Tutorial 9—Global Analysis
30.12.2021
1. Suppose M is a manifold and Dt : Qk(M)      Qk+Ti(M) for i = 1,2 a graded derivation of degree r« 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 u G Vtk{M) be a differential form and f/ cMan open subset. Show that u\u = 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.
(c) Suppose D and D are two graded derivations such that D(f) = D(f) and D(df) = D(df) for all / G C°°(M, R). Show that D = D.
2. Suppose M is a manifold and £, ?y 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, £^,iv:
(i) [d,£c] = 0.
(ii) [d, ^] = d o ^ + ^ o d = £%.
(iii) [£$,     = £[£,,,].
(iv) [£z,iv] = i[£,r,].
(v) fe^] = 0. Hint: Use (c) from 2.
3. Prove the Poincare Lemma: Suppose u G f2fc(IRn) is a closed £>form, where k > 1. Show that there exists r G f2fc_1(Mn) such that dr = u.
Hint:
Consider the vector field f G T(Mn) on Mn given by f (a;) = x G T^IR™ = Mn and let a : IR x W1 —> W1 be the smooth map a(t, x) = at(x) = tx. Then the flow of £ is given by Fl| = a(e*, rr).
1
2
• Show that (^Q*ti^u){x) is smooth in (t, x) for all £ G M and x G W1. Hence, £     ja*ti^(jj is a smooth family of [k — l)-forms on IRn.
• Show that J^ctJV = d{^a*ti^uj) and that uj = dr, where r =     ^a*ti^udt G
4. Show that n-dimensional projective space IRPn is orientable •<=>- n is odd.
5. Suppose (M, (?) C (IRn+1, geuc = (■, •)) is ahypersurface. Show that M is orientable if and only if it admits a global unit normal vector field.