MTH3400 — GEOMETRY 1 — SPRING 1997

HOMEWORK #2

1. Let p ∈ S^{n} ⊂ ℝ^{n+1} and consider the tangent space at p,

T_{p}(S^{n}) = {α′(0) | α : (−ϵ, ϵ) → ℝ^{n+1} is smooth, α(0) = p, im(α) ⊂ S^{n}}.

Prove that T_{p}(S^{n}) is the linear subspace of ℝ^{n+1} = T_{p}(ℝ^{n+1}) consisting of all vectors v ∈ ℝ^{n+1} such that v ⊥ p.

2. Let U ⊂ ℝ^{n}, V ⊂ ℝ^{m} be open subsets, and f : U → V a smooth map.

(i) Suppose there is a smooth map g : V → ℝ of rank 1 such that g ∘ f ≡ 0 on U. Show that the rank r of the Jacobian matrix Jf is everywhere < m.

(ii) Suppose that f does have constant rank r < m on U. Show that, on some open subset W ⊃ f(U), there is a smooth map g : W → ℝ of rank 1 such that g ∘ f ≡ 0 on U.

3. (See also Problem 33, p. 83, in Spivak’s book.) Consider the orthogonal group

O(n) = {A ∈ GL(n) | A^{⊤} · A = I},

where A^{⊤} denotes the transpose of A, and I is the identity n × n matrix. Consider the map Φ : GL(n) → GL(n), defined by Φ(A) = A^{⊤} · A. Prove:

(i) Show that O(n) is a compact subspace of GL(n). (ii) Φ is smooth. (iii) Relative to the standard identification T_{I}(GL(n)) = Mat(n), the differential d_{I}Φ :

T_{I}(GL(n)) → T_{I}(GL(n)) is given by:

d_{I}Φ(A) = A^{⊤} + A.

(iv) The map Φ has constant rank \frac{n(n+1)}{2}.

(v) using the above, conclude that the orthogonal group O(n) ⊂ GL(n) is a smooth, compact submanifold of dimension \frac{n(n−1)}{2}.

(vi) Show that the vector subspace T_{I}(O(n)) ⊂ Mat(n) is the space of skew-symmetric matrices.

4. Problem 34, pp. 83-84, in Spivak’s book.