§ 2.1
1 LetV be a vector space and letι:Vn→ ⊗nV be defined asι(v1, . . . , vn) = v1⊗ · · · ⊗vn, whereVn =V × · · · ×V (nfactors on the right hand side).
Prove that⊗nV satisfies the following universal property: for every vector spaceU and everyn-multilinear mapT : Vn → U, there exists a unique linear mapT˜:⊗nV →U such thatT˜◦ι=T.
⊗nV
Vn ι∧
T> U T˜ ...
...>...
2 Prove that⊗nV is canonically isomorphic to the dual space of the space n-multilinear forms onVn. (Hint: Use Problem 1.)
2.6. PROBLEMS 61 3 Let V be a vector space. Ann-multilinear map T : Vn → U is called alternatingifT(vσ(1), . . . , vσ(n)) = (sgnσ)T(v1, . . . , vn)for everyv1, . . . , vn∈ V and every permutationσof{1, . . . , n}, wheresgndenotes the sign±1of the permutation.
Letι:Vn→Λn(V)be defined asι(v1, . . . , vn) =v1∧· · ·∧vn. Note thatι is alternating. Prove thatΛnV satisfies the following universal property: for every vector spaceU and every alternatingn-multilinear mapT :Vn→U, there exists a unique linear mapT˜: Λn(V)→U such thatT˜◦ι=T.
ΛnV
Vn ι∧
T> U T˜ ...>
4 Denote the vector space of all alternating multilinear formsVn →Rby An(V). Prove thatΛn(V)is canonically isomorphic toAn(V)∗.
5 Prove thatv1, . . . , vk∈V are linearly independent if and only ifv1∧· · ·∧
vk6= 0.
6 LetV andW be vector spaces and letT :V →W be a linear map.
a. Show thatT naturally induces a linear mapΛk(T) : Λk(V)→Λk(W).
(Hint: Use Problem 3.)
b. Show that the mapsΛk(V)for variouskinduce an algebra homomor-phismΛ(T) : Λ(V)→Λ(W).
c. Let nowV =W andn= dimV. The operatorΛn(T)is multiplication by a scalar, asdim Λn(V) = 1; define the determinantofT to be this scalar. Anyn×nmatrixA= (aij)can be viewed as the representation of a linear operator onRnwith respect to the canonical basis. Prove that
detA=X
σ
(sgnσ)ai,σ(i)· · ·an,σ(n),
wheresgnσis the sign of the permutationσandσruns over the set of all permutations of the set{1, . . . , n}. Prove also that the determinant of the product of two matrices is the product of their determinants.
d. Using Problem 7(a) below, prove that the transpose mapΛk(T)∗ = Λk(T∗).
7 LetV be vector space.
a. Prove that there is a canonical isomorphism Λk(V∗)→Λk(V)∗ given by
v∗1∧ · · · ∧v∗k7→(u1∧ · · · ∧uk7→det(vi∗(uj)) ).
b. Letα,β∈V∗ ∼= Λ1(V∗)∼=A1(V). Show thatα∧β ∈Λ2(V∗), viewed as an element ofΛ2(V)∗ ∼=A2(V)is given by
α∧β(u, v) =α(u)β(v)−α(v)β(u) for allu,v∈V.
8 LetV be an Euclidean vector space, that is, a vector space equipped with a (positive-definite) inner producth,i. Prove that there is an induced inner product onΛk(V)given by
hu1∧ · · · ∧uk, v1∧ · · · ∧vki 7→det (hvi, uji). 9 LetV be a vector space.
a. In analogy with the exterior algebra, construct thesymmetric algebra Sym(V), a commutative graded algebra, as a quotient ofT(V).
b. Determine a basis of the homogeneous subspaceSymn(V).
c. State and prove thatSymn(V)satisfies a certain universal property.
d. Show that theSymn(V)is canonically isomorphic to the dual of the spaceSn(V)of symmetricn-multilinear formsVn→R.
In view of (d),Sym(V∗)is usually defined to be the spaceP(V)of polyno-mialsonV.
10 An element ofΛn(V)is calleddecomposableif it lies in the subsetΛ1(V)∧
· · · ∧Λ1(V)(nfactors).
a. Show that in general not every element ofΛn(V)is decomposable.
b. Show that, for dimV ≤ 3, every homogeneous element in Λ(V) is decomposable.
c. Letωbe a differential form. Isω∧ω= 0?
