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

a=1

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).

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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

a

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).

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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)

ia

bj .

Henceea

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

a.bea

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 = ta1a2...ar..............b1b2...bsea1 ea2 ....ear b1 b2 ... bs ,

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

T (1, ....r,v1, ...,vs) = ta1...ar..........b1...bs

1a1 ......

rarv1

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 = ta1a2...ar..............b1b2...bsea1 ea2 ....ear b1 b2 ... bs ,

R = ra1a2...ar..............b1b2...bsea1 ea2 ....ear b1 b2 ... bs ,

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

pT+qR = (pta1a2...ar..............b1b2...bs +qra1a2...ar..............b1b2...bs

)ea1ea2....earb1b2 ...bs .

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 = uar+1ar+2...ar+p.............. bs+1bs+2...bs+q

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

ta1a2...ar..............b1b2...bsuar+1ar+2...ar+p.............. bs+1bs+2...bs+q

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

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

ta1a2..ai1pai+1,..ar..........................b1b2bj1pbj+1...bs

ea1ea2...eai1eai+1 ....earb1b2..bj1bj2..bs .

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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.

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(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 sides of the following equation, that (u, v) = (u)(v). Compute the basis of V dual to the basis ofV formed by (u, v), and compute the corresponding new induced bases ofV V, V V , V V in terms of the bases induced from {ei} and {

i}.Compute the components of with respect to the new induced basis.

Exercise 2a:Let g be a tensor of type (0,2) on a vector space V . Let {ea) and {

a}be dual bases of V and V .. Let g = gab

a b, v = vaea and w = waea..

Compute g(ea, eb) and g(v, w). If the symmetric part of g is defined bysym g(v, w) := 1

2[g(v, w) + g(w, v)] for any v, w V , compute the com-

ponents of sym g with respect to the induced basis. Similarly computethe components of the skew (or anti) symmetric part of g defined by altg(v, w) := 1

2[g(v, w) g(w, v)]. Let h be a tensor of type (1,3) on V . Let

h(, u, v, w) = h(, u, w, v) for any V , u, v, w V . Express thisin terms of the components of h with respect to the induced basis for (1,3)tensors. Show that if h(, u, v, w) also changes sign under the interchange ofany u and v then h(, u, v, w) = 1

6[h(, u, v, w)+h(, v, w, u)+h(,w, u, v)

h(, v, u, w)h(,w, v, u)h(, u, w, v)]. Write this relation in terms of thecomponents of h.

1.2 Basic point set topology definitions

1. A topological space is a set S together with a collection, U , of subsets ofS such that

(a) S U and the empty set U(b) If ui U, i = 1..n then the finite intersection

ni=1ui U

(c) Arbitrary unions of elements of U lie in U .2. The elements of U are called open sets. The collection U is called a

topology on S.3. A set A S is said to be closed if and only if it is the complement of

an open set.4. A set B is called a basis for a topology on S if(a) B(b) bBb = S(c) If b1 and b2 belong to B then b1 b2 = bDb, for some subset D B.

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5. Let (S, U) be a topological space and let A S. Let UA = {A u,u U}. Then UA is a topology on A called the relative topology on A.

6. Let (S, U) and (T, V ) be topological spaces.(a) A function f : S T is said to be continuous if the inverse images

of open sets are open.(b) f is called a homeomorphism if f is bijective (one to one and onto,

that is injective and surjective) and both f and f1 are continuous.Two homeomorphic topological spaces are topologically equivalent.The above are all general topological definitions. The following are two

special topological properties.7. A topological space S is said to be connected if and only if the only

sets that are both open and closed are and S.From this definition it follows that S is connected if and only if it is not

the union of two disjoint non-empty open sets.8. A topological space S is called a Hausdorff space when, for any two

points s1 6= s2, there exist open sets u1 and u2 with s1 u1 and s2 u2 suchthat u1 u2 = .

