Math 6510 • Differentiable Manifolds

Let the curve \((f(s),g(s))\) be parameterized by arclength in
\(\mathbb{R}^2\) (\(f>0\)). Consider the surface of revolution \(S\) in \(\mathbb{R}^3\)
given parametrically by
\[
  X(s,\theta)=(f(s)\cos\theta,f(s)\sin\theta,g(s)).
\]

\textbf{(a)} Compute the mean curvature and Gauss curvature for points on \(S\).

\textbf{(b)} Find necessary and sufficient conditions for the ``latitudes'' obtained
when \(s\) is constant to be geodesics.