Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 513of the curvature form Ω of V , which is defined asΩ = dω + 1 [ω, ω],2with ω the connection form and d the exterior derivative, or via the sameexpression in which ω is a gauge form for the gauge group of V . Thescalar t is used here only as an indeterminate to generate the sum fromthe determinant, and I denotes the n × n identity matrix. To say that theexpression given is a representative of the Chern class indicates that ‘class’here means up to addition of an exact differential form. That is, Chernclasses are cohomology classes in the sense of de Rham cohomology. It canbe shown that the cohomology class of the Chern forms do not depend onthe choice of connection in V .For example, let CP 1 be the Riemann sphere: a 1D complex projectivespace. Suppose that z is a holomorphic local coordinate for the Riemannsphere. Let V = T CP 1 be the bundle of complex tangent vectors having theform a∂/∂z at each point, where a is a complex number. In the followingwe prove the complex version of the Hairy Ball Theorem: V has no sectionwhich is everywhere nonzero.For this, we need the following fact: the first Chern class of a trivialbundle is zero, i.e., c 1 (CP 1 × C) = 0. This is evinced by the fact that atrivial bundle always admits a flat metric. So, we will show that c 1 (V ) ≠ 0.Consider the Kähler metrich =dzd¯z(1 + |z| 2 ) ..One can show that the curvature 2–form is given byΩ =2dz ∧ d¯z(1 + |z| 2 ) 2 .Furthermore, by the definition of the first Chern classc 1 =i2π Ω..We need to show that the cohomology class of this is non–zero. It sufficesto compute its integral over the Riemann sphere:∫c 1 = i ∫dz ∧ d¯zπ (1 + |z| 2 ) 2 = 2,