Degree of a compact manifold

Question

Let $M$ be a compact manifold and $f$ a smooth function with isolated critical points. If $Q$ is a quadratic form of signature $(p,q)$, I define $sig(Q)=p-q$. I define $\chi(f)$ as:

$$\chi (f)= \sum_{x,df(x)=0} sig(Hess(f)(x))$$

where $Hess(f)(x)$ is the Hessian of $f$ at the point $x$.

Then we have, $\forall f$:

$$\chi (f)= Cst=\chi (M)$$

Have we defined a topological invariant of the manifold?

Answer 1

It is the theorem of Poincaré-Hopf, we obtain the Euler-Poincaré characteristic.