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Manifolds 8


  • Manifolds 2008-9 outline notes and homeworkexercises

    The examination will be based on my lectures and exercises. Exercisesmarked with an asterisk are examination questions set by me in the past.The solutions to these questions may sometimes include standard results thatcan be found in notes or books. Solutions will be placed on the course web-page before the end of the course. Please let me know if you find any mis-prints/errors during the course.

    Neil Lamberts notes for the course he taught a few years ago are on thecourse web-page and are often used. His notes, and the exercises in them,can be used as a useful supplement to the lectures and homework exercises.

    1 Review of background and notation

    1.1 Finite dimensional vector spaces over the real num-bers

    Let V be a n dimensional vector space of the real numbers R with additionand scalar multiplication denoted pv1 + qv2 for any two vectors v1 and v2and any scalars p and q R. Let {ea}, a = 1..n denote a basis of V . Then

    any vector v in V has a unique expansion v =n


    vaea, where its components

    with respect to this basis are the real numbers {va}.In this course we may use the Einstein index and summation convention.

    No (letter) index is repeated more than once, repeated indices are summedover a given range, and unrepeated (free) indices match on each side of anequation and are understood to take values in a given range. Of a pair ofrepeated indices one must be a superscript and the other a subscript. (Allthe letter indices in this section are understood to sum and range over thedimension of the relevant vector space.)

    Using this convention we write v = vaea. Under a change of basis,ea 7 e

    a = Mbaeb, where the n n matrix with real entries M

    ba is invertible,

    the components of the vector change, va 7 va = (M1)abvb, where v =

    vaea = vaea.

    Let W be an m dimensional vector space. A map L : V W is a linearmap if for any v1 and v2 in V, and any scalars p and q R,

    L(pv1 + qv2) = pL(v1) + qL(v2).


  • Let {bi}, i = 1..m be a basis of W , so that for any w W, w =wibi. Then

    the m n matrix representing L with respect to these bases has entries thereal numbers Lia where L(ea) = L

    iabi. It is straightforward to compute the

    change in the components representing L under change of bases.If n = m and L is invertible, then L is an isomorphism. If V = W then

    L is an endomorphism, an endomorphism which is also an isomorphism is anautomorphism.

    The set of linear maps L : L : V W can be given the structure of avector space, Hom(V,W ), of dimension mn, by defining addition and scalarmultiplication in the obvious way: for any v V

    (L1 + qL2)(v) :=L1(v) + qL2(v).

    Hom(V,R) is an important ndimensional vector space, denoted V ,isomorphic, but not canonically isomorphic to V . It is called the vectorspace dual to V . The vectors in V are called covariant vectors (co-vectors),linear forms or one-forms to distinguish then from vectors (sometimes calledcontravariant vectors) in V . If {ea} is a basis of V , it is always possible tochoose a basis of V , {a}, called the dual basis, which is such that a(eb) =ab . Here

    ab is the Kronecker delta, an indexed representation of the unit

    n n matrix.Let = a

    a V . Then

    (v) =aa(vbeb) = linearity = av

    ba(eb) = dual basis = avbab = av


    Alternative notations are (v) = ,v = vy.A change of basis of V , as above, induces the change of basis of V given

    by a 7 a = (M1)abb.

    Any linear map L : V W induces the adjoint linear map L : W V

    by L(),v := ,L(v).The direct product of V and W , V W , (more usual terminology for

    linear spaces is the direct sum V W ) is the n + m dimensional vector ofordered pairs (v,w) where

    p(v1,w1)+q(v2,w2) =(pv1 + qv2, pw1 + qw2),v1,v2 V and w1,w2 W.

    A map T : V W R is said to be bilinear if it is linear in each factor, e.g

    T (v,pw1+qw2)=pT (v,w1) + qT (v,w2),

    T (pv1+qv2,w)=pT (v1,w) + qT (v2,w).


  • In this course we shall only need to consider such maps where the vectorspaces W and V are V and V . The generalization to other vector spacesis obvious.

    A bilinear map T : V V R is called a tensor, over V , of type (orvalence) (1, 1). The vector space of such multilinear maps is called the tensorproduct space over V , T (1,1)V, and is often denoted V V . It has additionand scalar multiplication defined by (T1 + pT2)(,v) =T1(,v)+pT2(,v).The element of V V denoted v and called the tensor product of v Vand V , is by definition the bilinear map taking any (,v) V V tothe real number ,v , v. In particular then,

    ea b : (i, ej)


    bj .


    b : (,v) avb.

