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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 1

Let \(S^1\) denote the unit circle in \(\mathbb {C}\) and define \(f:S^1\times S^1\to \mathbb {R}\) by \(f(z,w)=\operatorname {Re}(z)+\operatorname {Re}(w)\). Show that \(f\) is a Morse function and compute the indices of the critical points of \(f\).

Raw LaTeX

Let \(S^1\) denote the unit circle in \(\C\) and define
\(f:S^1\times S^1\to\R\) by \(f(z,w)=\Reop(z)+\Reop(w)\). Show that \(f\) is a Morse
function and compute the indices of the critical points of \(f\).

Topic Tags

Morse Theory and Critical Points