Editing Chern class (section)

===The complex tangent bundle of the Riemann sphere===

Let <math>\mathbb{CP}^1</math> be the [[Riemann sphere]]: 1-dimensional [[complex projective space]]. Suppose that ''z'' is a [[Holomorphic function|holomorphic]] [[manifold|local coordinate]] for the Riemann sphere. Let <math>V=T\mathbb{CP}^1</math> be the bundle of complex tangent vectors having the form <math>a \partial/\partial z</math> at each point, where ''a'' is a [[complex number]].  We prove the complex version of the ''[[hairy ball theorem]]'':  ''V'' has no section which is everywhere nonzero.

For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e.,
<math display="block">c_1(\mathbb{CP}^1\times \Complex)=0.</math>

This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that
<math display="block">c_1(V) \not= 0.</math>

Consider the [[Kähler metric]]
<math display="block">h = \frac{dz d\bar{z}}{(1+|z|^2)^2}.</math>

One readily shows that the curvature 2-form is given by
<math display="block">\Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^2)^2}.</math>

Furthermore, by the definition of the first Chern class
<math display="block">c_1= \left[\frac{i}{2\pi} \operatorname{tr} \Omega\right] .</math>

We must show that this cohomology class is non-zero.  It suffices to compute its integral over the Riemann sphere:
<math display="block">\int c_1 =\frac{i}{\pi}\int \frac{dz\wedge d\bar{z}}{(1+|z|^2)^2}=2</math>
after switching to [[polar coordinates]]. By [[Stokes' theorem]], an [[exact form]] would integrate to 0, so the cohomology class is nonzero.

This proves that <math>T\mathbb{CP}^1</math> is not a trivial vector bundle.