Geometry Second Semester 2010/11 Homework 6

1. Describe the spherical image when n = 1 and when n = 2 of the n-surface f^{−1}(0) oriented by ∇f/∥∇f∥, where f is defined by

(a) f(x_{1}, x_{2}, …, x_{n+1}) = x_{2}^{2} + ⋯ + x_{n+1}^{2} − 1 (the cylinder)

(b) f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n+1}^{2} − r^{2} for r > 0 fixed (the sphere)

(c) f(x_{1}, x_{2}, …, x_{n+1}) = −x_{1} + x_{2}^{2} + ⋯ + x_{n+1}^{2} (the paraboloid)

(d) f(x_{1}, x_{2}, …, x_{n+1}) = −(x_{1}^{2}/a^{2}) + x_{2}^{2} + ⋯ + x_{n+1}^{2} − 1 for a > 0 fixed (the 1-sheeted hyperboloid)

2. What happens to the spherical image of f^{−1}(0) in part (d) of Question 1 as a → ∞? What happens as a → 0?

3. Show that the spherical image of the n-surface with orientation n is the reflection through the origin of the spherical image of the same n-surface with orientation −n.

4. Let a = (a_{1}, a_{2}, …, a_{n+1}) ∈ R^{n+1} be non-zero. Show that the spherical image of an n-surface S is contained in the n-plane {(x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1} | a_{1}x_{1} + a_{2}x_{2} + ⋯ + a_{n+1}x_{n+1} = 0} if and only if for every p ∈ S there is an open interval I containing 0 such that p + ta ∈ S for all t ∈ I. [Hint: For the ‘only if’ part, apply Corollary 5.3 to the constant vector field X(q) = (q, a).]

5. Let S = f^{−1}(c) for a smooth function f: R^{n+1} → R such that ∇f(p) ≠ 0 for all p ∈ S. Suppose that α: R → R^{n+1} is a parametrised curve such that ∇f(α(t)) · α̇(t) ≠ 0 for all t ∈ R such that α(t) ∈ S (when such a condition holds, we say α is nowhere tangent to S).

(a) Show that at each pair of consecutive crossings of S by α, the direction of the orientation ∇f/∥∇f∥ on S reverses relative to the direction of α. That is, show that if α(t_{1}) ∈ S and α(t_{2}) ∈ S where t_{1} < t_{2} and α(t) ∉ S for t_{1} < t < t_{2}, then ∇f(α(t_{1})) · α̇(t_{1}) < 0 if and only if ∇f(α(t_{2})) · α̇(t_{2}) > 0.

(b) Show that if S is compact and lim_{t→−∞} ∥α(t)∥ = lim_{t→∞} ∥α(t)∥ = ∞ then α crosses S an even number of times.

6. Let S be a compact n-surface in R^{n+1}. A point p ∈ R^{n+1} \ S is said to be outside S if there exists a continuous map α: [0, ∞) → R^{n+1} \ S such that α(0) = p and lim_{t→∞} ∥α(t)∥ = ∞. Let 𝒪(S) denote the set points outside S.

(a) Show that if β: [a, b] → R^{n+1} \ S is continuous and β(a) ∈ 𝒪(S) then β(t) ∈ 𝒪(S) for all t ∈ [a, b].

(b) Show that 𝒪(S) is a connected open subset of R^{n+1}.