Editing Atiyah–Singer index theorem (section)

===Chern-Gauss-Bonnet theorem===
Suppose that <math>M</math> is a compact oriented manifold of dimension <math>n = 2r</math>. If we take <math>\Lambda^\text{even}</math> to be the sum of the even exterior powers of the cotangent bundle, and <math>\Lambda^\text{odd}</math> to be the sum of the odd powers, define <math>D = d + d^*</math>, considered as a map from <math>\Lambda^\text{even}</math> to <math>\Lambda^\text{odd}</math>.  Then the analytical index of <math>D</math> is the [[Euler characteristic]] <math>\chi (M)</math> of the [[Hodge cohomology]] of <math>M</math>, and the topological index is the integral of the [[Euler class]] over the manifold. The index formula for this operator yields the [[Chern–Gauss–Bonnet theorem]].

The concrete computation goes as follows: according to one variation of the [[splitting principle]], if <math>E</math> is a real vector bundle of dimension <math>n = 2r</math>, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles <math>l_1,\, \ldots,\, l_r</math> such that <math>E \otimes \mathbb{C} = l_1 \oplus \overline{l_1} \oplus \dotsm l_r \oplus \overline{l_r}</math>. Therefore, we can consider the Chern roots <math>x_i (E \otimes \mathbb{C}) = c_1(l_i)</math>, <math>x_{r+i} (E \otimes \mathbb{C}) = c_1\mathord\left(\overline{l_i}\right) = -x_i(E \otimes \mathbb{C})</math>, <math>i = 1,\, \ldots,\, r</math>.

Using Chern roots as above and the standard properties of the Euler class, we have that <math display="inline">e(TM) = \prod^r_i x_i(TM \otimes \mathbb{C})</math>. As for the Chern character and the Todd class,<ref>{{citation|first= Mikio|last=Nakahara|title=Geometry, topology and physics|year=2003|isbn=0-7503-0606-8|publisher=Institute of Physics Publishing}}</ref>
:<math>\begin{align}
  \operatorname{ch}\mathord\left(\Lambda^\text{even} - \Lambda^\text{odd}\right)
    &= 1 - \operatorname{ch}(T^* M \otimes \mathbb{C}) + \operatorname{ch}\mathord\left(\Lambda^2 T^* M \otimes \mathbb{C}\right) - \ldots + (-1)^n \operatorname{ch}\mathord\left(\Lambda^n T^* M \otimes \mathbb{C}\right) \\
    &= 1 - \sum_i^n e^{-x_i}(TM \otimes \mathbb{C}) + \sum_{i<j} e^{-x_i}e^{-x_j}(TM \otimes \mathbb{C}) + \ldots + (-1)^n e^{-x_1} \dotsm e^{-x_n}(TM \otimes \mathbb{C}) \\
    &= \prod_i^n \left(1 - e^{-x_i}\right)(TM \otimes \mathbb{C}) \\[3pt]
  \operatorname{Td}(TM \otimes \mathbb{C})
    &= \prod_i^n \frac{x_i}{1 - e^{-x_i}} (TM \otimes \mathbb{C})
\end{align}</math>

Applying the index theorem, 
:<math>\chi(M) = (-1)^r \int_M \frac{\prod_{i}^n \left(1 - e^{-x_i}\right)}{\prod_i^r x_i} \prod_i^n \frac{x_i}{1 - e^{-x_i}}(TM \otimes \mathbb{C}) = (-1)^r \int_{M}(-1)^r\prod_i^r x_i(TM \otimes \mathbb{C}) = \int_M e(TM)</math>
which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying the [[Chern-Weil homomorphism]]).