Geometry Second Semester 2011/12 Homework 8

1. Compute ∇_{v}f(p) where f: R^{n+1} → R, v ∈ R_{p}^{n+1} and p ∈ R^{n+1} are given by

(a) n = 1, v = (1, 0, 2, 1) and f(x_{1}, x_{2}) = 2x_{1}^{2} + 3x_{2}^{2},

(b) n = 1, v = (1, 1, cos θ, sin θ) and f(x_{1}, x_{2}) = x_{1}^{2} − x_{2}^{2},

(c) n = 2, v = (1, 1, 1, a, b, c) and f(x_{1}, x_{2}, x_{3}) = x_{1}x_{2}x_{3}^{2}, and

(d) n ∈ N, v = (p, v) and f(q) = q · q.

2. Let U be an open set in R^{n+1} and let f: U → R be a smooth function. Show that if e_{i} = (0, …, 1, …, 0) ∈ R^{n+1} has a 1 in the i-th coordinate and 0 in all others, and e_{i} = (p, e_{i}) ∈ R_{p}^{n+1} for i = 1, 2, …, n + 1, then ∇_{e_{i}}f(p) = ∂_{i}f(p).

3. Compute ∇_{v}X(p) where v ∈ R_{p}^{n+1}, p ∈ R^{n+1} and X are given by

(a) n = 1, v = (1, 0, 0, 1) and X(x_{1}, x_{2}) = (x_{1}, x_{2}, x_{1}x_{2}, x_{2}^{2}),

(b) n = 1, v = (cos θ, sin θ, − sin θ, cos θ) and X(x_{1}, x_{2}) = (x_{1}, x_{2}, −x_{2}, x_{1}), and

(c) n ∈ N, v = (0, 0, …, 0, 1, 1, …, 1) and X(q) = (q, 2q).

4. Let p ∈ S, v ∈ S_{p}, and X and Y be smooth vector fields on S. Verify that differentiation of vector fields has the following properties.

(a) ∇_{v}(X + Y)(p) = ∇_{v}X(p) + ∇_{v}Y(p),

(b) ∇_{v}(fX)(p) = (∇_{v}f(p))X(p) + f(p)(∇_{v}X(p)), and

(c) ∇_{v}(X · Y)(p) = (∇_{v}X(p)) · Y(p) + X(p) · (∇_{v}Y(p)).

5. A smooth tangent vector field X on an n-surface S is said to be a geodesic vector field, or geodesic flow, if all integral curves of X are geodesics of S.

(a) Show that a smooth tangent vector field X on S is a geodesic field if and only if ∇_{X(p)}X(p) ⊥ S_{p} for all p ∈ S.

(b) Describe a geodesic flow on a 2-surface of revolution in R^{3}.

6. Choose an orientation and compute the Weingarten map for

(a) the hyperplane {(x_{1}, x_{2}, …, x_{n+1}) | a_{1}x_{1} + a_{2}x_{2} + ⋯ + a_{n+1}x_{n+1} = b} ⊂ R^{n+1} (with (a_{1}, a_{2}, …, a_{n+1}) ∈ R^{n+1} \ 0 and b ∈ R), and

(b) the cylinder {(x_{1}, x_{2}, x_{3}) | x_{2}^{2} + x_{3}^{2} = a^{2}} ∈ R^{3} (a ≠ 0).

7. Let S = f^{−1}(c) be an n-surface in R^{n+1} oriented by ∇f/∥∇f∥. Suppose that p ∈ S is such that ∇f(p)/∥∇f(p)∥ = e_{n+1}, where e_{i} are as in Question 2 (i = 1, 2, …, n + 1). Show that the matrix that corresponds to L_{p} with respect to the basis {e_{1}, e_{2}, …, e_{n}} for S_{p} is

(−\frac{1}{∥∇f(p)∥}∂_{i,j}^{2}f(p))_{i,j} .

8. Let S be an n-surface in R^{n+1} oriented by the unit normal vector field n. Suppose that X and Y are smooth tangent vector fields on S.

(a) Show that

∇_{X(p)}Y(p) · n(p) = ∇_{Y(p)}X(p) · n(p)

for all p ∈ S. [Hint: Show that both sides are equal to L_{p}(X(p)) · Y(p).]

(b) Conclude that the vector field [X, Y] defined on S by [X, Y](p) = ∇_{X(p)}Y(p)−∇_{Y(p)}X(p) is everywhere tangent to S. ([X, Y] is called the Lie bracket of the vector fields X and Y.)