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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Summer 2020 • Question 1

Question

Math 6510 • Differentiable Manifolds Summer 2020 • Question 1

Let \(M\) be a smooth manifold and \(\gamma :S^1\to M\) be a smooth embedding. Show that there exists a smooth vector field \(X\) on \(M\) that does not vanish on \(\gamma (S^1)\) and which is tangent to \(\gamma (S^1)\) at each point. (The last condition means that for each point \(p=\gamma (t)\), the vector \(X_p\) is a multiple of \(\gamma '(t)\).)

Raw LaTeX

Let \(M\) be a smooth manifold and \(\gamma:S^1\to M\) be a smooth embedding.
Show that there exists a smooth vector field \(X\) on \(M\) that does not vanish on
\(\gamma(S^1)\) and which is tangent to \(\gamma(S^1)\) at each point. (The last
condition means that for each point \(p=\gamma(t)\), the vector \(X_p\) is a
multiple of \(\gamma'(t)\).)

Topic Tags

Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Tangent Bundles and Vector Bundles Vector Fields and Flows