Morse Lemma

Critical Point

Let $f:M\to \mathbb{R}$ a smooth real-valued function on a manifold $M$. A point $p\in M$ is called a critical point of f if the induced map $f_{\ast }:T{M}_p\to T{R}_{f(p)}$ is zero. If we choose a local coordinate system $(x^1,\dots ,x^n)$ in a neighborhood $U$ of $p$, this means that $\frac{\partial f}{\partial x^1}(p)=\cdots =\frac{\partial f}{\partial x^n}(p)=0.$ The real number $f(p)$ is called a critical value of $p$.

We denote by $M^a$ the set of all points $x\in M$ such that $f(x)\le a$. If $a$ is not a critical value of $f$ then it follows from the implicit function theorem that $M^a$ is a smooth manifold-with-boundary. The boundary $f^{-1}(a)$ is a smooth submanifold of $M$.

Non-Degenerate Critical Point

A critical point $p$ is called non-degenerate if and only if the matrix $(\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(p))$ is not singular.

Hessian Matrix

If $p$ is a critical point of $f$ we define a symmetric bilinear functional $f_{\ast \ast }$ on $T{M}_p$, called the Hessian of $f$ at $p$. If $v,w\in T{M}_p$ then $v$ and $w$ have extensions $\tilde{v}$ and $\tilde{w}$ to vector fields. We let $f_{\ast \ast }(v,w)={\tilde{v}}_p(\tilde{w}(f))$, where ${\tilde{v}}_p=v$. It’s symmetric because ${\tilde{v}}_p(\tilde{w}(f))-{\tilde{w}}_p(\tilde{v}(f)=[\tilde{v},\tilde{w}{]}_p(f)=0.$ If $(x^1,\dots ,x^n)$ is a local coordinate system and $v=\sum a_i\frac{\partial }{\partial x^i}{|}_p,w=\sum b_j\frac{\partial }{\partial x^j}{|}_p$, we can take $\tilde{w}=\sum b_j\frac{\partial }{\partial x^j}$ where $b_j$ denote a constant function. So the matrix $(\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(p))$ represents the bilinear function $f_{\ast \ast }$ with respect to the basis $\frac{\partial }{\partial x^1}{|}_p,\dots ,\frac{\partial }{\partial x^n}{|}_p$.

Index and Nullity

The index of a bilinear functional $H$, on a vector space $V$, is defined to be the maximal dimension of a subspace of $V$ on which $H$ is negative definite. The nullity is the dimension of the null-space, i.e. the subspace consisting of all $v\in V$ such that $H(v,w)=0$ for every $w\in V$.

Lemma

The $f$ be a $C^{\infty }$ function in a convex neighbourhood $V$ of $0$ in ${\mathbb{R}}^n$, with $f(0)=0$. Then $f(x_1,\dots ,x_n)={\sum }_{i=1}^n{x}_i{g}_i(x_1,\dots ,x_n)$ for some suitable $C^{\infty }$ functions $g_i$ defined in $V$, with $g_i=\frac{\partial }{\partial x_i}(0)$.

Proof $f(x_1,\dots ,x_n)={\int }_0^1\frac{\text{d}f(t{x}_1,\dots ,t{x}_n)}{\text{d}t}\text{d}t={\int }_0^1{\sum }_{i=1}^n\frac{\text{d}f}{\text{d}{x}_i}(t{x}_1,\dots ,t{x}_n)\cdot x_i\text{d}t,$ therefore let $g_i(x_1,\dots ,x_n)={\int }_0^1\frac{\text{d}f}{\text{d}{x}_i}(t{x}_1,\dots ,t{x}_n)\text{d}t.$

Morse Lemma

Let $p$ be a non-degenerate critical point for $f$. Then there is a local coordinate system $(y^1,\dots ,y^n)$ in a neighborhood $U$ of $p$ with $y^i(p)=0$ for all $i$ and such that the identity $f=f(p)-(y^1{)}^2-\cdots -(y^{\lambda }{)}^2+(y^{\lambda +1}{)}^2+\dots (y^n{)}^2,$ holds throughout $U$, where $\lambda$ is the index of $f$ at $p$.

