Tutorial 6-8—Global Analysis
1. Suppose a* for i = 1,..., k and j = 1,..., n are smooth real-valued functions defined on some open set U C IRn+fc satisfying
£=1 £=1
where we write (x, z) = (x1,xn, z1,zk) for a point in Wl+k. Show that for any point (x0,z0) E U there exists an open neighbourhood V of x0 in IRn and a unique C°°-map / : V —> R.k such that
— (x1,...,xn) = a)(x1,...,xn,f1(x),...,fk(x))    and    f(x0) = z0.
In the class/tutorial we proved this for k = 1 and j = 2.
2. Which of the following systems of PDEs have solutions f(x, y) (resp. f(x, y) and g(x, y)) in an open neighbourhood of the origin for positive values of /(0, 0) (resp. /(0,0) and 0(0,0))?
(a) If = /cosy and     = -/log/tany.
(b) |£ =     and f£ = xe*f.
v 7   ot ay
(c) |£ = / and If = g; fa = 0 and & = /.
3. 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 IRfc —> M.
(a) Show that E —> M is trivializable •<=>- E M admits a global frame, i.e. there exist (smooth) sections s1;sk of E such that si(x),Sfc(^) sPan ^ for any x E 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) normed 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 IR8 is trivializable.
4. Let V 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:
1
2
(a) uj E ArV*
(b) For any vectors vi,vr E 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 E V twice.
(d) u(vi, ...,Vk) = 0, whenever vi, ...,vk E V are linearly dependent vectors.
5. 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) (uj A rj) A ( = uj A (77 A () for all w,)j,(G A*V*.
(b) 1 G 1 = A°V* satisfies 1Aw = wA1 = 1 for all uj E A*V*.
(c) ArV* A ASV* C Ar+sV*.
(d) ujAt] = (-l)rsr] A uj for uj E ArV* andr] E 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 f*(u A rj) = f*cu A f*rj.
6. Let V be a finite dimensional real vector space. Show that:
(a) If ui,ujt E V* and vi,vr E V, then
u3x 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*.
7. Let V be a finite dimensional real vector space. An element /1 E Lr(V, M) is called symmetric, if fi(vi,vr) = fi(va^, ...,va^) for any vectors v±, ...,vr E V^andany permutation a E Sr. Denote by SrV* C /1 E Lr(V, M) the subspace of symmetric elements in the vector space Lr(V, M).
(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.
3
(b) Consider the map Sym : Lr(V, R)     Lr(V, R) given by
Sym(/i)(wi,...,Wr) = ~Y1 KV<T(l),--,Va(r))-
Show that Image(Sym) = SrV* and that fj, E SrV* Sym(/i) = fi.
8. 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 \l E SrV* and v E SlV* define their symmetric product by
j2Qu:= Sym(/x 0 v) E S'+'V*.
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.
9. 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 ® F : = UxeMEx 0 Fx —>• M are again vector bundles over M.
10. 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 •<=>-Wj(£) = 0 for alH = 1,n — k.
(b) Show that E is involutive •<=>- whenever w1,wn_fc 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
QE(M) := {lu E tt(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 •••>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 •<=>- E is involutive.
11. Suppose M is a manifold and A : fifc(M)      fifc+r*(M) for i = 1,2 a graded derivation of degree   of (f2(M), A).
(a) Show that
[Du D2] := Dl0D2- (-l)rirW2 o ^ is a graded derivation of degree rx + 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.
Suppose M is a manifold and £, rj E Y(TM) vector fields.
(a) Show that the insertion operator    : Vtk{M) —> Vtk~l(M) is a graded derivation of degree -1 of (£l(M), A).
(b) Recall from class that [d, d] = 0. Verify (the remaining) graded-commutator relations between d, £^,iv:
(i) [d,^] = 0.
(ii) [d, i%\ = doi^ + i^od = £%.
(iii) [£$, £n] = £[t,n]-
(iv) [£z,ir,] = i[£,r,].
(v) fe^] = 0. Hint: Use (c) from 11.