"Recollects some basic properties as well as some fairly advanced results [which] is done with a spirit that allows one to understand that, even though the study of such manifolds has important differences from the flat case, some techniques come from the very elementary Euclidean geometry." --Mathematical Reviews

• 1 Introduction.- 1.1 Examples of Riemannian manifolds of negative or nonpositive sectional curvature.- 1.2 Mordell and Shafarevitch type problems.- 1.3 Geometric superrigidity.- 2 Spaces of nonpositive curvature.- 2.1 Local properties of Riemannian manifolds of nonpositive sectional curvature.- 2.2 Nonpositive curvature in the sense of Busemann.- 2.3 Nonpositive curvature in the sense of Alexandrov.- 3 Convex functions and centers of mass.- 3.1 Minimizers of convex functions.- 3.2 Centers of mass.- 3.3 Convex hulls.- 4 Generalized harmonic maps.- 4.1 The definition of generalized harmonic maps.- 4.2 Minimizers of generalized energy functional.- 5 Bochner-Matsushima type identities for harmonic maps and rigidity theorems.- 5.1 The Bochner formula for harmonic one-forms and harmonic maps.- 5.2 A Matsushima type formula for harmonic maps.- 5.3 Geometrie superrigidity.

"Recollects some basic properties as well as some fairly advanced results [which] is done with a spirit that allows one to understand that, even though the study of such manifolds has important differences from the flat case, some techniques come from the very elementary Euclidean geometry." --Mathematical Reviews

• 1 Introduction.- 1.1 Examples of Riemannian manifolds of negative or nonpositive sectional curvature.- 1.2 Mordell and Shafarevitch type problems.- 1.3 Geometric superrigidity.- 2 Spaces of nonpositive curvature.- 2.1 Local properties of Riemannian manifolds of nonpositive sectional curvature.- 2.2 Nonpositive curvature in the sense of Busemann.- 2.3 Nonpositive curvature in the sense of Alexandrov.- 3 Convex functions and centers of mass.- 3.1 Minimizers of convex functions.- 3.2 Centers of mass.- 3.3 Convex hulls.- 4 Generalized harmonic maps.- 4.1 The definition of generalized harmonic maps.- 4.2 Minimizers of generalized energy functional.- 5 Bochner-Matsushima type identities for harmonic maps and rigidity theorems.- 5.1 The Bochner formula for harmonic one-forms and harmonic maps.- 5.2 A Matsushima type formula for harmonic maps.- 5.3 Geometrie superrigidity.