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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2015 • Question 5

Question

Math 6510 • Differentiable Manifolds Spring 2015 • Question 5

Let \(f:M\to N\) be a differentiable map and \(V\) and \(W\) vector fields on \(M\) and \(N\), respectively, with \(f_*(x)V(x)=W(f(x))\) for all \(x\in M\). Assume that \(\phi _t\) and \(\psi _t\) are the respective flows of \(V\) and \(W\) and show that \(f\circ \phi _t=\psi _t\circ f\).

Raw LaTeX

Let \(f:M\to N\) be a differentiable map and \(V\) and \(W\) vector fields on \(M\)
and \(N\), respectively, with \(f_*(x)V(x)=W(f(x))\) for all \(x\in M\). Assume that
\(\phi_t\) and \(\psi_t\) are the respective flows of \(V\) and \(W\) and show that
\(f\circ\phi_t=\psi_t\circ f\).

Topic Tags

Left-Invariant Vector Fields Vector Fields and Flows