    It can be shown that {eab} forms a basis of VV which is of dimension

    n2. Any T V V can be expanded in terms of its components, the realnumbers ta.b, with respect to this (induced) basis as T = t


    b, whereta.b = T (

    a, eb). Consequently T (,v) = ta.bav

    b.It is straightforward to extend these ideas to more than two vector spaces.

    A tensor of type (r, s) is a multilinear map,

    T : V V .. V V V.... V R

    by(1, ....r,v1, ...,vs) T (

    1, ....r,v1, ...,vs).

    The vector space of tensors of type (r, s) over V is denoted T (r,s)V =rV s V where r denotes the r-fold tensor product of the vector spacewith itself.

    By definition, w1 ...wr 1 s is the element of T (r,s)V that maps

    (1, ....r,v1, ...,vs) to

    1, w1...r, wr

    1, v1...s, vs.

    In particular, T (o,o)V = R, T (1,0)V = V, T (0,1)V = V . The naturallyinduced basis from the above basis of V, and its dual basis, is written ea1 ea2 ....ear

    b1 b2 ... bs . Any T T (r,s)V can be expanded in termsof its components with respect to the induced basis,

    T = ea2 ....ear b1 b2 ... bs ,


  • and T (r,s)V has dimension nr+s. Hence

    T (1, ....r,v1, ...,vs) =

    1a1 ......


    b1 ......v1b1 ,

    where 1 = 1a1a1 , etc. These induced bases will always be the bases used

    for the vector spaces of tensors over V .If

    T = ea2 ....ear b1 b2 ... bs ,

    R = ea2 ....ear b1 b2 ... bs ,

    belong to T (p,q)V , then the vector space sum is just

    pT+qR = (

    )ea1ea2....earb1b2 .

    Further details, and basis indepependent formulations, can be found in thetextbooks or written down (as an exercise) by amplifying that brief summary.

    For tensors there are two further important operations. These are thetensor product and contraction, given below first in terms of basis expan-sions..These can, of course, be defined invariantly, that is without referenceto particular bases.

    Exercise: Do this as an exercise before looking at the definitions justbelow.

    1. If

    U =

    ear+1 ear+2 ....ear+p bs+1 bs+2 ... bs+q

    belongs to T (p,q)V , then the tensor product of T T (r,s)V and U , T U , isa tensor of type (r + p, s+ q) over V denoted by T U and it equals

    ea1ea2....earb1b2 ...bsbs+1bs+2 .

    Note that in general T U is not equal to U T but the tensor product isassociative.

    2. If T T (r,s)V the contraction of T (on the ith contravariant index orsuperscript, and the jth covariant index or subscript) is, when defined, thetensor C T (r1,s1)V , where C is given by


    ea1ea2...eai1eai+1 .


  • The basis independent formulation of these operations are:1. The tensor product

    (T U)(1, ....r+p,v1, ...,vs+q)

    = T (1, ....r,v1, ...,vs)U(r+1, ....r+p,vs+1, ...,vs+q).

    2. Contraction

    C(1, .., i1, i+1, ..r,v1, ..,vj1,vj+1, ..,vs)

    = T (1, .., i1, k, i+1..., r,v1, ..,vj1, ek,vj+1, ...,vs)

    where the summation over the k index of the dual basis vectors makes itclear that the definition is indeed basis independent.

    When the dual bases of V and V are changed, the induced bases of thetensor product spaces change in the obvious way (which should be writtendown) as do the components of tensors with respect to the bases.

    It is a straightforward matter to extend these ideas to vector spaces overother fields such as the complex numbers. The latter are needed in the studyof complex manifolds. However only real finite dimensional manifolds will bestudies in this course. It is also straightforward to extend them to modulesover rings. We shall use the latter when we study tensor fields on manifoldswhere the field of real numbers is replaced by the ring of real valued functionsand vector spaces are replaced by modules. The tensor product defines analgebra of tensors.

    More generally An algebra is a real vector space V with a product :V V V such that

    i) v 0 = 0 v = 0ii) (pv) u=v (pu) =p(v u)iii) v (u + w) = v u + v wiv)(u + w) v = u v + w ufor all u,v,w in V and any scalar p.Exercise 1a:Let {ei} be a basis of a n-dimensional vector space V , and let {

    i} andbe the dual basis of V .

    1) What is the dual space of V ?2) Evaluate i j(ea, eb), where i, j, a, b = 1..n.3) Let n = 2.(a) Prove that the tensor products {i j} do form a basis of the vector

    space V V of (0,2 ) [covariant] tensors.


  • (b) If = 31 + 62 and = 1 2, compute the components of with respect to the induced basis above. If u = 4e1 + e2 and v = e1 e2,verify, by explicitly computing both s