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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2008 • Question 7

Question

Math 6510 • Differentiable Manifolds Spring 2008 • Question 7

Let \(T=S^1\times S^1\subset \mathbb {C}\times \mathbb {C}\) be the torus. Let \(X\) be the vector field on \(T\) defined by \(X(z_1,z_2)=(iz_1,aiz_2)\), where \(a\in \mathbb {R}\). Compute the flow \(\Phi (t,p)\), \(p=(p_1,p_2)\) of \(X\). For \(a\) rational, show that \(t\to \Phi (t,p)\) is periodic.

Raw LaTeX

Let \(T=S^1\times S^1\subset\C\times\C\) be the torus. Let \(X\) be the vector
field on \(T\) defined by \(X(z_1,z_2)=(iz_1,aiz_2)\), where \(a\in\R\). Compute the
flow \(\Phi(t,p)\), \(p=(p_1,p_2)\) of \(X\). For \(a\) rational, show that
\(t\to\Phi(t,p)\) is periodic.

Topic Tags

Left-Invariant Vector Fields Surface Topology and Surface Groups Vector Fields and Flows