1.3 n dimensional Euclidean space

Let Rn (sometimes written En, depending on the context) be the set of allordered ntuples of real numbers, that is

Rn = {x = (x1, x2, ...xn) | x R, 1 n}. (1)

Here x is the -th coordinate of the point x. Rn can be given a vector spacestructure in the obvious way. It also has a standard topological structure.Introduce the metric d(x, y), for x, y Rn

d(x, y) =[(x

y)(x y)]1/2

, (2)

making Rm into a metric space and in fact a topological space, with the unionof open balls

bx,r = {y Rn | d(x, y) < r} (3)

as the open sets. The n dimensional vector space Rn with the above metricis called n dimensional Euclidean space.

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2 Differentiable manifolds (real)

Definition An n-dimensional chart (coordinate chart) on a Hausdorff topo-logical space (with a countable basis) M is a pair (U, ) where U is an openset of M and : U (U) Rn is a homeomorphism onto its image.

For any p U we can define the coordinates of p to be the coordinatesof x = (p) Rn i.e.

x = (p), = 1..n. (4)

The x or n-tuples (x1, x2, ...., xn) are called the local coordinates of p deter-mined by the coordinate chart.

Definition: A (smooth) ndimensional differentiable structure on M isa collection of ndimensional charts (Ui, i) i I such that

(i) M = iIUi(ii) For any pair of charts (Ui, i) and (Uj, j) with Ui Uj 6= the map

j 1i : i(Ui Uj) j(Ui Uj) is smooth, that is all partial derivatives

exist up to any order.(iii) We shall always take a differentiable structure to be a maximal set

of charts i.e. the union of all charts which satisfy (i) and (ii).The collection of charts (Ui, i) i I is called an atlas.Note:(a) Since this is a course in geometry diagrams can be very helpful, and

should be used, in understanding analytical statements.(b) As far as the calculus is concerned: i(UiUj) and j(UiUj) are two

non-empty open sets in Rn and the functions j 1i , functions on subsets

of Rn, are called transition functions (sometimes overlap maps). Locally,but not globally, a manifold looks like Rn. In addition to smooth (C)manifolds one can consider Ck manifolds, C0 or topological manifolds orC (analytic) manifolds. In this course, for simplicity and with not muchloss, we shall confine ourselves to smooth manifolds. The charts providecompatible coordinate systems for the manifolds, and are in fact often calledcoordinate charts. A local coordinate system about (or of) a point p onM refers to a coordinate system obtained from an admissable coordinatechart containing p. Local coordinate systems are often said to be related bycoordinate transformations. They enable one to calculate locally using theessential tool of multi-variable calculus of n variables, e.g. using the chainrule for ordinary and partial derivatives.

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(c) We shall also always consider, without loss of generality, only con-nected manifolds. Manifolds that are not connected can be considered asthe union of connected manifolds, and then (see Lamberts notes for theproof) all charts have the same value of n.

Definition: A (smooth) differentiable manifold of dimension n is a con-nected, Hausdorff topological space with a countable basis, together with a(smooth) n dimensional differentable structure.

Examples:(a) Rn is an n-dimensional manifold. A single chart that covers the whole

of Rn is (Rn, id) where id is the identity map id(p) = p. This provides thestandard differentiable structure on Rn.

(b) Any open sub-set U Rn is an n-dimensional manifold. In fact anyopen subset U of a manifold M with charts (Ui, i) is also a manifold with,for example, the charts (U Ui, i). Closed subsets of a manifold with theidentity map charts are not manifolds. The notion of a manifold can beextended to that of manifold with boundary.

(c) A set A is called an affine space, modelled on the real vector space V ,if there is a map, called an affine structure, A V A, by (p, v) p + vwhich has the properties:

(i) (p+ v) + w = p+ (v + w),p A and v, w, V(ii) p+ 0 = p,p A and 0 V(iii) for any pair of points p, q A, there is a unique element of V , denoted

(q p), such that p+ (q p) = q.By definition dimA = dimV (= n say) and A is some times written An.

