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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2017 • Question 3

Question

Math 6510 • Differentiable Manifolds Summer 2017 • Question 3

Let \(M\) be a compact manifold with a vector field \(X\in \Gamma (TM)\) and resulting one-parameter group \(\{\theta _t^X\}_{t\in \mathbb {R}}\le \operatorname {Diff}(M)\). For any \(f\in \operatorname {Diff}(M)\), let \(f_*X\in \Gamma (TM)\) be the pushforward of \(X\). Show that the one-parameter group for \(f_*X\) is \(\{f\circ \theta _t^X\circ f^{-1}\}_{t\in \mathbb {R}}\).

Raw LaTeX

Let \(M\) be a compact manifold with a vector field \(X\in\Gamma(TM)\) and
resulting one-parameter group \(\{\theta_t^X\}_{t\in\R}\le\operatorname{Diff}(M)\).
For any \(f\in\operatorname{Diff}(M)\), let \(f_*X\in\Gamma(TM)\) be the pushforward
of \(X\). Show that the one-parameter group for \(f_*X\) is
\(\{f\circ\theta_t^X\circ f^{-1}\}_{t\in\R}\).

Topic Tags

Homotopy and Degree on Manifolds Left-Invariant Vector Fields Tangent Spaces and Differential Maps Vector Fields and Flows