This book discusses various geometric and analytic aspects of nonpositive curvature, starting with a discussion of Riemannian examples and rigidity theorems. It then treats generalized notions of nonpositive curvature in metric geometry in the sense of Alexandrov and Busemann, as well as the theory of harmonic maps with values in such spaces.
It is intended for researchers and graduate students in Riemannian and metric geometry as well as calculus of variations.

The present book contains the lecture notes from a "Nachdiplomvorlesung", a topics course adressed to Ph. D. students, at the ETH ZUrich during the winter term 95/96. Consequently, these notes are arranged according to the requirements of organizing the material for oral exposition, and the level of difficulty and the exposition were adjusted to the audience in Zurich. The aim of the course was to introduce some geometric and analytic concepts that have been found useful in advancing our understanding of spaces of nonpos itive curvature. In particular in recent years, it has been realized that often it is useful for a systematic understanding not to restrict the attention to Riemannian manifolds only, but to consider more general classes of metric spaces of generalized nonpositive curvature. The basic idea is to isolate a property that on one hand can be formulated solely in terms of the distance function and on the other hand is characteristic of nonpositive sectional curvature on a Riemannian manifold, and then to take this property as an axiom for defining a metric space of nonposi tive curvature. Such constructions have been put forward by Wald, Alexandrov, Busemann, and others, and they will be systematically explored in Chapter 2. Our focus and treatment will often be different from the existing literature. In the first Chapter, we consider several classes of examples of Riemannian manifolds of nonpositive curvature, and we explain how conditions about nonpos itivity or negativity of curvature can be exploited in various geometric contexts.

Autorentext

Parvaneh Joharinad received her PhD in mathematics from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in March 2013. She worked as an assistant professor in the geometry group at the Institute for Advanced Studies in Basic Sciences (IASBS) in Zanjan, Iran, for seven years. She is interested in the use of geometry in data science and machine learning, and in particular in dimensionality reduction, a fundamental problem in topological and geometric data analysis. Her collaboration with Jürgen Jost began in 2017, via a project on a generalization of the concept of sectional curvature to datasets. In 2020, she received a grant from the Max-Planck society to continue her collaboration at the Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany. As of August 2022, she started a new position at the Center for Scalable Data Analytics and Artificial Intelligence, as a senior postdoc. Jürgen Jost worked as a Professor of Mathematics at Ruhr University Bochum from 1984 to 1996 and since 1996 has been director and a permanent member of the Max Planck Institute for Mathematics in the Sciences, Leipzig. In 1998 he became an Honorary Professor at the University of Leipzig. He is also an external member of the Santa Fe Institute for the Sciences of Complexity, New Mexico. He pursues both topical research in different fields of pure mathematics and theoretical physics (Riemannian and algebraic geometry, geometric analysis, calculus of variations, partial differential equations, dynamical systems, graph and hypergraph theory) and interdisciplinary research in complex systems, including evolutionary and theoretical molecular biology, mathematical and theoretical neuroscience, nonlinear dynamics and statistical physics, economics and social sciences, strategy science, history and philosophy of science. He directs a group of about 40 scientists, postdocs and PhD students, and has manyinternational cooperation partners.

Inhalt
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.