Geometry Second Semester 2010/11 Homework 5

1. Show that the unit sphere S^{n} defined in Example 4.2 is connected if n ≥ 1.

2. Show that if S is a connected n-surface in R^{n+1} and g: S → R is smooth and takes on only the values 1 and −1, then g is constant. [Hint: Let p ∈ S. For q ∈ S, let α: [a, b] → S be a continuous function such that α(a) = p and α(b) = q. Apply the Intermediate Value Theorem to the composition ——– f ∘ α g ∘ α.]

3. Find an example that demonstrates that Theorem 5.5 fails for S which are not connected.

4. Show that two orientations of the n-sphere S^{n}(r) of radius r, defined to be {(x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1} | x_{1}^{2} + x_{2}^{2} + ⋯ + x_{n+1}^{2} = r^{2}}, are given by n_{1}(p) = (p, p/r) and n_{2}(p) = (p, −p/r).

5. Consider the n-surface S = {(x_{1}, x_{2}, …, x_{n+1}) ∈ R^{n+1} | x_{n+1} = 0} = R^{n} × {0} (which can be identified with R^{n} in the obvious manner). Let n be the orientation on S defined by n(p) = (p, 0, 0, …, 0, 1) for each p ∈ R^{n+1}. Show that, with respect to this orientation,

(a) when n = 1, the positive tangent direction at p ∈ R is the direction (p, 1, 0),

(b) when n = 2, the positive θ-rotation in R_{p}^{2} is a anticlockwise rotation through the angle θ for each p ∈ R^{2}, and

(c) the ordered orthogonal basis {(p, 1, 0, 0, 0), (p, 0, 1, 0, 0), (p, 0, 0, 1, 0),[figure: (p, 0, 0, 0, 1)]} for R_{p}^{3} (with p ∈ R^{3}) is right-handed. [There should only be three elements in this basis, the fourth was a mistake in the first version.]

6. Let C be an oriented plane curve and let v be a non-zero vector tangent to the curve C at p ∈ C. Show that the basis {v} for C_{p} is consistent with the orientation on C if and only if the positive tangent direction at p is v/∥v∥. [Hint: Let θ denote the angle measured anticlockwise from (p, 1, 0) to the orientation n(p), so that n(p) = (p, cos θ, sin θ). Express both v and the positive tangent direction at p in terms of θ.]

7. Recall that the cross-product v × w of two vectors v = (p, v_{1}, v_{2}, v_{3}) and w = (p, w_{1}, w_{2}, w_{3}) in R_{p}^{3} is defined by

v × w = (p, v_{2}w_{3} − v_{3}w_{2}, v_{3}w_{1} − v_{1}w_{3}, v_{1}w_{2} − v_{2}w_{1}).

(a) Show that v ×w is orthogonal to both v and w, and that ∥v ×w∥ = ∥v∥∥w∥ sin θ, where θ = arccos (v · w/∥v∥∥w∥) is the angle between v and w.

(b) Show that if u = (p, u_{1}, u_{2}, u_{3}), then

u · (v × w) = v · (w × u) = w · (u × v) = | \begin{matrix}u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \\ w_{1} & w_{2} & w_{3} \end{matrix} | .

(c) Show that the only vector x such that u · x is equal to the determinant in (b) for all u ∈ R_{p}^{3} is x = v × w.

8. Let S be an oriented 3-surface in R^{4} and let p ∈ S. This question generalises the notion of the cross product in R_{p}^{3} to the tangent space S_{p} of a general oriented 3-surfaces S in R^{4}. Recall Question 5 shows us how R^{3} can be thought of as an 3-surface in R^{4}.

(a) Show that given vectors v = (p, v) and w = (p, w) in S_{p}, there is a unique vector, which we denote as v × w, belonging to S_{p} such that

u · (v × w) = det ( \begin{matrix}u \\ v \\ w \\ N(p) \end{matrix} )

for all u = (p, u) ∈ S_{p}, where n = (·, N(·)) is the orientation of S. The vector v × w is the cross product of v and w in S_{p}.

(b) Check that the cross product in S_{p} has the following properties. For u, v, w ∈ S_{p} and

c ∈ R we have: i. (v + w) × u = v × u + w × u; ii. v × (w + u) = v × w + v × u; iii. (cv) × w = c(v × w); iv. v × (cw) = c(v × w); v. v × w = −w × v; vi. u · (v × w) = v · (w × u) = w · (u × v); vii. v × w is orthogonal to both v and w; viii. u · (v × w) = 0 if and only if {u, v, w} is linearly —————– independent dependent; and ix. An ordered orthonormal basis {e_{1}, e_{2}, e_{3}} for S_{p} is right-handed if and only if e_{3} ·

(e_{1} × e_{2}) > 0.