Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2008 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2008 • Question 6

Show that the group of diffeomorphisms of a compact manifold \(M\) acts transitively on \(M\). That is, if \(p,q\in M\) there exists a diffeomorphism \(\phi :M\to M\) such that \(\phi (p)=q\). Hint: First assume \(p\) and \(q\) are in the same coordinate neighborhood \(U\) and construct \(\phi \) as the flow of a vector field supported in \(U\).

Raw LaTeX

Show that the group of diffeomorphisms of a compact manifold \(M\) acts
transitively on \(M\). That is, if \(p,q\in M\) there exists a diffeomorphism
\(\phi:M\to M\) such that \(\phi(p)=q\). Hint: First assume \(p\) and \(q\) are in the same
coordinate neighborhood \(U\) and construct \(\phi\) as the flow of a vector field
supported in \(U\).

Topic Tags

Compactly Supported Forms and Integration Diffeomorphisms and Isotopy Methods Left-Invariant Vector Fields Vector Fields and Flows