If I have a first fundamental form $ \mathrm{d}u^2+\cos^2 u \mathrm{d}v^2$, I am trying to show that the second fundamental form cannot equal $f(u,v)\mathrm{d}v^2$ for a smooth function $f(u,v)$.
I have calculated the Christoffel symbols and arrived at the following set of equations:
$\sigma_{uu} = L \cdot \mathbf{N}$
$\sigma_{uv} = -\tan u \cdot \sigma_v + M \cdot \mathbf{N}$
$\sigma_{vv} = \tan u \cdot \sigma_u + N \cdot \mathbf{N}$
I suppose I was expecting to arrive at a contradiction, but I can't really see where any problem occurs with the fundamental form only having a coefficient for $N$
I know that both fundamental forms of a sphere are $\mathrm{d} u^2+ \cos^2 u \mathrm{d}v^2$ but does having such a first fundamental form necessarily imply that the surface is a sphere?