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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2008 • Question 5

Question

Math 6510 • Differentiable Manifolds Spring 2008 • Question 5

Let \(M\) be a Riemannian manifold. Define \(d:M\times M\to \mathbb {R}\) by \[ d(x,y)=\inf \int _0^1 |\gamma '(t)|\,dt \] where the infimum is taken over piecewise \(C^1\) paths \(\gamma \) in \(M\) with \(\gamma (0)=x\) and \(\gamma (1)=y\). Prove that \(d\) is a metric on \(M\).

Raw LaTeX

Let \(M\) be a Riemannian manifold. Define \(d:M\times M\to\R\) by
\[
  d(x,y)=\inf\int_0^1 |\gamma'(t)|\,dt
\]
where the infimum is taken over piecewise \(C^1\) paths \(\gamma\) in \(M\) with
\(\gamma(0)=x\) and \(\gamma(1)=y\). Prove that \(d\) is a metric on \(M\).

Topic Tags

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