11 Let V be an oriented vector space equipped with a non-degenerate symmetric bilinear form (we do not require positive-definiteness from the outset). LetdimV =n.
a. Prove there exists an elementω ∈Λn(V)such that ω=e1∧ · · · ∧en
for every positively oriented orthonormal basis{e1, . . . , en}ofV (here orthonormalmeans thatei·ej =±δij (delta of Kronecker)).
b. Check that the bilinear form onV induces an isomorphismV →V∗, which induces an isomorphismΛk(V)→Λk(V∗)via Problem 6(a).
c. Show that the bilinear map
Λk(V)×Λn−k(V)→Λn(V), (α, β)7→α∧β together with the isomorphism
R→Λn(V), a7→a ω
2.6. PROBLEMS 63 define a canonical isomorphism
(Λk(V))∗ →Λn−k(V).
d. Combine the isomorphisms of (b) and (c) with that in Problem 7(a) to get a linear isomorphism
∗: Λk(V)→Λn−k(V) for0≤k≤n, called theHodge star.
e. Show that
α∧ ∗β=hα, βiω
for allα,β ∈Λk(V), where we use the inner product of Problem 8.
f. Assume the inner product is positive definite and let{e1, . . . , en}be a positively oriented orthonormal basis ofV. Show that
∗1 =e1∧ · · · ∧en, ∗(e1∧ · · · ∧en) = 1, and
∗(e1∧ · · · ∧ek) =ek+1∧ · · · ∧en. Show also that
∗∗= (−1)k(n−k) onΛk(V).
§ 2.2
12 LetM be a smooth manifold. ARiemannian metricgonMis an assign-ment of positive definite inner productgpon each tangent spaceTpMwhich is smooth in the sense thatg(X, Y)(p) = gp(X(p), Y(p))defines a smooth function for everyX,Y ∈X(M). ARiemannian manifoldis a smooth mani-fold equipped with a Riemannian metric.
a. Show that a Riemannian metricgonM is the same as a tensor fieldg˜ of type(0,2)which issymmetric, in the sense that˜g(Y, X) = ˜g(X, Y) for every X, Y ∈ X(M), with the additional property of positive-definiteness at each point.
b. Fix a local coordinate system(U, x1, . . . , xn)onM.
(i) Letgbe a Riemannian metric onM. Show thatg|U =P
i,jgijdxi⊗ dxj where gij = g(∂x∂
i,∂x∂
j) ∈ C∞(U),gij = gji and the matrix (gij)is everywhere positive definite.
(ii) Conversely, given functions gij = gji ∈ C∞(U) such that the matrix(gij)is positive definite everywhere inM, show how to define a Riemannian metric onU.
c. Use part (b)(ii) and a partition of unity to prove that every smooth manifold can be equipped with a Riemannian metric.
d. On a Riemannian manifoldM there exists a natural diffeomorphism T M ≈T∗M taking fibers to fibers. (Hint: There exist linear isomor-phismsv∈TpM 7→gp(v,·)∈TpM∗).
§ 2.3
13 Consider R3 with coordinates (x, y, z). In each case, decide whether dω= 0or there existsηsuch thatdη=ω.
a. ω=yzdx+xzdy+xydz. b. ω=xdx+x2y2dy+yzdz.
c. ω= 2xy2dx∧dy+zdy∧dz.
14 (The operatordonR3) Identify1- and2-forms onR3with vector fields onR3, and0- and3-forms onR3with smooth functions onR3, and check that:
don0-forms is the gradient;
don1-forms is the curl;
don2-forms is the divergent.
Also, interpretd2 = 0is those terms.
§ 2.4
15 LetMandNbe smooth manifolds whereMis connected, and consider the projection π : M ×N → N onto the second factor. Prove that a k-formωonM ×N is of the formπ∗ηfor somek-formηonN if and only if ιXω=LXω = 0for everyX∈X(M×N)satisfyingdπ◦X= 0.
16 LetMbe a smooth manifold.
a. Prove thatιXιX = 0for everyX∈X(M).
b. Prove that ι[X,Y]ω = LXιYω−ιYLXω for everyX, Y ∈ X(M) and ω∈Ωk(M).
§ 2.5
17 TheWhitney sumE1⊕E2of two vector bundlesπ1:E1→M,π2:E2→ M is a vector bundleπ :E =E1⊕E2 → M whereEp = (E1)p⊕(E2)p for allp∈M.
a. Show that E1⊕E2 is indeed a vector bundle by expressing its local trivializations in terms of those ofE1andE2and checking the condi-tions of Definition 2.5.2.
b. Similarly, construct the tensor product bundleE1⊗E2 and the dual bundleE∗.
