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Aspects of Differential Geometry II (Paperback)
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Aspects of Differential Geometry II (Paperback)
Series: Synthesis Lectures on Mathematics & Statistics
Expected to ship within 10 - 15 working days
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Differential Geometry is a wide field. We have chosen to
concentrate upon certain aspects that are appropriate for an
introduction to the subject; we have not attempted an encyclopedic
treatment. Book II deals with more advanced material than Book I
and is aimed at the graduate level. Chapter 4 deals with additional
topics in Riemannian geometry. Properties of real analytic curves
given by a single ODE and of surfaces given by a pair of ODEs are
studied, and the volume of geodesic balls is treated. An
introduction to both holomorphic and Kahler geometry is given. In
Chapter 5, the basic properties of de Rham cohomology are
discussed, the Hodge Decomposition Theorem, Poincare duality, and
the Kunneth formula are proved, and a brief introduction to the
theory of characteristic classes is given. In Chapter 6, Lie groups
and Lie algebras are dealt with. The exponential map, the classical
groups, and geodesics in the context of a bi-invariant metric are
discussed. The de Rham cohomology of compact Lie groups and the
Peter--Weyl Theorem are treated. In Chapter 7, material concerning
homogeneous spaces and symmetric spaces is presented. Book II
concludes in Chapter 8 where the relationship between simplicial
cohomology, singular cohomology, sheaf cohomology, and de Rham
cohomology is established. We have given some different proofs than
those that are classically given and there is some new material in
these volumes. For example, the treatment of the total curvature
and length of curves given by a single ODE is new as is the
discussion of the total Gaussian curvature of a surface defined by
a pair of ODEs.
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