Some Examples

Now, we take $f$ to be some special function to obtain some classical classes.

Sometimes, we would prefer to normalize the function by conceder
\[
\tr\left[f\left(\frac{\sqrt{-1}}{2\pi}R^E\right)\right],
\]
since $R^E$ is a anti-symmetric matrix, we want to make the eigenvalue be real, and $2\pi$ just a unitization, such that it becomes rational even integer.
 

Excise 1. the product of closed forms is still a closed form.

Example 1 (Chern form and Chern classes). Let $E$ be a complex vector bundle, and set
\[
c(E,\nabla^E)=\det\left(\I+\frac{\sqrt{-1}}{2\pi}R^E\right)
=\exp\left(\tr\left[\log\left(\I+\frac{\sqrt{-1}}{2\pi}R^E\right)\right]\right),
\]
note that
\[
\I+\frac{\sqrt{-1}}{2\pi}R^E
\]
is invertible, and
\[
\exp(\tr A)=\det(\exp(A)).
\]
Note also that the power series
\begin{align*}
\log(1+x)&=x+\frac{x^2}{2}+\cdots\\
\exp(x)&=1+x+\frac{x^2}{2!}+\cdots,
\end{align*}
substitute $x$ with $\frac{\sqrt{-1}}{2\pi}R^E$, then it only has finite terms and this shows that for any integer $k\geq0$, $\tr[(R^E)^k]$ is a linear combination of various products of $c_i(E,\nabla^E)$’s, this established the fundamental importance of Chern classes in the theory of characteristic classes of complex vector bundles.

We write
\begin{align*}
c(E,\nabla^E)&=\exp\left(\tr\left[\log\left(\I+\frac{\sqrt{-1}}{2\pi}R^E\right)\right]\right)\\
&=1+c_1(E,\nabla^E)t+c_2(E,\nabla^E)t^2+\cdots,
\end{align*}
the $c_i(E,\nabla^E)$ are closed $2$-form, called Chern form, and $c(E,\nabla^E)\bigoplus c_i(E,\nabla^E)$ called the total Chern form, and the cohomology class associated to $k$-th Chern form $c_k(E,\nabla^E)$ are called the $k$-th Chern classes of second type, denoted as $c_k(E)$, and $\bigoplus c_k(E)$ are called total Chern classes.


Example 2 (Pontrjagin classes of real vector bundle). Let $E$ be a real vector bundle of $M$, define
\[
p(E,\nabla^E)=\det\left(I-\left(\frac{\sqrt{-1}}{2\pi}\right)^2\right)^{\frac{1}{2}}.
\]
Expand $\sqrt{1-x}$, we can write
\[
p(E,\nabla^E)=1+P_1(E,\nabla^E)t+\cdots,
\]
here $p_k(E,\nabla^E)$ is closed $4k$-form.

Since any real vector bundle can be complexificated to be $E\otimes\C$, and $\nabla^E$ can be extend to a complex-linear operator $\nabla^E_\C$, then
\[
c_{2k}(E\otimes \C)=(-1)^kp_k(E).
\]
Excise 2. Prove that claim that
\[
c_{2k}(E\otimes \C)=(-1)^kp_k(E).
\]
Hint: try the consider it from their froms.

Similarly, if we write
\[
\log\left(\det\left(I-\left(\frac{R^E}{2\pi}\right)^2\right)^{\frac{1}{2}}\right)
=\tr\left(\frac{1}{2}\log\left(I-\left(\frac{R^E}{2\pi}\right)^2\right)\right),
\]
and from the power series expansion formulas for $\log(\sqrt{1-x})$, one deduces that for any integer $k\geq0$, $\tr[(R^E)^{2k}]$ can be written as a linear combination of various products of $p_i(E,\nabla^E)$’s.

This establishes the fundamental importance of Pontrjagin classes in the theory of characteristic classes of real vector bundles.

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