Affine spaces are smooth manifolds and admit global coordinate systemswhich enable A to be identified with Rn. A choice of affine coordinates on Acorresponds to achoice of a point p0 A (the origin) plus a choice of basis, say{ea} for V (the axes). Then for any p A, p = p0+(pp0) = p0+x

aea,where(p p0) = x

aea V, using property (iii) above. The affine coordinates ofA with respect to the origin p0 and the axes {ea} are given by {x

a}. Theyclearly define a single chart that covers A, (A, ), with : A Rn, byp (x1, ..., xn). They are also clearly not unique. An important physicalexample of an affine space is four dimensional Minkowski space-time andPoincare transformations are affine transformations. Extensive discussionsof the geometry of affine spaces can be found in the text by Crampin andPirani.

(d) The n-spheres, Sn, (x1)2+(x2)2+(x3)2+...(xn+1)2 = 1, (x1, x2, ....xn+1) R

n are n-dimensional manifolds that require at least two charts.

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(e) RP n, the n dimensional real projective space is an n-dimensionalmanifold.

(f) If M and N are m and ndimensional manifolds respectively thenthe product M N is an (m+n)dimensional manifold - the product man-ifold. For example the n-torus T n = S1 S1.. S1, the n-fold product ofthe circle.

(g) etc. etc. etc. All these are discussed in the texts and some inLamberts notes.

The notion of a real manifold is simply extended to that of a n-dimensionalcomplex manifold where (Ui, i) is replaced by C

n and the transition func-tions are required to be holomorphic on their domains. Such a complexmanifold is also a real 2n-dimensional manifold. In addition to finite dimen-sional manifolds one can consider infinite dimensional manifolds, for examplethose modelled on Banach spaces. Here, of course, analysis plays a moreimportant role. (See texts by e.g. Lang, Marsden et al).

Exercise 1b*Give the definition of a smooth manifold, M , and explain what a chart

and atlas are. Define the product manifold M M.Consider the circle of unit radius, S1, in R2. Show that S1 is a smooth

one dimensional manifold by constructing an atlas of two charts and showingthat the overlap map is smooth. Hence or otherwise show that the torus T 2

can be given the structure of a smooth two dimensional manifold.Exercise 2b*Consider the 2-sphere S2 in R3 given by (y1)2+(y2)2+(y3)2 = 1. Let the

complex function be given by y1+iy2

1+cy3, where the denominator is non-zero

and c is either 1 or 1. Show that the coordinates of any point on S2 are

given by (y1, y2, y3) = ( +1+

, i()1+

, c(1)1+

), as long as y3 is not equal to c.

Show that S2 is a smooth, two dimensional manifold by constructing andidentifying an atlas of two charts, and by showing that the overlap map issmooth.

Exercise 3bThe group of 2 2 matrices with determinant equal to one is denoted

SL(2,R). Show that SL(2,R) is a three dimensional manifold.Next consider maps between manifolds and the geometrical equivalence

of manifolds.Exercise 4bRead a textbook (or Neil Lamberts notes) about real projective spaces

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and then check your understanding by working through this problem.Consider the real projective space

RP n = (Rn+1 {0})/

where xx for any non-zero real number where x = (x1, x2, ...xn+1) R

n+1 {0}. Give a geometrical description of RP 1. Define homogeneouscoordinates. Show that RP 2 is a smooth two dimensional manifold byconstructing an atlas of three charts and showing that the transition functionsare smooth. Construct an atlas for RP 2 RP 2. Why is RP 2 RP 2 notdiffeomorphic to either R4 {0} or T 3?

Definition: Let M, (Ui, i), i I and N, (V, ), A be two mani-folds of dimensions m and n respectively. A map from M to N , f : M N,is smooth if and only if

f 1i : i(f

1(V)) Rn

is smooth for all i I and A.By using the chain rule it is straightforward to show thatTheorem: Let M , N and P be manifolds with f : M N and g : N

P smooth. Then g f : M P is also smooth.Definition: If f : M N is a bijection (one to one and onto or both

an injection and a surjection), with both f and f1 smooth then f is calleda diffeomorphism.

Two diffeomorphic manifolds (that is there is a diffeomorphism betweenthem) are geometrically equivalent. Diffeomorphisms play the role in dif-ferential geometry that isomorphisms do in group theory or vector spacetheory.

