Tutorial 7-8—Global Analysis
9723. 11.2021
1. Suppose E —> M is a (smooth) vector bundle of rank k over a manifold M. Then E is called trivializable, if it isomorphic to the trivial vector bundle M x Rk —> M.
(a) Show that E —> M is trivializable •<=>- E M admits a global frame, i.e. there exist (smooth) sections si,Sk of E such that si(x),Sfc(^) sPan -^r for any rr G M.
(b) Show that the tangent bundle of any Lie group G is trivializable.
(c) Recall that IRn has the structure of a (not necessarily associative) division algebra over IR for n = 1, 2,4, 8. Use this to show that the tangent bundle of the spheres S1 C IR2, S3 C IR4 and S7 C Rs is trivializable.
2. Let 7 be a finite dimensional real vector space and consider the subspace of r-linear alternating maps ArV* = L^t(V, M) of the vector space of r-linear maps Lr(V,R) = (V*)®r. Show that for lu e Lr(V,R) the following are equivalent:
(a) uj G ArV*
(b) For any vectors vi,vr G V one has
U(VU ...,V,t, ...,Vj,...,Vk) = -U(VU ...,Vj, ...,v,t, ...,vk)
(c) u is zero whenever one inserts a vector v G V twice.
(d) u(vi, ...,Vk) = 0, whenever vi, ...,vk G V are linearly dependent vectors.
3. Let V be a finite dimensional real vector space. Show that the vector space A*V* : = 0r>o ArV* is an associative, unitial, graded-anticommutative algebra with respect to the wedge product A, i.e. show that the following holds:
(a) (u A rj) A ( = uj A (77 A () for all w,)j,(G A*V*.
(b) 1 G M = A°V* satisfies 1Aw = wA1 = 1 for all uj G A*V*.
(c) ArV* AASV* C Ar+S\/*.
(d) ujAt] = (-l)rsr] A uj for uj G ArV* and 77 G ASV*.
Moreover, show that for any linear map f : V —?■ W the linear map /* : A*W* —?■ A*V* is a morphism of graded unitial algebras, i.e. f*l = 1, f*(ArW*) C ArV* and        A rj) = f*cu A f*rj.
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4. Let V be a finite dimensional real vector space. Show that:
(a) If ui,ur E V* and vi,vr E V, then
lui A ... A u}r(vt, ...,vr) = det((wj(^))i<ij<r)-
In particular, ui, ...,ur are linearly independent •<=>- wi A ... A wr ^ 0.
(b) If {Ai,An} is a basis of V*, then
{\h A ... A \ir : 1 < «! < ... < ir < n}
is a basis of ArV*.
5. Let V be a finite dimensional real vector space. An element /i E Lr(V, R) is called symmetric,if fi(vi, ...,vr) = fi(va^, ...,va^) for any vectors v1} ...,vr E Vandany permutation a E Sr. Denote by SrV* C /i E Lr(V, R) the subspace of symmetric elements in the vector space Lr(V, R).
(a) For j2E Lr(V,R) show that
fi E SrV* fi(v±, ...,Vi, ...,Vj, ...,vk) = fi(v±, ...,Vj, ...,Vi, ...,vk),
for any vectors vi,vr E V.
(b) Consider the map Sym : Lr(V, R)     Lr(V, R) given by
1 X
Sym(fj)(v1,...,vr) = ~Y1 MMi)'-'Mo)-
Show that Image(Sym) = SrV* and that fj, E SrV* Sym(/i) = fi.
6. Let V be a finite dimensional real vector space and set S(V*) := ©^0S'rl/* with the convention S°V* = R and S^* = V*. For fj, E SrV* and v E SlV* define their symmetric product by
j2Qu:= Sym(/x 0 v) E Sr+tV*.
By blinearity, we extend this to a IR-bilinear map 0 : S(V*) x S(V*) S(V*). Show that S(V*) is an unitial, associative, commutative, graded algebra with respect to the symmetric product 0.
7. 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 0 F : = U^gM-Er 0 —^ M are again vector bundles over M.
8. Suppose E 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.
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(b) Show that E is involutive •<=>- whenever u1,to11 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) := {to E fl(M) : u\E = 0} C ft(M)
is an ideal of the algebra (f2(M), A). Here, cu\E = 0 for a £-form w means that ...,£<?) = 0 for any sections £i, ...^ of E.
(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.
9. Suppose M is a manifold and A : fifc(M)      fifc+r*(M) for i = 1,2 a graded derivation of degree r« of (f2(M), A).
(a) Show that
[£>i, 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 E 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 / e C°°(M, R). Show that D = D.
10. Suppose M is a manifold and £, ?y G T(TM) vector fields.
(a) Show that the insertion operator    : Vtk{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^,in:
(i) [d,£c] = 0.
(ii) [d, ^] = d o   + ^ o d =
(iii) Cv] = C[£,v]-
(iv) = i[£,v]-
(v) fe^] = 0. Hint: Use (c) from 2.