Differential Analysis on Complex Manifolds pp 65-107 | Cite as

# Differential Geometry

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 *H*^{2}(*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].

## Keywords

Vector Bundle Line Bundle Differential Form Cohomology Class Chern Class## Preview

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