I'm reading a paper where the following form of Chern-Gauss-Bonnet theorem is proved:
Let $R$ denote the Riemannian curvature tensor associated to the Levi-Civita connection (the Riemannian connection) on a closed Riemannian manifold $X$ of dimension $n$, and let $\text{Pf}(R)$ be the Pfaffian density of the curvature. Then \begin{equation*} \frac{1}{(2\pi)^{\frac{n}{2}}}\int_X \text{Pf}(R)= \text{Index}(\nabla h), \end{equation*} where $h$ is any Morse function on $X$ and $\text{Index}(\nabla h)$ is the Hopf index of the gradient vector field $\nabla h$ of $h$.
Before stating this, the paper stated in its introduction as a side effect that
$$ \frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\underline{\text{SM}}(\mathbb{R}^{0|2},X)}\exp(-\mathcal{S}_0(\phi)= \chi(X), $$ without proof, where $\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\underline{\text{SM}}(\mathbb{R}^{0|2},X)}\exp(-\mathcal{S}_0(\phi)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_X \text{Pf}(R)$ by later computation, giving half the equality of the above Chern-Gauss-Bonnet theorem.
I've read before that in Bott & Tu's Differential Forms in Algebraic Topology, the Hopf index theorem is proved using the Euler class, giving
$$\chi (X)=\int_X e(TX)=\text{Index}(\nabla h). $$
My advisor told me that $\frac{1}{(2\pi)^{n/2}}\text{Pf}(R)$ is exactly the Euler class $e(TX)$, but he does not remember clearly how this can be deduced.
I haven't yet figured out what the formula of $\text{Pf}(R)$ is, but I'm confused by the relation between the Chern-Gauss-Bonnet theorem and Hopf index theorem. Under the assumption that $\frac{1}{(2\pi)^{n/2}}\text{Pf}(R)=e(TX)$, it seems that they are exactly the same -- then what's the point of distinguishing the two theorems?
To spell out, my question is:
Can the fact that $\frac{1}{(2\pi)^{n/2}}\text{Pf}(R)=e(TX)$ be deduced by direct computation? Or does it follow as a corollary from the two theorems that $$\frac{1}{(2\pi)^{\frac{n}{2}}}\int_X \text{Pf}(R)= \text{Index}(\nabla h)=\int_Xe(TX)$$ and the fact that $H_{DR}^n(X)=\mathbb{R}$?
Is there any good reason to distinguish the Chern-Gauss-Bonnet theorem and the Hopf index theorem, which makes sense even if $\frac{1}{(2\pi)^{n/2}}\text{Pf}(R)=e(TX)$ could be given by direct computation?
