Diffgeo 1 blatt 05 PDF

Preview

Summary

Download Diffgeo 1 blatt 05 PDF

Description

Dr. Robert Helling Prof. Dr. Sebastian Hensel Dr. Jan Swoboda

Differentiable Manifolds Problem Set 5

1. (15 points) (a) Let the complex projective space CP1 be defined as the set of lines in the complex vector space C2 , i.e., the set of equivalence classes of the equivalence relation z∼w

⇐⇒

w = λz

for some λ ∈ C ∖ {0}

on C2 ∖ {0}, cf. exercise 2.4 for the analogous construction of RPd . We endow CP1 with the quotient topology induced from the natural topology on C2 . Homogeneous coordinates are defined as before. Show that these give rise to a smooth atlas on CP1 with charts defined by ϕ0 ∶ {[z0 ∶ z1 ] 󳈌 z0 ≠ 0} → C,

(z 0 ∶ z 1 ) ↦

z1 z0

ϕ1 ∶ {[z0 ∶ z1 ] 󳈌 z1 ≠ 0} → C,

(z 0 ∶ z 1 ) ↦

z0 . z1

and

(b) A smooth map F ∶ M → N between smooth manifolds is called a smooth submersion if it is surjective and if its differential dx F at every x ∈ M is surjective. Let S 3 = {(z0 , z1 ) ∈ C2 󳈌 󳈌z0 󳈌2 + 󳈌z1 󳈌2 = 1} denote the unit sphere in C2 . Show that the Hopf map H∶ S 3 → CP 1 , (z0 , z1 ) ↦ [z0 ∶ z1 ]. is a smooth submersion. (c) Prove that for every p ∈ CP1 the preimage H −1 (p) is a smooth manifold diffeomorphic to S 1 . 2. (10 points) Let E1 , E2 be two vector bundles over a manifold M. Prove that there is a vector bundle E1 ⊗ E2 over M, so that −1 (u) = π1−1(u) ⊗ π2−1(u) πE 1 ⊗E2

for every u ∈ M and the following holds: (a) There is a bundle map B∶ E1 ⊕E2 → E1 ⊗ E2 which covers the identity (i.e., which satisfies πE1 ⊗E2 ○ B = πE1 ⊕E2 ) and such that for every u ∈ M the restriction (E 1 ⊕ E 2 )u → (E 1 ⊗ E 2 )u −1 to the fibre πE (u) is bilinear. 1 ⊕E2

(b) For every vector bundle F over M and every bilinear bundle map B 󰐞 ∶ E1 ⊕E2 → F which covers the identity, there is a unique bundle map B 󰐞󰐞 ∶ E1 ⊗ E2 → F such that B 󰐞 = B 󰐞󰐞 ○ B . 3. (8 points) Let E be a smooth vector bundle of rank r over a smooth manifold M . Prove that E is trivial if and only if there exist smooth sections s1 , . . . , sr of E which are everywhere linearly independent (i.e., for every p ∈ M the set {s1 (p), . . . , sr (p)} is a linearly independent subset of the vector space Ep ). 4. (8 points) Let G be a Lie group with identity element e, i.e., G is a smooth manifold such that the group multiplication µ∶ G×G → G, µ(g, h) = gh and inversion ι∶ G → G, ι(g) = g−1 are smooth maps. Prove that the tangent bundle T G is diffeomorphic to G × Te G, hence trivial.

Please hand in your solutions by Wednesday 2018/11/21, 12 a.m....