The Morse Lemma

Norbert A'Campo

The following separation of variables theorem is a key step in the proof of the Morse Lemma. In this exposition the term smooth stands for infinitely many times differentiable.

Theorem 1   Let $U$ be a neihgborhood of $0$ in ${\bf R}^n$. Let $f:U \to {\bf R}$ be a smooth function with $f(0)=0$ and $(Df)_0=0$. We assume, that some second partial derivative at $0$ of $f$ is non zero. Then there exists a neihgborhood $U_0 \subset U$ of $0$ and a system of smooth coordinate functions $x_1,x_2, \dots ,x_n$ on $U_0$ with $x_i(0)=0$ and a sign $\sigma=\pm 1$, such that for $p\in U_0$ the expression $g(x(p)):=f(x(p))-\sigma x_1(p)^2$ is a smooth function of the coordinate functions $x_2,x_3, \dots ,x_n$.

The Morse Lemma is the following theorem, which follows from the above reduction with separation of variables.

Theorem 2   Let $U$ be a neihgborhood of $0$ in ${\bf R}^n$. Let $f:U \to {\bf R}$ be smooth function with $f(0)=0$ and $(Df)_0=0$. We assume, that the Hessian $H_f:{\bf R}^n \to ({\bf R}^n)^*$ at $0$ is invertible. Then there exists a neihgborhood $U_0 \subset U$ of $0$ and a system of smooth coordinate functions $x_1,x_2, \dots ,x_n$ on $U_0$ with $x_i(O)=0$ and a system of signs $\sigma_i=\pm 1, i=1..n$, such that for $p\in U_0$ we have

$$f(x(p)=\sigma_1 x_1(p)^2+\sigma_2 x_2(p)^2+ \dots +\sigma_n x_n(p)^2.$$

Proof of the separation of variables theorem. By assumption, the $n \times n$ matrix $M$ of second derivatives at $0$ is non zero. So, there exist an invertible $n \times n$ matrix $P$ such that the entry $(1,1)$ of the matrix $^tP \circ M \circ P$ is $\pm 1$. Let $x=(x_1,x_2, \dots ,x_n)$ be the linear system of coordinates on ${\bf R}^n$ obtained by changing the standard coordinates with $P$. Now we have that at $0$ the second partial derivative with respect to $x_1$ of $f$ equals $\sigma_1=\pm 1$. The restriction of the function $f$ to the intersection $X_1$ of $U$ with the $x_1$-coordinate axis has at $x_1=0$ a non-degenerate local maximum or minimum, according to the sign of ${\partial^2}/{\partial{x_1}^2}f$ at $0$. In particular, the restiction to the interval $-\epsilon \leq x_1 \leq \epsilon$ on $X_1$ is strictly convex or concave, if we choose $\epsilon>0$ to be sufficiently small. Having chosen $\epsilon$, we may choose $\eta > 0$, such that for every $x_2,x_3, \dots ,x_n$ with $\vert x_i\vert \leq \eta,i=2 \dots n,$ the restriction of $f$ to the interval $-\epsilon \leq x_1 \leq \epsilon$ is defined and is strictly convex or concave with a non-degenerate critical point in the interior. The position of this critical point is at $x_1=x_{1,critical}$, where $x_{1,critical}$ is implicitely given by

$$g(x_{1,critical},x_2, \dots ,x_n):={\partial}/{\partial{x_1}}f(x_{1,critical},x_2, \dots ,x_n)=0$$

Since

$${\partial}/{\partial{x_1}}g={\partial^2}/{\partial{x_1}^2}f\not=0$$

for $\vert x_1\vert \leq \epsilon, \vert x_i\vert \leq \eta, i=2 \dots n,$, we can express $x_{1,critical}=x_{1,critical}(x_2,x_3, \dots ,x_n)$ as a smooth function of the coordinates $x_2,x_3, \dots ,x_n$. Put

$$y_1=sign(x_1-x_{1,critical})\sqrt{\vert f(x_1,x_2, \dots ,x_n)-f(x_{1,critical},x_2, \dots ,x_n)\vert})$$

The functions $y_1,x_2, \dots ,x_n$ build a smooth coordinate system in a neihgborhood $U_0$ of $0$ in $U$. This finishes the proof, since we have

$$f(y_1,x_2, \dots ,x_n)-\sigma_1 y_1^2=f(x_{1,critical}(x_2,x_3, \dots ,x_n),x_2, \dots ,x_n)$$

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