Middle East Technical University Fall 1998 Math 541 Differentiable Topology

PROBLEM LIST 1

Differentiable Manifolds, Differentiable Maps and Submanifolds

Problem 1. Assume that f: ℝ → ℝ is a homepmorphism and D_{0} is the standard smooth structure on ℝ.

(1) Construct a smooth structure, D_{1}, on ℝ, so that f: (ℝ, D_{0}) → (ℝ, D_{1}) is a diffeomorphism.

(2) Construct another smooth structure, D_{2}, on ℝ, so that f: (ℝ, D_{0}) → (ℝ, D_{2}) is differential map, but not a diffeomorphism.

Problem 2. Prove that the groups GL(n), SL(n), O(n), SO(n) are differential submanifolds in ℝ^{n×n}, considered as the space of all n × n matrices. Determine the dimentions of these submanifolds.

Problem 3. Check, if the closure Cl(A) ⊂ ℝP^{2} of the curve A ⊂ ℝ^{2} defined by the equation y^{2} = x^{3} − x is a smooth curve ?

Answer the same question for the complexification of the curve A.

Problem 4. Check that (1) a polynomial, p(z), defines a smooth mapping ℂP^{2} → ℂP^{2}.

(2) a ratio, \frac{p(z)}{q(x)} of polynomials define a smooth mapping ℂP^{2} → ℂP^{2}.

Problem 5. Consider function f(x, y) = \frac{x^{2}+y+1}{x^{2}+y^{2}−1}. Extend f to a function ϕ: ℝP^{2} → ℝP^{1}, so that f becomes a presentation of ϕ in the standard charts on ℝP^{2} and ℝP^{1}. Check, if ϕ is well defined on the whole ℝP^{2} ?

Problem 6. Denote by G_{n,k} the set of k-dimensional subspaces in ℝ^{n}. (1) Define a manifold structure in G_{n,k}.

(2) Define a differential structure on G_{n,k}. Remark: G_{n,k} is called a Grassman manifold.

Problem 7. Assume that τ: M → M is a smooth fixed point free involution (i.e. τ ∘ τ = id) on a manifold M. Consider the quotient space N = M/τ.

(1) Prove that N is a manifold.

(2) Provide N with a differential structure so that the quotient map f: M → N is smooth.

PROBLEM LIST 1

Tangent and normal vector bundles, immersion, submersion, embedding

Problem 8. Check that the following vector bundles are trivial (1) the tangent bundle to S^{1};

(2) the normal bundle to S^{n−1} ⊂ ℝ^{n};

(3) the normal bundle to a smooth closed curve in R^{2};

(4) the normal bundle to the boundary of any smooth solid body (i.e., a 3dimensional compact submanifold with a boundary) in R^{3}.

(5) (more difficult) the normal bundle to a smooth closed curve in R^{3}, and more generally, in R^{n}.

Give an example of immersed curve in R^{2} with a non-trivial normal bundle.

Problem 9. Prove that the Veroneze map f: ℙ^{2} → ℙ^{5}, f([x : y : z]) = [x^{2} : y^{2} : z^{2} : xy : xz : yz]), is an embedding. ℙ^{k} denotes here a projective space over ℝ or ℂ.

More excercises

Problem. Denote by S_{n,k}, n ⩾ k, the set of (n×k)-matrices having rank k. Prove that S_{n,k} ⊂ ℝ^{nk} is a smooth submanifold.

Problem. Given differentiable manifolds, X_{1}, X_{2}, (1) define a differential structure on their product X_{1} × X_{2};

(2) prove that the projections, p_{i}: X_{1} × X_{2} → X_{i}, i = 1, 2, and the slice maps, X_{i} → X_{1} × X_{2}, are differentiable;

(3) prove that a map f: Z → X_{1}×X_{2} is differentiable if and only if the products p_{i} ∘ f, i = 1, 2 are differentiable;

(4) prove that for any differentiable map g: X_{1} → X_{2} its graph, Γ_{g} is a differential submanifold of X_{1} × X_{2}.

Problem. Consider the tangent vector v ∈ T ℝP^{2} at the point p = [2 : 1 : −1] with the coordinates (3, 1) in one of the standard charts, (U_{1}, ϕ_{1}). Find the coordinates of v in the other standard charts.

Problem. Prove that a continuous map f: ℝP^{n} → M is smooth if and only if the product f ∘ p: S^{n} → M with the projection S^{n} → ℝP^{n} is smooth.

Problem. Prove that a pair of points, x, y, in a connected smooth manifold can be connected by a smooth simple curve.