Let \(M\subset \mathbb{R}^3\) be an embedded surface. Prove the following:

1. The principal curvatures can be expressed in terms of the mean curvature and Gaussian curvature. That is, compute an explicit formula for \(\kappa_1\) and \(\kappa_2\) in terms of \(H\) and \(K.\)

2. If the second fundamental form vanishes identically, then \(M\) is a plane.

A surface \(M\subset \mathbb{R}^3\) is said to be ruled if there exists a family \(\{r_{\alpha}\}_{\alpha\in\mathbb{R}}\) of (open segments of) straight lines such that \(M=\bigcup_{\alpha\in\mathbb{R}}r_{\alpha}.\) Examples include the plane, helicoid, hyperbolic paraboloid, and hyperboloid of one sheet. Prove the following:

1. The cylinder is a ruled surface.

2. If \(M\) is a smoothly embedded ruled surface, then \(M\) does not contain elliptic points.

Consider \(M\subset\mathbb{R}^3\) is a smoothly embedded surface.

1. Suppose the shape operator is trace-free. What can you deduce about the determinant of the shape operator? Does this allow you to conclude whether \(M\) admits umbilical, planar, parabolic, elliptic, or hyperbolic points?

2. Suppose the shape operator is trace-free and nondegenerate. Does this allow you to conclude whether \(M\) admits umbilical, planar, parabolic, elliptic, or hyperbolic points?

3. For what points \(p\in M\) (e.g. umbilical, planar, parabolic, elliptic, and hyperbolic) does the normal curvature at \(p,\) viewed as a real-valued function on unit vectors in \(T_pM,\) admit a zero?

4. What can be said about the normal curvature at an umbilical point?