Proof We first show that if there is any such expression for $f$, then $\lambda$ must be the index of $f$ at $p$. For any coordinate system $(z^1,\dots ,z^n)$, if $f(q)=f(p)-(z^1(q){)}^2-\cdots -(z^{\lambda }(q){)}^2+(z^{\lambda +1}(q){)}^2+\cdots +(z^n(q){)}^2$ then we have $\frac{{\partial }^{2}f}{\partial z^i\partial z^j}(p)=\{\begin{array}{ll}-2& \text{if }i=j\le \lambda ,\\ 2& \text{if }i=j>\lambda ,\\ 0& \text{otherwise},\end{array}$ which shows that the matrix representing $f_{}$ with respect to the basis ${\frac{\partial }{\partial z^1}|}_p,\dots ,{\frac{\partial }{\partial z^n}|}_p$ is

$\left(\begin{array}{cccccc}-2& & & & & \\ & \ddots & & & & \\ & & -2& & & \\ & & & 2& & \\ & & & & \ddots & \\ & & & & & 2\end{array}\right)$

Therefore there is a subspace of $T{M}_p$ of dimension $\lambda$ where $f_{}$ is negative definite, and a subspace $V$ of dimension $n-\lambda$ where $f_{}$ is positive definite. If there were a subspace of $T{M}_p$ of dimension greater than $\lambda$ on which $f_{}$ were negative definite then this subspace would intersect $V$, which is clearly impossible. Therefore $\lambda$ is the index of $f_{}$.

We now show that a suitable coordinate system $(y^1,\dots ,y^n)$ exists. Obviously we can assume that $p$ is the origin of ${\mathbb{R}}^n$ and that $f(p)=f(0)=0$. By previous lemma we can write $f(x_1,\dots ,x_n)={\sum }_{j=1}^n{x}_j{g}_j(x_1,\dots ,x_n)$ for $(x_1,\dots ,x_n)$ in some neighborhood of $0$. Since $0$ is assumed to be a critical point: $g_j(0)=\frac{\partial f}{\partial x^j}(0)=0.$ Therefore, we have $g_j(x_1,\dots ,x_n)={\sum }_{i=1}^n{x}_i{h}_{ij}(x_1,\dots ,x_n)$ for certain smooth functions $h_{ij}$. It follows that $f(x_1,\dots ,x_n)={\sum }_{i,j=1}^n{x}_i{x}_j{h}_{ij}(x_1,\dots ,x_n).$

We can assume that $h_{ij}=h_{ji}$, since we can write ${\bar{h}}_{ij}=\frac{1}{2}(h_{ij}+h_{ji})$, and then have ${\bar{h}}_{ij}={\bar{h}}_{ji}$ and $f=\sum x_i{x}_j{\bar{h}}_{ij}$. Moreover the matrix $(h_{ij}(0))$ is equal to $(\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(0))$, and hence is non-singular.

There is a non-singular transformation of the coordinate functions which gives us the desired expression for $f$, in a perhaps smaller neighborhood of $0$. To see this we just imitate the usual diagonalization proof for quadratic forms. (See for example, Birkhoff and MacLane, “A survey of modern algebra,” p. 271.) The key step can be described as follows.

Suppose by induction that there exist coordinates $u_1,\dots ,u_n$ in a neighborhood $U_1$ of $0$ so that

$f=\pm (u_1{)}^2\pm \cdots \pm (u_{r-1}{)}^2+{\sum }_{i,j\ge r}{u}_i{u}_j{H}_{ij}(u_1,\dots ,u_n)$

throughout $U_1$; where the matrices $(H_{ij}(u_1,\dots ,u_n))$ are symmetric. After a linear change in the last $n-r+1$ coordinates we may assume that $H_{rr}(0)\ne 0$. Let $g(u_1,\dots ,u_n)$ denote the square root of $|H_{rr}(u_1,\dots ,u_n)|$. This will be a smooth, non-zero function of $u_1,\dots ,u_n$ throughout some smaller neighborhood $U_2\subset U_1$ of $0$. Now introduce new coordinates $v_1,\dots ,v_n$ by

$v_i=u_i\text{for }i\ne r$

$v_r(u_1,\dots ,u_n)=g(u_1,\dots ,u_n)\left[u_r+{\sum }_{i>r}{u}_i{H}_{ir}(u_1,\dots ,u_n)/H_{rr}(u_1,\dots ,u_n)\right].$

It follows from the inverse function theorem that $v_1,\dots ,v_n$ will serve as coordinate functions within some sufficiently small neighborhood $U_3$ of $0$. It is easily verified that $f$ can be expressed as

$f={\sum }_{i\le r}\pm (v_i{)}^2+{\sum }_{i,j>r}{v}_i{v}_j{H}_{ij}^{\prime }(v_1,\dots ,v_n)$

Corollary

Non-degenerate critical points are isolated.