C H A P T E R 3
Lie groups
Lie groups are amongst the most important examples of smooth manifolds.
At the same time, almost all usually encountered examples of smooth man-ifolds are related to Lie groups, in a way or another. A Lie group is a smooth manifold with an additional, compatible structure of group. Here compati-bility refers to the fact that the group operations are smooth (another point of view is to regard a Lie group as a group with an additional structure of manifold...). The reader can keep in mind the matrix groupGL(n,R)of non-singular realn×nmatrices (Examples 1.2.7) in which then2matrix co-efficients form a global coordinate system. The conjunction of the smooth and the group structures allows one to give a more explicit description of the differential invariants attached to a manifold. For this reason, Lie groups form a class of manifolds suitable for testing general hypotheses and conjectures. The same remarks apply to homogeneous spaces, which are certain quotients of Lie groups.
3.1 Basic definitions and examples
ALie groupGis a smooth manifold endowed with a group structure such that the group operations are smooth. More concretely, the multiplication mapµ : G×G→ Gand the inversion mapι :G → Gare required to be smooth.
3.1.1 Examples (a) The Euclidean spaceRn with its additive vector space structure is a Lie group. Since the multiplication is commutative, this is an example of aAbelian(orcommutative) Lie group.
(b) The multiplicative group of nonzero complex numbersC×. The sub-group of unit complex numbers is also a Lie sub-group, and as a smooth mani-fold it is diffeomorphic to the circleS1. This is also an Abelian Lie group.
(c) IfGandH are Lie groups, the direct product group structure turns the product manifoldG×Hinto a Lie group.
65
(d) It follows from (b) and (c) that then-torusTn=S1×· · ·×S1(ntimes) is a Lie group. Of course, Tn is a compact connected Abelian Lie group.
Conversely, we will see in Theorem 3.5.3 that every compact connected Abelian Lie group is ann-torus.
(e) If Gis a Lie group, the connected component of the identity of G, denoted byG◦, is also a Lie group. Indeed,G◦ is open inG, so it inherits a smooth structure fromGjust by restricting the local charts. Sinceµ(G◦× G◦)is connected andµ(1,1) = 1, we must haveµ(G◦×G◦)⊂G◦. Similarly, ι(G◦)⊂G◦. SinceG◦ ⊂Gis an open submanifold, it follows that the group operations restricted toG◦are smooth.
(f) Any finite or countable group endowed with the discrete topology becomes a0-dimensional Lie group. Such examples are calleddiscrete Lie groups.
(g) We now turn to some of the classical matrix groups. The general linear groupGL(n,R)is a Lie group since the entries of the product of two matrices is a quadratic polynomial on the entries of the two matrices, and the entries of inverse of a non-singular matrix is a rational function on the entries of the matrix.
Similarly, one defines thecomplex general linear group of ordern, which is denoted byGL(n,C), as the group consisting of all nonsingularn×n com-plex matrices, and checks that it is a Lie group. Note thatdimGL(n,C) = 2n2andGL(1,C) =C×.
We have already encountered the orthogonal group O(n) as a closed embedded submanifold ofGL(n,R)in 1.4.14. SinceO(n)is an embedded submanifold, it follows from Theorem 1.4.9 that the group operations of O(n)are smooth, and henceO(n)is a Lie group.
Similarly toO(n), one checks that the
SL(n,R) = {A∈GL(n,R)| det(A) = 1} (real special linear group) SL(n,C) = {A∈GL(n,C)| det(A) = 1} (complex special linear group)
U(n) = {A∈GL(n,C)|AA∗=I} (unitary group)
SO(n) = {A∈O(n)| det(A) = 1} (special orthogonal group) SU(n) = {A∈U(n)| det(A) = 1} (special unitary group)
are Lie groups, whereA∗ denotes the complex conjugate transpose matrix ofA. Note thatU(1) =S1.
Lie algebras
For an arbitrary smooth manifold M, the space X(M) of smooth vector fields onM is an infinite-dimensional vector space over R. In addition, we have already encountered the Lie bracket, a bilinear map[·,·] :X(M)× X(M)→X(M)satisfying:
a. [Y, X] =−[X, Y];
3.1. BASIC DEFINITIONS AND EXAMPLES 67 b. [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0(Jacobi identity);
for everyX,Y ∈X(M). In general, a vector space with a bilinear operation satisfying (a) and (b) above is called aLie algebra. SoX(M) is Lie algebra overR.
