Differential Geometry: Final Exam Problem List
1. The space of n matrices, M at(n, R), is a vector space of dimension n2 and therefore a smooth manifold. Let GL(n, R) be the subset of invertible n × n matrices, and O(n, R) be the set of orthogonal matrices:
O(n, R) := {A ∈ Mat(n, R) | AAT = Id}.
(a) Prove that GL(n, R) is a smooth manifold. What is its dimension?
(b) Prove that O(2, R) is a smooth manifold. What is its dimension?
2. Given a closed 1-form ω on a smooth manifold M ,
(a) prove that if M is simply connected, γ1 and γ2 be two paths starting and ending at the same points, then
Z
γ1
ω = Z
γ2
ω.
(b) show the conclusion of part (a) is false if M is not simply connected.
3. Let M be a compact manifold, N a connected manifold, and F : M → N is smooth.
(a) Prove that F is a closed map.
(b) Prove that if F is a submersion, then N must also be compact.
4. Prove that T S1 is trivial.
5. Show that every manifold M admits a smooth embedding into some Euclidean space.
6. Let U = (0, π) × (0, 2π) and f : R2 → R3 be given by
f (u, v) = (sin u cos v, sin u sin v, cos u).
Compute R
Uf∗ω for
ω = xdy ∧ dz + ydz ∧ dx + zdx ∧ dy.
7. Let
α = 3x2cos y + exy
dx ∧ dy + 17x3dx ∧ dz + 3 x + yz7 + cos z
dy ∧ dz be a two form on R3. Compute
Z
S2
α,
where S2 is given by standard orientation determined by outward normal vector.
8. Let M be a compact, oriented and connected manifold with boundary. Let ι : ∂M → M be inclusion, α and β be k form and n − k − 1 form on M , respectively. Moreover assume that ι∗β = 0. Prove that
Z
M
dα ∧ β = (−1)k+1 Z
M
α ∧ dβ.
9.
10.
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