Book
Differential Geometry.
| Title: | Differential Geometry. |
|---|---|
| Authors: | Axler, S., Ribet, K. A., Wells, Raymond O. |
| Source: | Differential Analysis on Complex Manifolds; 2008, p65-107, 43p |
| Abstract: | This chapter is an exposition of some of the basic ideas of Hermitian differential geometry, with applications to Chern classes and holomorphic line bundles. In Sec. 1 we shall give the basic definitions of the Hermitian analogues of the classical concepts of (Riemannian) metric, connection, and curvature. This is carried out in the context of differentiable C-vector bundles over a differentiable manifold X. More specific formulas are obtained in the case of holomorphic vector bundles (in Sec. 2) and holomorphic line bundles (in Sec. 4). In Sec. 3 is presented a development of Chern classes from the differential-geometric viewpoint. In Sec. 4 this approach to characteristic class theory is compared with the classifying space approach and with the sheaf-theoretic approach (in the case of line bundles). We prove that the Chern classes are primary obstructions to finding trivial subbundles of a given vector bundle, and, in particular, to the given vector bundle being itself trivial. In the case of line bundles, we give a useful characterization of which cohomology classes in H2(X, Z) are the first Chern class of a line bundle. Additional references for the material covered here are Chern [2], Griffiths [2], and Kobayashi and Nomizu [1]. [ABSTRACT FROM AUTHOR] |
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| DOI: | 10.1007/978-0-387-73892-5_3 |
| Database: | Complementary Index |
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