Riemannian geometry of non-Riemannian spaces

Two problems from Riemannian geometry are considered in the more general context of metrically complete, locally compact inner metric spaces: (1) controlling topology through geometry and (2) understanding the geometry and topology of Gromov-Hausdorff limits. The questions are of interest because the limits of Riemannian manifolds are inner metric spaces whose structure has recently received attention.;Basic definitions are recalled (including one of bounded curvature) and some new geometric concepts introduced, including geodesic terminal (a point through which some geodesic cannot be continued) and geodesic completeness (no geodesic terminals exist). The local topological structure of a space X with bounded curvature and nowhere dense set T of geodesic terminals is established: x is topological manifold with boundary, and ;Applications of the above results are generalizations of Cartan's classification of simply connected space forms, and the Cartan-Hadamard theorem.;Geometric properties inherited by a Gromov-Hausdorff limit of inner metric spaces are also addressed. Curvature conditions are given for which a geodesic in the limit can be approximated by geodesics in the converging spaces; from such approximations many geometric properties in the limit (e.g. bounded curvature, geodesic completeness) can be deduced.;Finally, the "lifting" of the metric described in the second paragraph above gives rise to a "local action" in the tangent space. A rigorous theory of local actions and their extensions to global actions is developed. The essential difference between the local and global cases is shown to be the associative law. The general theory is then applied, together with local Path and Homotopy Lifting Lemmas, to construct a local tangent space action without resorting the theorems from differential geometry. Various consequences of the local action are briefly discussed.