There is in fact a one-parameter
family of integral expressions. As the parameter goes to infinity, a
topological version of the heat equation proof of the Lefschetz fixed
submanifold formula is obtained. As the parameter goes to zero and under a
transversality assumption, a lower bound for the number of points mapped into
their cut locus is obtained. For diffeomorphisms with Lefschetz number unequal
to the Euler characteristic, this number is infinite for most metrics, in
particular for metrics of non-positive curvature.

Differential Geometry