MathaiQuillen forms and Lefschetz theory
Abstract
MathaiQuillen 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 ChernGaussBonnet theorem. The formula is computed explicitly for constant curvature metrics. There is in fact a oneparameter 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 nonpositive curvature.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 1998
 DOI:
 10.48550/arXiv.math/9802061
 arXiv:
 arXiv:math/9802061
 Bibcode:
 1998math......2061F
 Keywords:

 Differential Geometry;
 57R35 (Primary) 53C20 (Secondary)
 EPrint:
 44 pages, Latex