Geometry Second Semester 2011/12 Homework 7

1. Find the velocity, acceleration and speed of each of the following parametrised curves α: R → R^{n+1}.

(a) n = 1 and α(t) = (t, t^{2}),

(b) n = 1 and α(t) = (cos t, sin t),

(c) n = 1 and α(t) = (cos 3t, sin 3t),

(d) n = 2 and α(t) = (cos t, sin t, t), and

(e) n = 3 and α(t) = (cos t, sin t, 2 cos t, 2 sin t).

2. Show that if α: I → R^{n+1} is a parametrised curve with constant speed, then α̈(t) ⊥ α̇(t) for all t ∈ I.

3. Let α: I → R^{n+1} be a parametrised curve such that α̇(t) ≠ 0 for all t ∈ I. Show that there exists a unit speed reparametrisation β of α. That is, show that there exists an interval J and a smooth surjective function h: J → I such that h′ > 0 and β = α ∘ h has unit speed. [Hint: Set h = s^{−1}, where s(t) = ∫_{t_{0}}^{t} ∥α̇(τ)∥dτ for some t_{0} ∈ I.]

4. Let X and Y be smooth vector fields along the parametrised curve α: I → R^{n+1} and let f: I → R be a smooth function along α. Verify the following identities, which we claimed to hold in class:

(a) (X + Y)˙ = Ẋ + Ẏ;

(b) (fX)˙ = f′X + fẊ; and

(c) (X · Y)′ = Ẋ · Y + X · Ẏ.

5. Let S be an n-surface in R^{n+1}, p ∈ S, v ∈ S_{p} and let α: I → S be the maximal geodesic in S passing through p with velocity v. Show that the maximal geodesic β in S with β(0) = p and β̇(0) = cv, with c ∈ R, is give by the formula β(t) = α(ct) for all t belonging to some interval Ĩ.

6. Let α: I → S be a geodesic on an n-surface S and let β = α ∘ h, where h: Ĩ → I is a smooth surjective function, be a reparametrisation of α. Show that β is a geodesic on S if and only if h(t) = at + b for some a, b ∈ R and all t ∈ Ĩ.

7. An n-surface in R^{n+1} is said to be geodesically complete if every maximal geodesic on S has domain R. Which of the following n-surfaces are geodesically complete?

(a) The n-sphere S^{n} ⊂ R^{n+1} (see Example 4.2).

(b) f^{−1}(1) ⊂ R^{n+1} where f: U → R has domain U = {(x_{1}, x_{2}, …, x_{n+1}) | x_{n+1} < 1} and is given by f(x_{1}, x_{2}, …, x_{n+1}) = x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n+1}^{2} for each (x_{1}, x_{2}, …, x_{n+1}) ∈ U. This is the n-sphere S^{n} \ {(0, …, 0, 1)} with the north pole deleted.

(c) f^{−1}(0) ⊂ R^{3} where f: U → R has domain U = {(x_{1}, x_{2}, x_{3}) | x_{3} > 0} and is given by f(x_{1}, x_{2}, x_{3}) = x_{1}^{2} +x_{2}^{2} −x_{3}^{2} for each (x_{1}, x_{2}, …, x_{n+1}) ∈ U. This is a cone with the vertex deleted.

(d) The 2-surface S in R^{3} which is the cylinder over the plane curve S^{1} (see Examples 4.2 and 4.5).