Math 545 Topology and Geometry of Manifolds Winter 2000
Assignment #7
Due 2/25/2000
I. Required problems.
1. Prove the following refinement of Lemma 5.1: If V:M→TM is a (not necessarily
continuous) map such that Vp∈TpMfor each p∈M,thenVis a smooth vector
field if and only if Vf is a smooth function on Mfor every f∈C∞(M).
2. If N⊂Mis a closed embedded submanifold and V∈T(N), show that there is a
smooth vector field Won Msuch that V=W|N.
3. Let Ebe a smooth rank-kvector bundle over a smooth manifold M, with projec-
tion π:E→M,andletU⊂Mbe an open set. A local f rame for Eover Uis an
ordered k-tuple (σ1,...,σ
k)whereeachσiis a smooth section of Eover U(i.e., a
smooth map σi:U→Esuch that π◦σi=Id
U), and such that (σ1|p,...,σ
k|p)isa
basis for the fiber π−1(p) for each p∈U. It is called a global frame if U=M. Show
that Eadmits a local frame over Uif and only if it admits a local trivialization
over U,andEadmits a global frame if and only if it is trivial.
4. Find all integral curves of the following vector fields on the plane.
(a) V=2x∂
∂x −y2∂
∂y.
(b) W=y∂
∂x −x∂
∂y.
II. Optional problems.
5. A smooth manifold Mis said to be parallelizable if its tangent bundle admits a
global frame, which is equivalent by Problem 3 to TM being a trivial bundle.
Show that the Tn=S1×···×S1and S3are parallelizable.