Contents

# Contents

## Definition

Consider a real-valued smooth function

(1)$f \colon M \longrightarrow \mathbb{R}$

on a smooth manifold $M$.

###### Definition

A point $p\in M$ is a critical point of $f$ (1) if for any smooth curve $\gamma : (-\epsilon, \epsilon)\to M$ with $\gamma(0)=p$, the tangent vector

$\frac{d(f\circ\gamma)}{dt} |_{t=0} = 0 \,.$

The critical point is regular if for one (or equivalently any) chart $\phi : U^{\open}\to \mathbb{R}^n$, where $p\in U$ and $\phi(p) = 0\in \mathbb{R}^n$, the Hessian matrix

$\left(\frac{\partial^2 (f\circ \phi^{-1})}{\partial x^i\partial x^j}(0)\right)_{i,j=1,\ldots, n}$

is a nondegenerate (i.e. maximal rank) matrix.

###### Definition

The function $f$ is called a Morse function if every critical point of $f$ is regular (Def. ).

A choice of a Morse function on a compact manifold is often used to study topology of the manifold. This is called the Morse theory.

## Properties

One of the basic tools of Morse theory is the Morse lemma.