It turns out in case of a Lie groupG, we can single out a finite dimen-sional subalgebra ofX(M). For that purpose, let us first introduce transla-tions inG. The left translationdefined byg ∈ Gis the mapLg : G → G, Lg(x) = gx. It is a diffeomorphism ofG, its inverse being given by Lg−1. Similarly, theright translation defined byg ∈ Gis the map Rg : G → G, Rg(x) = xg. It is also a diffeomorphism ofG, and its inverse is given by Rg−1.
The translations in Gdefine canonical identifications between the tan-gent spaces toG at different points. For instance, dLg : ThG → TghGis an isomorphism for everyg,h ∈ G. This allows us to consider invariant tensors, the most important case being that of vector fields. A vector field XonGis calledleft-invariantifd(Lg)x(Xx) =Xgxfor everyg,x∈X. This condition is simplydLg◦X = X◦Lg for everyg ∈ G; equivalently,X is Lg-related to itself, or yetLg∗X=X(sinceLgis a diffeomorphism), for all g ∈G. We can similarly defineright-invariantvector fields, but most often we will be considering the left-invariant variety. Note that left-invariance and right-invariance are the same property in case of an Abelian group.
3.1.2 Lemma Every left invariant vector fieldXinGis smooth.
Proof. Letf be a smooth function defined on a neighborhood ofginG, and letγ : (−ǫ, ǫ) → Gbe a smooth curve withγ(0) = 1andγ′(0) = X1. Then the value ofXonfis given by
Xg(f) =dLg(X1)(f) =X1(f◦Lg) = d dt
t=0f(gγ(t)) = d dt
t=0f ◦µ(g, γ(t)),
and hence, it is a smooth function ofg.
Letgdenote the set of left invariant vector fields onG. It follows thatg is a vector subspace ofX(M). Further,gis a subalgebra ofX(M), for given X,Y ∈g, we have by Proposition 1.6.18 that
Lg∗[X, Y] = [Lg∗X, Lg∗Y] = [X, Y],
for everyg ∈ G. Finally, we explain whygis finite-dimensional: the map X∈g7→X1defines a linear isomorphism betweengand the tangent space toGat the identityT1G, since any left invariant vector field is completely defined by its value at the identity.
The discussion above shows that to any Lie groupGis naturally asso-ciated a (real) finite-dimensional Lie algebra g of the same dimension as G, consisting of the left invariant vector fields onG. This Lie algebra is
the infinitesimal object associated to G and, as we shall see, completely determines its local structure. Also, it is often convenient to view g as T1Gequipped with the Lie algebra structure such that the evaluation map g → T1Gis a Lie algebra isomorphism, and at times we shall follow this practice without further comment.
3.1.3 Examples (The Lie algebras of some known Lie groups)
(i) The left-invariant vector fields onRnare precisely the constant vec-tor fields, namely, the linear combinations of coordinate vecvec-tor fields (in the canonical coordinate system) with constant coefficients. The bracket of two constant vector fields onRnis zero. It follows that the Lie algebra ofRn isRnitself with the null bracket. In general, a vector space equipped with the null bracket is called anAbelianLie algebra.
(ii) The Lie algebra of the direct productG×His the direct sum of Lie algebrasg⊕h, where the bracket is taken componentwise.
(iii) Owing to the skew-symmetry of the Lie bracket, every one-dimensional Lie algebra is Abelian. In particular, the Lie algebra ofS1is Abelian. It fol-lows from (ii) that also the Lie algebra ofTnis Abelian.
(iv) G and G◦ have canonically isomorphic Lie algebras. In fact the differential of the inclusion ι : G◦ → G defines a linear isomorphism T1G◦ ∼= T1Gand the corresponding left-invariant vector fields onG◦ and Gareι-related.
(v) The Lie algebra of a discrete group is{0}. 3.1.4 Examples (Some abstract Lie algebras)
(i) Let Abe any real associative algebra and set [a, b] = ab−ba fora, b∈A. It is easy to see thatAbecomes a Lie algebra.
(ii) The cross-product × on R3 is easily seen to define a Lie algebra structure.
(iii) IfV is a two-dimensional vector space andX,Y ∈ V are linearly independent, the conditions[X, X] = [Y, Y] = 0,[X, Y] = X define a Lie algebra structure onV.
(iv) If V is a three-dimensional vector space spanned by X, Y, Z, the conditions[X, Y] = Z, [Z, X] = [Z, Y] = 0define a Lie algebra structure onV, called the(3-dimensional) Heisenberg algebra. It can be realized as a Lie algebra of smooth vector fields onR3as in Example 1.6.15(b).
3.1.5 Exercise Check the assertions of Examples 3.1.3 and 3.1.4.