Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2014 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2014 • Question 4

Let \(U\subset \mathbb {R}^n\) be a connected open set with \(p,q\in U\). Show that there exists a diffeomorphism \(\phi \) of \(\mathbb {R}^n\) such that \(\phi (p)=q\) and \(\phi \) is 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 set with \(p,q\in U\). Show that there
exists a diffeomorphism \(\phi\) of \(\R^n\) such that \(\phi(p)=q\) and \(\phi\) is 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