Second fundamental form question.

Question

Honestly, I dont have any idea for that question I posted. Please can someone help me solving the question. Thank you.

Answer 1

The second fundamental form of a surface, parametrized as $\pmb r(u,v)$ with normal $\pmb n(u,v)$,is the expression $$Ldu^2 + 2M dudv + N dv^2$$ where $L = \pmb r_{uu}\cdot\pmb n,\, M = \pmb r_{uv} \cdot \pmb n, N = \pmb r_{vv} \cdot \pmb n$.

If the second fundamental form vanishes, $$\pmb r_{u} \cdot \pmb n_u = 0 = \pmb r_v \cdot \pmb n_u = \pmb r_u \cdot \pmb n_v = \pmb r_v \cdot \pmb n_v$$ so that $$\pmb n_u = \pmb n_v = 0$$ since $\pmb n_u, \pmb n_v$ are orthogonal to $\pmb n$ and hence linear combinations of $\pmb r_u, \pmb r_v$. Thus $\pmb n$ is constant. This means $$(\pmb r \cdot \pmb n)_u = \pmb r_u \cdot \pmb n = 0,\; (\pmb r \cdot \pmb n)_v = \pmb r_v \cdot \pmb n = 0$$ and so $$\pmb r \cdot \pmb n = \text{const}$$ which is the equation of a plane.