Homework 5—Global Analysis
Due date: 1.15.2020
1. Suppose p : E —>• M and q : F —> M are vector bundles over M. Show that their direct sum E © F := UxeMEx ® Fx ^ M and their tensor product E <g> F : = UxeMEx ® Fx ^ M are again vector bundles over M.
2. Suppose C TM is a smooth distribution of rank A; on a manifold M of dimension n and denote by Vt{M) the vector space of differential forms on M.
(a) Show that locally around any point x E M there exists (local) 1-forms cu1,..., w such that for any (local) vector field £ one has: £ is a (local) section of E •<=>-u)i(£) = 0 for alH = 1,n — k.
(b) Show that E is involutive •<=>- whenever w1,cun~k are local 1-forms as in (a) then there exists local 1-forms /i1^ for i, j = 1,n — k such that
n—k
(c) Show
Q,E{M) := {cu E fl(M) : w|E = 0} C ft(M)
is an ideal of the algebra (f2(M), A). Here, cu\E = 0 for a £-form w means that •••>6?) = 0 for any sections £1; ...^ of E1.
(d) An ideal J" of (fi(M), A) is called differential ideal, if d(J) C J. Show that ^(M) is a differential ideal •<=>-     is involutive.
3. Suppose M is a manifold and A : &k(M) Qk+Ti(M) for i = 1,2 a graded derivation of degree   of (f2(M), A).
(a) Show that
[Du D2] := DxoD2- (-l)r^D2 o Dx is a graded derivation of degree r\ + r2.
(b) Suppose D is a graded derivation of (f2(M), A). Let cu E Vtk{M) be a differential form and f/ cMan open subset. Show that lu\jj = 0 implies D{u)\u = 0.
Hint: Think about writing 0 as feu for some smooth function / and use the defining properties of a graded derivation.
1
(c) Suppose D and D are two graded derivations such that D(f) = D(f) and D(df) = D(df) for all / e C°°(M, R). Show that D = D.
Suppose M is a manifold and £,   G T(TM) vector fields.
(a) Show that the insertion operator    : f2fc(M) —>• Vtk~l(M) is a graded derivation of degree -1 of (fi(M), A).
(b) Recall from class that [d, d] = 0. Verify (the remaining) graded-commutator relations between d, C^,iv:
(i) [d,£c] = 0.
(ii) [d, ^] = d o   + ^ o d =
(iii) Cn] = C[£,v]-
(iv) = i[£,v]-
(v) fe^] = 0. Hint: Use (c) from 2.