Definition: A (real-valued) function f : M R is smooth if and onlyif for each chart (Ui, i) in the differentiable structure of M

f 1i : (Ui, i) Rn R

is smooth. The set (ring) of such functions on M is denoted C(M).If f : M N is smooth and h : N R is a smooth function on N then

h f is a smooth function on M . On the other hand if h : M R is asmooth function on M there is not necessarily a smooth induced function onN .

Often a shorthand local coordinate presentation is used e.g. for f , f(x1, ....xn).

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Theorem: C(M) is an algebra with addition and multiplication de-fined point wise.

(f + g)(p) = f(p) + g(p),

(f)(p) = f(p)

f g(p) = f(p)g(p)

As an illustration it is easy to verify that f g is in C(M). For

(f g) 1i = (f 1i )(g

1i ),

and f 1i and g 1i are both smooth, so their pointwise product is too.

Again using the general idea of maps between manifolds we can define acurve.

Definition: Let (a, b) R be an open interval. A smooth map C :(a, b) M is called a smooth (parametrized) curve on M .

When (Ui, i) is a chart and UiC(a, b) 6= its corresponding coordinatepresentation is given by the n real valued functions of one variable determinedby i C. The image of C in M is called a path. A curve provides apath with a parametrization and an orientation. A path is determined bymore than one curve, for example ones obtained by reparametrization. Areparametrization may be orientation preserving or orientation reversing.

3 The tangent space TpM , tangent vectors

and tensors over TpM

Here the introduction of tangent vectors to a manifold at a point p is consid-ered. It will be seen that these naturally form a n = dim M vector space,TpM , the tangent space at p. In this course this will be the basic vectorspace from which the dual (cotangent) space and tensor product spaces willbe constructed. Two familiar concepts from Rn underlie the two standardintroductory approaches to tangent vectors. We shall consider them in theorder, first the directional derivative - and second the tangent vector to acurve. The first is more algebraic than the second which is more geomet-rical. Often the reverse order is followed. The calculations underlying thetheory here make use of conventional multi-variable calculus lifted from Rn

using charts. First we introduce a definition that provides a concept and bitof nomenclature that are particularly handy when discussing vectors.

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Definition: For a point p M let C(p) be the set of functions suchthat

i) f : U R where p U M and U is an open set.ii) f C(U).Definition: A tangent vector at a point p M is a mapXp : C

(p) Rwhich satisfies

i) Xp(f + g) = Xp(f)+ Xp(g)ii) Xp(constant map) = 0iii) Xp(fg) = f(p)Xp(g) +Xp(f)g(p) this is the Leibniz rule (c.f. differ-

entiation of a product).Note: In general objects that satisfy these (essentially algebraic) proper-

ties are called derivations.Definition: The set of tangent vectors at a point p M is called the

tangent space to M at p and is denoted TpM.With this definition a tangent vector is a linear map from the infinite di-

mensional vector space (algebra) C(p) to the reals so it is an element of theinfinite dimensional dual vector space. However, as will be seen, the secondand third conditions restrict the tangent space to be finite dimensional.

Example: Consider Rn with the usual single chart, (Rn, id). Thenthe directional derivative operator in the direction of a vector (1, 1, ..., 1),X =

x, where {}, ( = 1..n) are real numbers, is a tangent vector.

Example: Consider a smooth curve in Rn, C : (0, 1) Rn by t (C1(t), C2(t), ..., Cn(t)). Recall from elementary calculus and geometry thatthe tangent vector to a point p = C(t1) has components

(d

dtC1 |t=t1 ,

d

dtC2 |t=t1 , .....,

d

dtCn |t=t1).

Now if f C(p), then Xp : C(p) R defined by

Xp(f) =d

dtf(C(t)) |t=t1

is, according to the above definition, a tangent vector to Rn at p = C(t1).On the other hand, using the chain rule

Xp(f) =d

dtC |t=t1 f(x

) |x=C(t1) .

Notation: f =fx

etc.

