Differential Geometry Homework #10 due 12/21 1. Let Γ be a discrete group acting smoothly on the manifold M . Show that Γ acts properly discontinuously on M if and only if any two points x, x

Differential Geometry Homework #10 due 12/21

1. Let Γ be a discrete group acting smoothly on the manifold M . Show that Γ acts properly discontinuously on M if and only if any two points x, x⁰ in M have neighborhoods U and U⁰ such that the set {h ∈ Γ : hU ∩ U⁰ = ∅} is finite.

2. Let G be a Lie group acting smoothly on a manifold M . Prove that each orbit is an immersed submanifold of M .

3. (Boothby, p.103 #2) Show that π₂ : ˜M /˜Γ → M as defined in class is a covering map.

4. (Boothby, p.103 #3) Show that the covering transforms form a group and that if x, y ∈ ˜M , a covering manifold of M , then there is at most one covering transform taking x to y. Show further that if ˜Γ is transitive on π⁻¹(p) for some p ∈ M , then it is transitive for every p.

5. (Boothby, p.103 #5) Let π : ˜M → M be a covering and F : [a, b] → M a continuous curve from F (a) = p to F (b) = q. If x₀ ∈ π⁻¹(p), show that there is a unique continuous curve ˜F : [a, b] → ˜M such that ˜F (a) = x0 and π ◦ ˜F = F .

6. Let π : ˜M → M be a covering map. Show that π is open.

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