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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2012 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 2012 • Question 10

Let \(M\) be a smooth compact manifold and let \(X\) be a smooth vector field with a corresponding 1-parameter flow group \(\{\theta _t^X:M\to M\}_{t\in \mathbb {R}}\). Let \(f\) be a diffeomorphism of \(M\) and let \(f_*X\) be its pushforward. Show that the flow group for \(f_*X\) is \(f\circ \theta _t^X\circ f^{-1}\).

Raw LaTeX

Let \(M\) be a smooth compact manifold and let \(X\) be a smooth vector field with
a corresponding 1-parameter flow group \(\{\theta_t^X:M\to M\}_{t\in\R}\). Let \(f\)
be a diffeomorphism of \(M\) and let \(f_*X\) be its pushforward. Show that the flow
group for \(f_*X\) is \(f\circ\theta_t^X\circ f^{-1}\).

Topic Tags

Diffeomorphisms and Isotopy Methods Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Vector Fields and Flows