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Part of: Synthesis digital library of engineering and computer science.

Includes bibliographical references (pages 133-137) and index.

4. Additional topics in Riemannian geometry -- 4.1 Curves and surfaces in Rn given by ODEs -- 4.2 Volume of geodesic balls -- 4.3 Holomorphic geometry -- 4.4 Kahler geometry --

5. de Rham cohomology -- 5.1 Basic properties of de Rham cohomology -- 5.2 Clifford algebras -- 5.3 The Hodge decomposition theorem -- 5.4 Characteristic classes --

6. Lie groups -- 6.1 Basic concepts -- 6.2 Lie algebras -- 6.3 The exponential function of a matrix group -- 6.4 The classical groups -- 6.5 Representations of a compact lie group -- 6.6 Bi-invariant pseudo-Riemannian metrics -- 6.7 The killing form -- 6.8 The classical groups in low dimensions -- 6.9 The cohomology of compact lie groups -- 6.10 The cohomology of the unitary group --

7. Homogeneous spaces and symmetric spaces -- 7.1 Smooth structures on coset spaces -- 7.2 The isometry group -- 7.3 The lie derivative and killing vector fieldS -- 7.4 Homogeneous pseudo-Riemannian manifolds -- 7.5 Local symmetric spaces -- 7.6 The global geometry of symmetric spaces --

8. Other cohomology theories -- 8.1 Homological algebra -- 8.2 Simplicial cohomology -- 8.3 Singular cohomology -- 8.4 Sheaf cohomology -- Bibliography -- Authors' biographies -- Index.

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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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Title from PDF title page (viewed on May 20, 2015).