MTH3400 — GEOMETRY 1 — SPRING 1997

HOMEWORK #3

1. Let p : E → M be a vector bundle.

(i) Show there is a map of vector bundles (over M) f : M ×ℝ^{n} → E which is surjective on each fiber, if and only if there are sections s_{1}, …, s_{n} of E over M such that, for each x ∈ M, s_{1}(x), …, s_{n}(x) span the fiber p^{−1}(x).

(ii) Show there is a map of vector bundles (over M) f : M × ℝ^{n} → E which is injective on each fiber, if and only if there are sections s_{1}, …, s_{n} of E over M such that, for each x ∈ M, s_{1}(x), …, s_{n}(x) are linearly independent in the fiber p^{−1}(x).

(iii) Show that E is isomorphic to the trivial rank n bundle M × ℝ^{n} → M if and only if there exist sections s_{1}, …, s_{n} of E over M such that, for each x ∈ M, s_{1}(x), …, s_{n}(x) form a basis for the fiber p^{−1}(x).

2. (i) Let u = x^{2} − y^{3}, v = 3xy + y^{2} − x^{2}. For which (a, b) in ℝ^{2} is there a neighborhood

U of (a, b) such that (U, (u, v)) is a coordinate system? (ii) For which real numbers c is the locus y^{2} − x(x − 1)(x − c) = 0 a submanifold of ℝ^{2}?

3. Let ℝℙ^{2} = {x = (x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} | x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1}/x ∼ −x be the real projective space, with the standard differentiable structure. Let Φ : ℝℙ^{2} → ℝ^{5} be given by

Φ([x]) = (x_{1}^{2}, x_{2}^{2}, x_{1}x_{2}, x_{1}x_{3}, x_{2}x_{3})

Show that Φ is a well-defined, smooth embedding. Compute Φ_{∗} : T (ℝℙ^{2}) → T (ℝ^{5}).

4. (i) Prove that the group SO(3) of orthogonal 3 × 3 matrices with determinant 1 is an

embedded submanifold of the 9-dimensional Euclidean space of all 3 × 3 matrices. (ii) Prove that SO(3) with the differentiable structure described above is diffeomorphic

to the real projective space ℝℙ^{3} with the standard differentiable structure.

5. Describe the structure of differentiable manifold on the complex projective space ℂℙ^{n} by explicitly defining coordinate charts and calculating transition functions.

6. Let M be a differentiable manifold and f : M → M a diffeomorphism. Consider the direct product M × [0, 1] with the identification of pairs of points (f(x), 0) and (x, 1), for all x ∈ M. Let M_{f} = M × [0, 1]/ ∼ be the quotient space (called the mapping torus of f, or, the suspension construction on f.)

(i) Show that M_{f} possesses a natural structure of differentiable manifold. (ii) Show that M_{f} is a locally trivial fiber bundle with base S^{1} and fiber M. (iii) Apply the suspension construction to the following three cases:

(a) M = ℝ, f(x) = −x.

(b) M = S^{1}, f(z) = −z.

(c) M = S^{1}, f(z) = \overline{z}. Identify the resulting mapping tori. In which of the three cases is the bundle M_{f} → S^{1} trivial?