In July 2001, the Centre de Recerca Matematica organised at the Universitat Jaume I, in Castello de la Plana, a 20 hour Advanced Course on global Riemannian geometry: curvature and topology. We focused our talks on two main topics: a) The comparison theory for distance functions in spaces which have well defined bounds on their curvature. In this setting we obtained information about diffusion processes, isoperimetric inequalities, transience, and effective resistance of the spaces in question. b) The study of Gromov's invariants to measure the K-theoretic size of a Rie­ mannian manifold by using the famous Lichnerowicz formula for Dirac operators. After the course was finished we extended and smoothed out the material presented in the lectures, and integrated it with the background material furnished to the participants and with their many interesting comments. It is our great pleasure to thank Professor Vicent Palmer, the local organisers Ximo Gual, Ana Lluch and Vicent Miquel, the Centre de Recerca Matematica (CRM) and the Universitat Jaume I for the invitation, the cordial hospitality, the favourable planning of the course and the stimulating working atmosphere. Contents Distance Geometric Analysis on Manifolds Steen Markvorsen 1 Appetizer and Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The Comparison Setting and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 Analysis of Restricted Distance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 Concerning the Setting and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5 Green's Formulae and the Co-area Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 The First Dirichlet Eigenvalue Comparison Theorem . . . . . . . . . . . . . . . . . . . . . 13 7 Isoperimetric Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7. 1 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .