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\newcommand{\id}{\mathop{\rm id}\nolimits} \)
CHERN-WEIL AND GAUSS-BONNET
ISHAN LEVY
Given a manifold, Chern-Weil theory says that we can obtain characteristic classes by applying
invariant polynomials on the curvature of a connection. We will see here an explicit proof (without
using the Chern-Weil homomorphism) of the Gauss-Bonnet theorem for vector bundles, which is
an example of the phenomenon.
Let \(E \to M\) be a rank \(2p\) real vector bundle with a metric and a metric connection \(\nabla \), and let \(\Omega _E\) be its
curvature \(2\)-form. Then we can take the Pfaffian \(\Pf (\Omega _E)\) multiplied by a normalizing constant \((\frac{-1}{2\pi })^{p}\) of the
curvature to get a \(d\)-form, whose cohomology class we should interpret by Chern-Weil theory as a
characteristic class of the bundle. Indeed, we can call this class the geometric Euler class (\(e_g(E)\)), and we
can prove that it indeed coincides with the topological Euler class (\(e_t(E)\)). This can be viewed as a
generalization of Gauss-Bonnet:
Theorem 0.1 (Gauss-Bonnet). Given an even dimensional Riemannian manifold \(M^{2p}\), if \(\Omega \) is
the curvature, then \(\int _{M} (\frac{-1}{2\pi })^{p} \Pf (\Omega ) = \chi (M)\).
In the case that the bundle is the tangent bundle, and the metric is a Riemannian metric, this
becomes the Gauss-Bonnet theorem. Indeed, the Euler class integrates to the Euler characteristic,
and the geometric Euler class is an integral of the Pfaffian of the Riemann curvature tensor (up to
a constant).
The first thing to note is that the geometric Euler class is natural. It is easy to check that it
commutes with pullbacks, and that \(e_g(E_1\oplus E_2) = e_g(E_1)\wedge e_g(E_2)\) (Note: here the notation is abused since \(e_g\) seems to depend on
the connection). Then by the splitting principle, it suffices to show that \(e_g = e_t\) for oriented plane
bundles, for which we can more explicitly calculate.
For a plane bundle \(E \xrightarrow{\pi } M\), let the connection be given in local neighborhood \(U_\alpha \) by the skew-symmetric
matrix of 1-forms \((\theta _\alpha )_i^j=\omega _\alpha \). The curvature \(\Omega _\alpha = d \omega _\alpha - \omega _\alpha \wedge \omega _\alpha \) is given by the matrix \(\begin{pmatrix} (\theta _\alpha )_1^2 \wedge (\theta _\alpha )_1^2 & d (\theta _\alpha )_1^2 \\ -d (\theta _\alpha )_1^2 & (\theta _\alpha )_1^2 \wedge (\theta _\alpha )_1^2 \end{pmatrix}\) so that the Pfaffian is \(d(\theta _\alpha )_1^2\).
Now suppose we have a partition of unity \(\gamma _\alpha \) subordinate to the choice of local coordinate cover \(U_\alpha \),
and let \(g_{\alpha \beta }\) be the transition functions with values in \(\SOr (2)\) that define the vector bundle. Then by
identifying \(\SOr (2) = \RR /2 \pi \ZZ \), we can think of the \(g_{\alpha \beta }\) as the angle the transition function rotates counterclockwise. By
one construction (eg. in Bott and Tu’s book) \(e_t\) is given by \(\frac{-1}{2\pi } \sum _\beta d \gamma _\beta d g_{\alpha \beta }\). If \(r_\alpha , r_\alpha '\) make up the local frame in \(U_\alpha \), since the
connection is a metric connection, we have that \(d r_\alpha = (\theta _\alpha )_1^2 r'_\alpha \) (here we view the connection as on the frame
bundle).
On the bundle since \(g_{\alpha \beta }\) is the transition function, we have \(d \pi ^* r_\alpha = (\pi ^* d r_\beta + \pi ^* g_{\alpha \beta }) \pi ^* r_\alpha '\). By injectivity of \(\pi ^*\) we obtain \(d r_\alpha = d r_\beta + d g_{\alpha \beta }r'_\alpha \). Thus we
must have \(d g_{\alpha \beta } = (\theta _\alpha )_1^2- (\theta _\beta )_1^2\).
Then we have:
\[ \frac{-1}{2\pi } \sum _\beta d( \gamma _\beta d g_{\alpha \beta }) = \frac{-1}{2\pi } \sum _\beta d (\gamma _\beta ( (\theta _\alpha )_1^2- (\theta _\beta )_1^2)) = \frac{-1}{2\pi } d (\theta _\alpha )_1^2 + \frac 1{2\pi } d(\sum _\beta \gamma _\beta (\theta _\beta )_1^2) \]
The second resulting term defines a global form which is clearly exact, and we get that \(e_t\) is
cohomologous to \(-\frac{1}{2\pi }d(\theta _\alpha )_1^2\), which is exactly \(e_g\).