Tutorial 9—Global Analysis
1. Prove the Poincare Lemma: Suppose to E f2fc(IRm) is a closed fc-form, where k > 1. Show that there exists r E f2fc_1(Mm) such that dr = uj.
Hint: Show that for any fc-form to = J2il<: <ik cu^^^dx11 A ... A dx%k on IRm,
n--1   „\ S-i, L-/ 0
xladxl1 A...Adxia A...Adxlk.
a=l ii<...<ik
is a (A; — l)-form on IRm satisfying
w = rf(PH) + P(duj).
Here,       means that this term is omitted.
2. Show that any manifold with a parallelizable tangent bundle is orientable.
3. Suppose M C N is a submanifold of codimension 1 (i.e. dim M = dim N — 1) of an oriented manifold N. Suppose there exists a smooth vector field along M that is transverse everywhere to M, that is, a smooth map v : M —> TN such that for all x E M one has
(i) v(x) E TXN and
(ii) v(x) and TXM span T^iV.
Show that M is orientable. Deduce that a hypersurface
(M,(?)c(Mm+1,(?) = (Mm+1,/uc)
in Euclidean space is orientable if and only if M admits a globally defined unit normal vector field.
4. Consider Sm C IRm+1 the unit sphere and the global unit normal vector field v(x)
Q = dx1 A ...Adxm+1
Y^T=i x%^i f°r Sm- Show that for the nowhere vanishing m + 1-form
on Rm+\
u(x) := (ivQ)(x) = Q(x)(v(x),_ ,) for x E Sr'
1
2
restricts to a nowhere vanishing m-form on Sm that satisfies
A*uj = (-l)m+V where A : Sm —?■ Sm is the antipodal map A(x) = —x.
5. Show that n-dimensional projective space IRPm is orientable •<=>- m is odd.
Hint: For , consider the natural projection n : Sm —?■ BLPm, given by n(x) = [x], and use the previous exercise. For , <=' construct an oriented atlas.
6. Suppose M and N are connected, compact, oriented manifolds of the same dimension m. Let Jo, fi '■ M —> N be smooth maps that are homo topic to each other, i.e. there exists a smooth map F : M x [0,1] —> N such that F(x, 0) = fo(x) and F(x, 1) = /i(ar). Show that for any u e ttm(N) one has
/ fo" = [ />•
J M J M
Hint: M x [0,1] is an oriented manifold with boundary dM = — (M x {0}) U M x {1}, where the minus indicates that the orientation on M x {0} is reversed. Use Stokes' Theorem.
7. Use the previous exercise to show that, if the antipodal map A : Sm —?■ Sm on the sphere Sm is homotopic to the identity Id^m on Sm, then m is odd.
8. Show that on a sphere S2m of even dimension any smooth vector field £ G X(S2m) has a zero.
Hint: Show that if £ G X(S2m) is nowhere vanishing, then there exists a homotopy between the antipodal map and the identity.