Gaussian Curvature, Normal Curvature, and the Shape Operator

[i1100]

Homework Statement

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

Homework 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 $$K= det S$$.

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.

 

[mighty2000]

Thank you for your question. Let me first clarify the geometric information that can be deduced from each of the conditions on S at point p.

a) If S(u_1) \bullet u_2 = 0, this means that the shape operator S is perpendicular to the tangent vector u_2. This could indicate that the surface M has a point of inflection at p, where the tangent plane is perpendicular to the surface.

b) If S(u_1) + S(u_2) = 0, this means that the shape operator S is orthogonal to both u_1 and u_2. This could indicate that the surface M is a minimal surface at p, where the mean curvature is zero.

c) If S(u_1) \times S(u_2) = 0, this means that the cross product of the shape operators S(u_1) and S(u_2) is zero. This could indicate that the surface M is either a cylinder or a cone at p, where the principal curvatures are constant and perpendicular to each other.

d) If S(u_1) \bullet S(u_2) = 0, this means that the shape operators S(u_1) and S(u_2) are perpendicular to each other. This could indicate that the surface M has a point of inflection at p, where the principal curvatures are equal in magnitude but opposite in sign.

Now, in regards to your solutions:

a) Your solution is correct. If S(u_1) = au_1, then the surface M is indeed a sphere at p, where a is the radius of the sphere. This can also be seen from the fact that the shape operator S is scalar multiplication, which means that the principal curvatures are equal in magnitude and opposite in sign.

c) Your solution is also correct. Since K(p) = 0, this means that the surface M is either a cylinder or a cone at p. As you mentioned, the specific type of surface depends on the principal curvatures, which we do not have information about. However, we can say that the principal curvatures are parallel to each other at p.

b) and d) These conditions do not provide enough information to deduce any specific geometric information about the surface M at p. However, we can say that in both cases, the shape operator S is orthogonal to both