Curvature and Homology: Enlarged Edition (Dover Books on Mathematics) [ペーパーバック]

Chapter 1 is a review of elementary differential geometry that is to be used in the rest of the book. Then in chapter 2 the author begins with a review of singular homology and de Rham cohomology. The key point, proved in an appendix, is the de Rham theorem which establishes an isomorphism between de Rham and singular cohomology. The pth Betti number is then the number of linearly independent closed differential forms of degree p modulo the exact forms of degree p. The rest of the chapter is devoted to showing how this result was extended by the mathematician W.V.D Hodge to a restricted class of forms, the famous "harmonic forms". Now called Hodge theory, it is a homology theory based on the Laplace-Beltrami operator, which generalizes, as expected, Laplace's equation.

Chapter 3 is devoted to finding an explicit expression for the Laplace-Beltrami operator in local coordinates. This expression is dependent on the Riemannian curvature of the Riemannian manifold, and so the homology of a compact and orientable manifold will depend on its curvature. The issue then is finding harmonic forms of a given degree. The obstruction to the existence of these is given by a particular quadratic form involving the curvature tensor. The absence of harmonic forms of degree p gives that the pth Betti number is zero. In particular the author shows that the Betti numbers of a compact, orientable, conformally flat Riemannian manifold of positive definite Ricci curvature are all zero. The author then applies these results to compact Lie groups in chapter 4. The harmonic forms on compact Lie groups are those differential forms that are invariant under both left and right translations of the group. The author shows that the first and second Betti numbers of compact Lie groups are zero and shows the existence of a harmonic 3-form, the latter proving that the third Betti number is greater than or equal to one.

The author turns his attention to complex manifolds in chapter 5. He approaches these objects from the standpoint of first defining complex structures on separable Hausdorff spaces. The complex structures then allow a definition of a Riemannian metric on these spaces. If the metric does have any torsion, then one can associate a particular 2-form with the metric and the complex structure that is closed. This 2-form is the famous "Kaehler metric", and the resulting space is called a "Kaehler manifold". The local geometry of Kaehler manifolds is referred to as "Hermitian geometry", and the author studies in detail this geometry in this chapter. Loosely speaking, a Kaehler metric can be viewed as a generalization of "flatness" in the usual Riemannian case, for the author shows that at each point of a Kaehler manifold there exists a system of local complex coordinates which is geodesic. He also introduces the important concept of a holomorphic p-form, and shows that on a Kaehler manifold these are harmonic.

In chapter 6, the author studies in detail how curvature and homology are related for the case of Kaehler manifolds. The results in this chapter could be viewed as a generalization of the classical results concerning compact Riemann surfaces, namely that the universal covering space of a complex n-dimensional compact Kaehler manifold of constant holomorphic curvature K is a projective space for K > 0, the interior of a unit sphere for k < 0, and the space of complex variables for K = 0. After defining the holomorphic curvature, the author shows that the pth Betti number of a compact Kahler manifold M with positive constant holomorphic curvature is zero if p is odd and 1 if p is even. In addition, he shows that any holomorphic form of degree p, for p > 0 and p less than or equal to n, on a compact Kaehler manifold with positive definite Ricci curvature is zero. The author also gives the reader a taste of sheaf theory, in which he discusses briefly the Kodaira vanishing theorems.

In the last chapter, the author generalizes what was done in chapter 3 regarding conformal transformations on Riemannian manifolds, namely that an infinitesimal holomorphic transformation of a compact Kaehler manifold can be viewed as the solution of a system of differential equations which involve the Ricci curvature. Conditions are given for making this transformation an isometry, and the author shows that for a compact Kaehler manifold of complex dimension greater than 1, an infinitesimal conformal transformation is holomorphic if and only if it is an infinitesimal isometry. This leads him to consider the groups of holomorphic transformations, and he gives conditions under which a compact complex manifold cannot admit a transitive Lie group of holomorphic transformations. The author also studies the most general class of Riemannian manifolds for which an infinitesimal conformal transformation is also an infinitesimal isometry. These are the famous "almost Kaehler" manifolds, and the author shows that an infinitesimal conformal transformation of a compact almost Kaehler manifold of dimension 2n for n > 1 is an infinitesimal isometry.