INTRODUCTION TO MANIFOLDS — I

Supplementary problems

Matrix manifolds

♣ Problem 1. Let M = Mat_{n×n} ≃ ℝ^{n^{2}} be the set of all square matrices. Prove that the map

Ad_{C}: A ↦ C^{−1}AC, det C ≠ 0,

defines a diffeomorphism of the manifold M onto itself.

♣ Problem 2. The same question about the manifold N ⊂ M of matrices of determinant 1.

♣ Problem 3. The same question but for the map

M ∋ A ↦ B^{−1}AC,  det B, det C ≠ 0.  (1)

♣ Problem 4. Prove that for any two matrices of the same rank there exists a diffeomorphism M → M of the form (1) taking one into the other.

♣ Problem 5. Prove that

det(E + εB) = 1 + ε tr B + O(ε^{2}).

Using the previous problem, deduce the formula for the first order term in the expansion det(A + εB), when det A ≠ 0.

♣ Problem 6. Prove that the set of matrices M_{r} ⊂ M of the rank r ⩽ n is a smooth submanifold in M. Is this submanifold closed? Compute its dimension. (Answer: n^{2} − (n − r)^{2} = 2nr − r^{2}.)

Partition of unity

Everywhere below M stands for a smooth n-dimensional manifold.

♣ Problem 7. Construct a C^{∞}-smooth function f: ℝ → ℝ such that f(x) > 0 if x > 0 and f(x) = 0 when x ⩽ 0. Why such an example is impossible in the analytic category?

♣ Problem 8. Construct a function which will be positive only on the interval (0, 1) ⊂ ℝ, and identically zero outside.

♣ Problem 9. Construct a smooth nonnegative function which is equal to 1 on (−1, 1) and vanishes outside (−2, 2).

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♣ Problem 10. The same question about a function φ : ℝ^{n} → ℝ_{+}, equal to 1 for ∥x∥ < 1 and vanishing for ∥x∥ ⩾ 2.

♣ Problem 11. Prove that on a smooth manifold M^{n} for any two points x ≠ y there exists a nonnegative function which is identically equal to 1 near x and to zero near y.

♣ Problem 12. Prove that for any point x there is a neighborhood U, M ⊇ U ∋ x, diffeomorphic to ℝ^{n}.

♣ Problem 13. Construct a diffeomorphism ℝ^{n+1} ⊃ 𝕊^{n} − (North Pole) → ℝ^{n}. (Answer: stereographic projection.)

♣ Problem 14. Prove that for any point x ∈ M there exists a neighborhood U and a smooth map f: M → 𝕊^{n} which is a diffeomorphism between U and 𝕊^{n} − (North Pole).

♣ Problem 15. Prove that for any compact manifold M there exists an injective smooth map f: M → ℝ^{N} for a sufficiently big N, which has rank n everywhere on M (the Whitney embedding theorem in the weakest form).

♣ Problem 16. Prove that for any discrete set of points x_{1}, x_{2}, ⋯ ∈ M there exist a smooth function f: M → ℝ which has nondegenrate local minima at these points, f(x_{i}) = 0, and positive outside of the set.

   /black/users2/yakov/pub, filename 2.ppt

E-mail address: yakov@wisdom.weizmann.ac.il, mtwiener@weizmann.weizmann.ac.il