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# 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 be a neihgborhood of in . Let be a smooth function with and . We assume, that some second partial derivative at of is non zero. Then there exists a neihgborhood of and a system of smooth coordinate functions on with and a sign , such that for the expression is a smooth function of the coordinate functions .

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

Theorem 2   Let be a neihgborhood of in . Let be smooth function with and . We assume, that the Hessian at is invertible. Then there exists a neihgborhood of and a system of smooth coordinate functions on with and a system of signs , such that for we have

Proof of the separation of variables theorem. By assumption, the matrix of second derivatives at is non zero. So, there exist an invertible matrix such that the entry of the matrix is . Let be the linear system of coordinates on obtained by changing the standard coordinates with . Now we have that at the second partial derivative with respect to of equals . The restriction of the function to the intersection of with the -coordinate axis has at a non-degenerate local maximum or minimum, according to the sign of at . In particular, the restiction to the interval on is strictly convex or concave, if we choose to be sufficiently small. Having chosen , we may choose , such that for every with the restriction of to the interval 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 , where is implicitely given by

Since

for , we can express as a smooth function of the coordinates . Put

The functions build a smooth coordinate system in a neihgborhood of in . This finishes the proof, since we have