EAN: 9783031012808

SYNTHESIS LECTURES ON MATHEMATICS & STATISTICS ...

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 Kähler
geometry is given. In Chapter 5  the basic properties of de Rham cohomology are discussed  the
Hodge Decomposition Theorem  Poincaré duality  and the Künneth 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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