Middle East Technical University Fall 2003 Math 541 Differential Topology

HOMEWORK I. EXERCISES TO THE

TOPIC DIFFERENTIAL STRUCTURES

Problem 1. Show that the following examples present smooth manifolds, determine their dimension and describe local coordinates in the charts of an atlas.

(1) a sphere in ℝ^{3},

(2) a set

3x + y − z + u^{2} = a
x + y + 2z + u = 0
2x + 2y − 3z + 2u = 0

(for which values a it is a manifold ?)

(3) SL(n, ℝ) ⊂ ℝ^{n^{2}} and SL(n, ℂ) ⊂ ℝ^{2n^{2}},

(4) O_{n} ⊂ ℝ^{n^{2}} and U_{n} ⊂ ℝ^{2n^{2}} (Hint: determine the rank of the map f: A → AA^{∗}, first at the point E),

(5) Stiefel manifold S_{n,k} consisting of orthonormal k-frames in ℝ^{n},

(6) subset in ℂ^{2} defined by the equations

z_{1}^{2} + z_{2}^{2} = a
|z_{1}|^{2} + |z_{2}|^{2} = 1

(for which a is a manifold ?)

Problem 2. Describe the charts and the transition functions in (1) ℝP^{n},

(2) S^{2} × S^{1}

Problem 3. Determine the dimensions of the Stiefel and the Grassmann manifolds S_{n,k} and G_{n,k}.

Problem 4. Write explicitly the coordinate change formulae for the tangent and the cotangent vector bundles in ℝP^{2}.

Problem 5. Prove that f: X → Y is smooth iff it transforms smooth functions into smooth, that is f^{∗} maps C^{∞}(Y) into C^{∞}(X).

Problem 6. Smooth functions in ℝP^{2} are even smooth functions in S^{2} (justify !). Prove that there exists a smooth embedding ℝP^{2} → ℝ^{4}, or in the other words, there exist four smooth functions f_{1}, …, f_{4} such that ∀x, y ∈ ℝP^{2}∃i : f_{i}(x) ≠ f_{i}(y).

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