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Nonpositive Curvature: Geometric and Analytic Aspects (Paperback, 1997 ed.)
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Nonpositive Curvature: Geometric and Analytic Aspects (Paperback, 1997 ed.)
Series: Lectures in Mathematics. ETH Zurich
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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."
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