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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2016 • Question 4

Question

Math 6510 • Differentiable Manifolds Spring 2016 • Question 4

Let \(U\subset \mathbb {R}^n\) be a connected, open subset. For all \(p,q\in U\) show that there exists a diffeomorphism \(\phi :\mathbb {R}^n\to \mathbb {R}^n\) with \(\phi (p)=q\) and \(\phi \) the identity outside of \(U\). (Hint: First assume that \(U\) is convex with compact closure and find a vector field with support in \(U\) whose flow takes \(p\) to \(q\).)

Raw LaTeX

Let \(U\subset\R^n\) be a connected, open subset. For all \(p,q\in U\) show
that there exists a diffeomorphism \(\phi:\R^n\to\R^n\) with \(\phi(p)=q\) and
\(\phi\) the identity outside of \(U\). (Hint: First assume that \(U\) is convex with
compact closure and find a vector field with support in \(U\) whose flow takes
\(p\) to \(q\).)

Topic Tags

Diffeomorphisms and Isotopy Methods Left-Invariant Vector Fields Vector Fields and Flows