Geodesics and Invariants of Riemannian Manifolds
Final Report Abstract
The goal of the project was to find connections between the global behaviour of geodesic curves and topological and geometric invariants of smooth manifolds. These connections were studied in particular through string topology and topological and geodesic complexity. In joint work with Philippe Kupper we studied the behaviour of the intersection multiplicity of homology classes on loop spaces and the connections with the string topology coproduct. In particular, we show that the coproduct on the based loop space of a manifold is trivial if this manifold is the total space of a fiber bundle with a section. Moreover, one can show with a simple argument that the coproduct on the free loop space of a product manifold vanishes if both factors have vanishing Euler characteristic. Based on this observation it was studied in joint work with Nathalie Wahl how the coproduct behaves on arbitrary product manifolds. It seems that the behaviour of the coproduct on product manifolds is determined by the Sullivan-Frobenius relation - a relation, which does not hold in general in string topology, but is applicable in the case of product manifolds. It yields a formula for the coproduct on a product manifold in terms of the individual factors. Moreover, the connection between topological complexity and the behaviour of critical points on a Banach manifold was studied in a joint project with Stephan Mescher. We prove inequalities between the sequential topological complexities of a space, respectively between the parametrized sequential topological complexities of a fiber bundle and certain quantities related to the number of critical points on the considered space. These results are formulated in such a way that they could in particular be applied to variational problems, e.g. to the energy functional on the free loop space of a closed manifold whose critical points are the closed geodesics.
Publications
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”Approaches to critical point theory via sequential and parametrized topological complexity”.
Stephan Mescher & Maximilian Stegemeyer
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”Intersection Multiplicity in Loop Spaces”. To appear in: Springer Tohoku Series in Mathematics, 2025.
Philippe Kupper & Maximilian Stegemeyer
