Math 545 Topology and Geometry of Manifolds Winter 2000
Assignment #8
Due 3/3/2000
I. Required problems.
1. If N⊂Mis an embedded submanifold and γ:J−→ Mis a smooth curve whose
image happens to lie in N, show that γ0(t) is in the subspace Tγ(t)Nof Tγ(t)Mfor
all t∈J. Give a counterexample if Nis not embedded.
2. Let F:M−→ Mbe a diffeomorphism, let Vbe a smooth vector field on M,and
let θbe the flow generated by V. Show that Vis invariant under Fif and only
if θcommutes with F, in the sense that θt◦F=F◦θtwhenever either side is
defined. [Hint: use uniqueness of integral curves.]
3. Prove Lemma 5.12.
4. If Vis a vector field on a smooth manifold M,thesupport of Vis defined to be
the closure of the set {p∈M:Vp6=0}. Show that every smooth vector field with
compact support is complete.
II. Optional problems.
5. All the systems of differential equations that we have considered have been of the
form
γi0(t)=Vi(γ(t)),
in which the functions Vido not depend explicitly on the independent variable t.
(Such a system is said to be autonomous.) If instead Vis a function of (t, x)in
some subset of R×Rn, the resulting system is called nonautonomous;itcanbe
thought of as a “time-dependent vector field” on a subset of Rn. This problem
shows that local existence, uniqueness, and smoothness for a nonautonomous
system follow from the corresponding results for autonomous ones.
Suppose U⊂Rnis an open set, J⊂Ris an open interval, and V:J×U−→ Rnis
a smooth map. For any (t0,x
0)∈J×Uand any sufficiently small ε>0, show that
there exists a neighborhood U0of x0in Uand a smooth map θ:(t0−ε, t0+ε)×U0
−→ Usuch that for each x∈U0,thecurveγ(t)=θ(t, x) is the unique solution on
(t0−ε, t0+ε) to the nonautonomous initial-value problem
γi0(t)=Vi(t, γ(t)),
γi(t0)=xi.
[Hint: replace this system of ODEs by an autonomous system in Rn+1.]