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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2007 • Question 1

Question

Math 6510 • Differentiable Manifolds Winter 2007 • Question 1

Let \(S^n\) be the unit sphere in \(\mathbb {R}^{n+1}\) and \(h:\mathbb {R}^{n+1}\to \mathbb {R}\) the projection to the last coordinate. Prove that the restriction of \(h\) to \(S^n\) is a Morse function and find all critical points and their indices.

Raw LaTeX

Let \(S^n\) be the unit sphere in \(\mathbb{R}^{n+1}\) and
\(h:\mathbb{R}^{n+1}\to\mathbb{R}\) the projection to the last coordinate. Prove that the
restriction of \(h\) to \(S^n\) is a Morse function and find all critical points and their
indices.

Topic Tags

Morse Theory and Critical Points