## 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.

## Chern-Weil Theorem

Recall that given a vector bundle on $M$, there exists (many) connections $\nabla^E\mathpunct{:}\Gamma(E)\to\Gamma(T^\ast M\otimes E)$, which can be extended to $\nabla^E\mathpunct{:}\Omega^\cdot(M;E)\to\Omega^{\cdot+1}(M;E)$, and we defined the curvature (operator) $R^E$ of $\nabla^E$ as
$R^E=(\nabla^E)^2\in\Omega^{\cdot+2}(M;\End(E)),$
it can be viewed as a matrix of 2-forms. What is more, it satisfy the Bianchi identity $[\nabla^E,R^E]=0$.

## Connections and Curvatures on Vector Bundles

Let $E\to M$ be a smooth complex vector bundle over a smooth compact manifold $M$. Denote $\Omega^\cdot(M;E):=\Gamma(\Lambda^\cdot(T^\ast M)\otimes E)$ be the space of smooth sections of the tensor product vector bundle $\Lambda^\cdot(T^\ast M)\otimes E$.

A connection on $E$ is an extension of exterior differential operator $\rd$ to include the coefficient $E$.

$\nabla^E(fs)=(\rd f)\otimes s+f\nabla^Es.$
Let $M$ be a smooth closed (i.e., compact and without boundary) manifold. Let $TM$ and $T^\ast M$ denote the tangent and cotangent (vector) bundle of $M$, respectively. Denote $\Lambda^\cdot(T^\ast M)$ the complex exterior algebra bundle of $T^\ast M$ and $\Omega^\cdot(M,\C)=\Gamma(\Lambda^\cdot(T^\ast M))$ be the space of smooth sections of $\Lambda^\cdot(T^\ast M)$. Particularly, for any integer $p$ with $0\leq p\leq n=\dim M$, we denote by
$\Omega^p(M)=\Gamma(\Lambda^p(T^\ast M))$
be the space of smooth $p$-forms over $M$.