Morse index and Euler characteristic

Question

I found the following problem and I couldn't solve it. Let $X$ be a compact manifold and $f$ a Morse function (all of its critical points are non degenerate) on $X$. Prove that the sum of the Morse indices of $f$ at its critical points equals the Euler characteristic of $X$. The Morse index $ind_{x}(f)$ is defined as the sign of the determinant of the Hessian of $f$ at $x$, where $x$ is a critical non degenerate point. Does anyone have an idea? Thank you.

Answer 1

I don't know what book you are using, but this is not a standard exercise. This is a big-name theorem for which the proof, or at least the one I am aware of, is ten levels of cleverness above a standard exercise. In fact, this theorem is probably one of the brightest achievements of a standard Morse theory course. This is the Poincare-Hopf theorem--you can find more here.