Effective universal coverings and local minima of the length functional on loop spaces
The author describes a construction of effective universal coverings of Riemannian manifolds and of path metric spaces. Moreover, he gives two applications of this construction. In the first application the author demonstrates that if the length functional on the space \(\Omega_xM^n\) of all loops based at a point \(x\in M^n\) has a non-trivial very deep local minimum, then it has many deep local minima. The second application is a version of a theorem proved by M. Anderson. If \(M^n\) is a closed \(n\)-dimensional manifold with volume \(\geq v> 0\), diameter \(d\) and Ricci curvature \(\geq -(n-1)\) then there exist explicitly given positive \(\varepsilon = \varepsilon(n,v,d)\) and \(N=N(n,v,d)\) such that for each closed curve \(\gamma\) of length smaller than \(\varepsilon\) one of its first \(N\) iterates is contractible. The author finds an explicit upper bound for the length of curves in homotopy contracting one of the first \(N\) iterates of \(\gamma\).
scientific article; zbMATH DE number 5800300