Index of Morse function

Question

What is the definition of the index of a Morse function in dimension one?

Answer 1

There's no difficulty in making a general definition, so I'll answer this for all dimensions, not just dimension $1$.

Let $M$ be a smooth $n$-dimensional manifold and suppose $f: M \longrightarrow \Bbb R$ is a Morse function on $M$. Let $p \in M$ be a critical point of $f$, i.e. $df_p$ is the zero map. Now choose local coordinates $(x^1, \dots, x^n)$ near $p$. Then we can define the Hessian of $f$ at $p$ with respect to these coordinates to be the $n \times n$ matrix whose $(i,j)$-entry is $$\frac{\partial^2 f}{\partial x^i \partial x^j}(p).$$ The Morse index of $f$ at the critical point $p$ is then defined to be the number of negative eigenvalues of this matrix.

One can show that since $f$ is Morse and $p$ is a critical point of $f$, the Morse index of $f$ at $p$ is independent of the choice of coordinates $(x^1, \dots, x^n)$.