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Manifolds of nonpositive curvature. (English) Zbl 0811.53038
Greene, Robert (ed.) et al., Differential geometry. Part 3: Riemannian geometry. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 3, 179-227 (1993).
This is a thorough review article on the development of the theory of nonpositively curved manifolds during the last decade. In the first part (Ch. 1-5), the authors explain the essential notions, e.g. duality condition, stable and unstable foliation, Anosov property, Kanai connection, Tits and Hamenstaedt metrics and Liouville, harmonic and Bowen-Margulis measures on the ideal boundary of the universal cover. The remaining chapters 6-11 contain results in various branches: Fundamental group, entropy, rigidity of various types, conjugacy of the geodesic flow and marked length spectrum, Margulis lemma, arithmeticity of lattices in semisimple Lie groups. The article tries to keep balance between the major ideals which have influenced the field. It is also an excellent source for references.
MSC:
53C20Global Riemannian geometry, including pinching
53C35Symmetric spaces (differential geometry)
53-02Research monographs (differential geometry)
22E40Discrete subgroups of Lie groups