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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2022 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2022 • Question 3

Let \(X\) and \(Y\) be smooth vector fields on smooth manifolds \(M\) and \(N\) that are \(f\)-related by a smooth map \(f:M\to N\). If \(\psi _t\) is the flow generated by \(X\) and \(\phi _t\) is the flow generated by \(Y\) show that

\[ f\circ \psi _t=\phi _t\circ f. \]

Raw LaTeX

Let \(X\) and \(Y\) be smooth vector fields on smooth manifolds \(M\) and
\(N\) that are \(f\)-related by a smooth map \(f:M\to N\). If \(\psi_t\) is the
flow generated by \(X\) and \(\phi_t\) is the flow generated by \(Y\) show that

\[
f\circ\psi_t=\phi_t\circ f.
\]

Topic Tags

Homotopy and Degree on Manifolds Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Vector Fields and Flows