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Definition: Let p M . Consider a chart (Ui, i), with p Ui, defininglocal coordinates about p, that is for all q Ui,

i(q) = (1i (q), ...,

ni (q)) = (x

1(q), ..., xn(q))

. Define n maps ( = 1..n)

x|p: C

(p) R, by f

x|p f

= (f 1i )(x

1(p), ..., xn(p)) = (f 1i )(i(p)).

Theorem: Each of the maps x

|pis a tangent vector to M at p.The proof of this result follows directly from the usual properties of the

derivative in Rn.Theorem: TpM is an ndimensional real vector space and a set of basis

vectors is the nmaps x

|p.Hence a general element of TpM is of the form Xp =

x

|p,where {}

are real numbers.The proof of the theorem is in two parts. The first to define addition and

scalar multiplication in the obvious way so that it is clear that TpM is a realvector space. The second, and intuitively obvious but formally not trivial,is to show that {

x|p} forms a basis of this vector space, and consequently

that it is ndimensional. To do this it must be shown that { x

|p} arelinearly independent and span the vector space. The first is straightforward,just let the maps act on the coordinate functions. The second can be shownby making use of a Taylor-type expansion result contained in the followinglemma.

Lemma: Let (x1, ..., xn) be a coordinate system about p M . Then forevery function f C(p) there exits n functions f1, f2, ..., fn in C

(p) with

f(p) =

x|p f , and

f(q) = f(p) + (x(q) x(p))f(p),

where the latter expression is defined.Definition: Let C : (0, 1) M by t C(t) be a smooth curve on M

and let p = C(t1), for t1 (0, 1). The tangent vector Tp(C) TpM to thecurve at p is defined by

Tp(C)(f) =d

dtf(C(t)) |t=t1 .

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Let (Ui, i) be a chart with p Ui defining local coordinates for all q Ui,by

i(q) = (1i (q), ...,

ni (q)) = (x

1(q), ..., xn(q)).

Then (by effectively using the chain rule) we have the explicit expansion orcoordinate presentation

Tp(C) =d

dt(i C(t))

x|p .

All smooth curves through p M define a tangent vector to M at p.Conversely all tangent vectors to M at p can be realized as the tangentvector to a (non-unique) curve. For example, if Tp = T

x

|p for constantsT this tangent vector at p is tangent to the curve C : (, ) M withcoordinate presentation i C(t) = x

(p) + tT . This latter point leads tothe following theorem, which we just note, the content of which is often usedas a starting point for the discussion of vectors.

Theorem: TpM is isomorphic to the set of all curves through p Mmodulo the equivalence relation that C(t) C (t) iff

d

dt(f C) |t=t1=

d

dt(f C ) |t=t1

for all f C(p) where C(t1) = C(t1) = p.

It is important to understand how the components of a vector changeunder a change of basis. Consider, in particular, a change of a coordinatebasis to another coordinate basis. Essentially by the chain rule we have

Theorem C: Let (x1, ..., xn) = 1 and (y1, ..., yn) = 2 be two coordinate

systems at a point p M with U1 U2 6= and suppose that Xp TpM . If

Xp = A

x|p and Xp = B

y|p,

thenB = A/x(21)

evaluated at 1(p). Here 21 = 2 11 : 1(U1) R

n 2(U2) Rn is

the smooth function with coordinate presentation, written variously, y =21(x

1, ...., xn) = y(x1, ...., xn). One usually just writes and calculates thiscoordinate transformation rule for vector components more simply, e.g. asB = A y

x.

15

Since TpM is a vector space we can apply our general vector space theoryin this particular case and construct tensors and tensor product spaces.

Definition: The cotangent space to M at p M is the dual vector spaceto TpM and is denoted T

pM .Definition: Let p M . Consider a chart (Ui, i), with p Ui, defining

local coordinates about p, for all q Ui,

i(q) = (1i (q), ...,

ni (q)) = (x

1(q), ..., xn(q)).

The dual basis (of T pM) to the coordinate basis for TpM ,

x|p, is denoted

dx |pand satisfies the defining condition

dx |p,

x|p =

.

