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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2003 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2003 • Question 2

Let \(f:X\to Y\) be a smooth map between smooth manifolds that is one-to-one on a compact submanifold \(Z\) of \(X\). Suppose that for all \(x\in Z\) \[ df_x:T_x(X)\to T_{f(x)}(Y) \] is an isomorphism. Prove that there is a neighborhood \(U\) of \(Z\) such that \[ f|_U:U\to f(U) \] is a diffeomorphism.

Raw LaTeX

Let \(f:X\to Y\) be a smooth map between smooth manifolds that is one-to-one on a
compact submanifold \(Z\) of \(X\). Suppose that for all \(x\in Z\)
\[
  df_x:T_x(X)\to T_{f(x)}(Y)
\]
is an isomorphism. Prove that there is a neighborhood \(U\) of \(Z\) such that
\[
  f|_U:U\to f(U)
\]
is a diffeomorphism.

Topic Tags

Diffeomorphisms and Isotopy Methods Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds