Does this surface exist

Question

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? This look somehow very similar to a sphere, but I could not find an explicit representation of this surface. I.e. maybe anybody of you could prove or disprove the existence of this surface as a an immersion $f: U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$?

Answer 1

Look at the Fundamental Theorem of Surface Theory. Locally, the Codazzi and Gauss equations are necessary and sufficient to get such an immersion. Here they fail, as I leave it to you to check.