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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 17

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 17

Let \[ X(s,\theta )=(\varphi (s)\cos (\theta ),\varphi (s)\sin (\theta ),\gamma (s)) \] be a surface of revolution such that \(\varphi '(s)^2+\gamma '(s)^2=1\). Prove that the meridians \(\theta =\mathrm {const}\) are geodesics. What condition on the lattitude \(s=\mathrm {const}\) is necessary and sufficient to made it geodesic?

Raw LaTeX

Let
\[
  X(s,\theta)=(\varphi(s)\cos(\theta),\varphi(s)\sin(\theta),\gamma(s))
\]
be a surface of revolution such that \(\varphi'(s)^2+\gamma'(s)^2=1\). Prove that the
meridians \(\theta=\mathrm{const}\) are geodesics. What condition on the lattitude
\(s=\mathrm{const}\) is necessary and sufficient to made it geodesic?

Topic Tags

Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups