Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.