Michael Taylor : Anderson-Cheeger limits of smooth Riemannian manifolds, and other Gromov-Hausdorff limits- Geometry and Topology ( 141 Views )
If you take a surface in Euclidean space that is locally the graph of a C2 function, this induces a local coordinate system in which the metric tensor is merely C1. Geodesic flows are well defined when the metric tensor is C2, but there are lots of examples of metric tensors of class C^(2-epsilon) for which geodesics branch. Nevertheless, for the C2 surface mentioned above, the geodesic flow is well defined. This result has been noted several times. It has several proofs. One uses the fact that geodesic flows are well defined whenever the Ricci tensor is bounded. An important class of Gromov-Hausdorff limits of smooth Riemannian manifolds studied by Anderson and Cheeger puts a lower bound on the Ricci tensor (and the injectivity radius), and obtains a limiting manifold whose metric tensor is not quite C1. We will explore the question of whether the geodesic flow is well defined on such a limit, and also look at some other limits of smooth manifolds, with wilder behavior.