Homework Help: Gaussian Curvature, Normal Curvature, and the Shape Operator

1. Mar 30, 2010

i1100

1. The problem statement, all variables and given/known data
Let $$u_1, u_2$$ be orthonormal tangent vectors at a point p of M. What geometric information can be deduced from each of the following conditions on S at p?

a) $$S(u_1) \bullet u_2 = 0$$

b) $$S(u_1) + S(u_2) = 0$$

c) $$S(u_1) \times S(u_2) = 0$$

d) $$S(u_1) \bullet S(u_2) = 0$$

2. Relevant equations
If v and w are linearly independent tangent vectors at a point p of M, then $$S(v) \times S(w) = K(p)v \times w$$, where $$K= det S$$.

3. The attempt at a solution

a) Since $$u_1, u_2$$ form a basis to $$T_p (M)$$, we can write $$S(u_1)=au_1 + bu_2$$. Then $$(au_1 + bu_2) \bullet u_2 = 0$$. Since the dot product is linear we can write $$au_1 \bullet u_2 + bu_2 \bullet u_2 = 0 = bu_2 \bullet u_2 = 0 = b$$. Hence $$S(u_1)=au_1$$, so the shape operator is just scalar multiplication. Does this mean M is a sphere?

c) By the given formula, we know that $$K(p) = 0$$ since $$u_1 \times u_2 \neq 0$$. But when $$K(p)=0$$, there are two cases, depending on the principal curvature, which I don't have any information about.

I don't have any information on b) or d). Thanks for any input.