Math Help - Normal of a surface

1. Normal of a surface

The normal line at a point p of a surface S is the straight line passing through p parallel to the normal N of S at p. Prove:

If the normal lines are all parallel, then S is an open subset of a plane

If all the normal lines pass through some fixed point, then S is an open subset of a sphere

Any help will be appreciated

2. Re: Normal of a surface

For the first question, note that the normal to the surface must be a constant vector, and so the coefficients of the second fundamental form are zero,
which in turn implies that the Gaussian curvature is zero.

For the second question, express the point in the basis ${X_u,X_v,N}$ and differentiate to show that the fundamental forms are a constant multiple of each other.
This shows that the Gaussian curvature is constant and non-zero on $S$.