1998/02/12 by Mihail Frumosu, Frumosu, Mihail, Steven Rosenberg +1
Computer Science · Mathematics · #57R35 (Primary) 53C20 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.DG #msc:53C20 #msc:57R35
paper · pdf · doi:10.48550/arxiv.math/9802061
44 pages, Latex
arxiv created 1998/02/12 · openalex publication_date 1998/02/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.