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Geometric Structures on Manifolds (Paperback)
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Geometric Structures on Manifolds (Paperback)
Series: Graduate Studies in Mathematics
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The theory of geometric structures on manifolds which are locally
modeled on a homogeneous space of a Lie group traces back to
Charles Ehresmann in the 1930s, although many examples had been
studied previously. Such locally homogeneous geometric structures
are special cases of Cartan connections where the associated
curvature vanishes. This theory received a big boost in the 1970s
when W. Thurston put his geometrization program for 3-manifolds in
this context. The subject of this book is more ambitious in scope.
Unlike Thurston's eight 3-dimensional geometries, it covers
structures which are not metric structures, such as affine and
projective structures. This book describes the known examples in
dimensions one, two and three. Each geometry has its own special
features, which provide special tools in its study. Emphasis is
given to the inter-relationships between different geometries and
how one kind of geometric structure induces structures modeled on a
different geometry. Up to now, much of the literature has been
somewhat inaccessible and the book collects many of the pieces into
one unified work. This book focuses on several successful
classification problems. Namely, fix a geometry in the sense of
Klein and a topological manifold. Then the different ways of
locally putting the geometry on the manifold lead to a ""moduli
space"". Often the moduli space carries a rich geometry of its own
reflecting the model geometry. The book is self-contained and
accessible to students who have taken first-year graduate courses
in topology, smooth manifolds, differential geometry and Lie
groups.
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