Middle East Technical University Spring 1996 Math 505 Differentiable Manifolds

PROBLEM LIST 2

Partition of Unity, Vectors, Tensor Fields

Problem 1. (3 pts) Prove that any function f: F → ℝ defined on a closed subset F of a smooth manifold X can be extended to a smooth function f̄: X → ℝ, if it can be extended to a smooth function on some open set U ⊃ F.

Formulate and give an outline of the proof for similar statements

(1) if ℝ is replaced by an arbitrary smooth manifold,

(2) if the function f is replaced by a vector field,

(3) if the function is replaced by a 1-form,

(4) if the function is replaced by a tensor field of type (p, q).

Problem 2. (1 pts) Given a smooth map between manifolds f: X → Y which differential is identically zero prove that f is constant on the connected components of X.

Problem 3. (2 pts) 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^{i} denotes here a projective space over ℝ (or ℂ).

Problem 4. (2 pts) Construct an isomorphism T^{∗}(V ⊕ W) ≅ T^{∗}(V) ⊗ T^{∗}(W) for arbitrary vector spaces V and W (T^{∗}(…) denotes here the tensor algebra).

Construct an algebra isomorphism T^{∗}(ℝ) ≅ ℝ[[x]].

Problem 5. (3 pts) Let X, Y, Z denote vector fields on a smooth manifold M, and f ∈ C^{∞}(M).

(1) Prove that for any X, Y there exist and unique Z, such that for any f Z(f) = X(Y (f)) − Y (X(f)) (that is the Lie product, Z = [X, Y]).

(2) Prove skew-symmetry of the Lie product that is [X, Y] = −[Y, X].

(3) Prove the “Leibnitz rule”, [X, fY] = f[X, Y] + Y (f)X.

(4) Prove Jacobi identity that is [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] for any X, Y, Z.

(5) Given coordinates (X^{i}), (Y^{i}) of X and Y in some chart determine the

coordinates (Z^{i}) of their Lie product, Z. Think yourself, do not look for a solution in books !

Problem 6. (3 pts) Consider a tensor field T of type (2, 1) on ℝp^{2} which has coordinates T_{k}^{i,j} = (i − 1)u_{1} + (j − k)u_{2}, i, j, k ∈ {1, 2}, in the chart (U_{x}, ϕ_{x}), U_{x} = \{[x : y : z] ∈ ℝp^{2} | x ≠ 0\}, ϕ_{x}([x : y : z]) = (u_{1}, u_{2}), u_{1} = \frac{y}{x}, u_{2} = \frac{z}{x}.

(1) Is T symmetric ? Skew-symmetric ?

(2) Determine T_{1}^{1,2} in the chart (U_{y}, ϕ_{y}), U_{y} = {[x : y : z]} ∈ ℝp^{2} | y ≠ 0}, ϕ_{y}([x : y : z]) = (v_{1}, v_{2}), v_{1} = \frac{x}{y}, v_{2} = \frac{z}{y}.