Hence if we have a vector and a co-vector expanded in the dual coordinatebases

Xp = X(p)

x|p, p = (p)dx

|p

thenp, Xp = (p)X

(p).

If we consider again the coordinate transformation in Theorem C above,and if a co-vector has the expansions p = dx

|pand p = dy |pin the

two corresponding coordinate bases then

=

x|p

21,

this is usually written more simply as = y

x.

Since coordinate transformations are invertible we can also think in termsof the inverse transition function

12 = 1 12 : 2(U2) R

n 1(U1) Rn written variously as

x = 12(y1, ...., yn) = x(y1, ...., yn) = x(y).

The (Jacobian) matrices ( y

x) and (x

y) are inverses of each other so we also

have, and write, = x

yetc.

It should be noted that while coordinate bases are often very convenient,at least locally, and are often used, it is also the case that there are circum-stances where the use of other bases is more helpful. For example when

16

metrics are considered it is often useful to orthonormal bases. Coordinatebases are sometimes called holonomic bases and non-coordinate bases arethen called anholonomic bases.

All this, and the results of section A, extend in the expected way totensors over TpM and their expansion in bases. Very Briefly:

Definition: An (r,s) -tensor Tp at a point p M is an element of

T(r,s)p M = (rTpM) (

sT pM).Hence, from section A, we can write Tp in terms of its components with

respect to the (induced) coordinate basis as:

Tp = T1...r..........1....s.

(p)

x1|p ...

xr|p dx

1 |p .... dxs |p .

Exercise 1c: The first problem on page 31 of Neil Lamberts notes-henceforth NLN- (action of a tensor, as a multilinear map, on r co-vectorsand s-vectors expressed in terms of components with respect to coordinatebases).

Exercise 2c: The second problem on page 31 of NLN (change of compo-nents of a tensor under change of coordinates, and hence the correspondingcoordinate bases).

Exercise 3c: Read the definitions of symmetric and anti-symmetric (orskew symmetric) tensors of pages 32 and 33 of NLN. Prove that the totalcontraction of a symmetric (2, 0) tensor and an anti-symmetric (0, 2) tensoris zero. Here total contraction means contraction, pairwise on all indices.

4 Tensor fields; vector fields, integral curves

and flows

4.1 Tensor fields

A (smooth) (r, s) tensor field assigns, in a smooth way a (r, s) tensor to eachpoint of the manifold. Important examples include functions - (0, 0) fields,vector fields -(1, 0) fields, covector fields or one-forms - (0, 1) fields, differ-entiable exterior forms - totally anti-symmetric covariant or (0, s) tensors,metrics tensors- totally symmetric (0, 2) tensors, etc.

Definition: The union of all tangent (respectively cotangent, respec-tively type (r, s) tensor) spaces toM is called the tangent (respectively cotan-

17

gent, respectively type (r, s) tensor) bundle TM = pMTpM , (respectively

T M = pMT

pM , respectively T(r,s)M = pMT

(r,s)p M).

These are examples of fibre bundles, in particular vector and tensor prod-uct bundles, and are only mentioned here. Further details, such as the dif-ferential structure, can be found in the text books, e.g. Nakahara and, iftime, later in the course.

The dimensions of the manifolds TM and T M are 2 dimM and thesemanifolds are each locally homeomorphic to Rn Rn.

Definition: A smooth vector field is a map X : M T (1,0)M = TMsuch that X(p) = Xp TpM , and for all f C

M the mapping M R byp X(f)(p) is C.

Definition: A smooth co-vector (or one form) field is a map : M T (0,1)M = T M such that (p) = p T

pM and (X) : M R is inC(M) for all smooth vector fields X.

Definition: A smooth (r, s) tensor field is a map T : M T (r,s)M such

that T (p) = Tp T(r,s)p M and T (1...r, V1, ...Vs) : M R is in C

(M) forany r smooth co-vector fields 1...r and s smooth vector fields V1, ...Vs.

For vector fields (and other tensor fields) we shall drop the explicit sub-script p and, for example, write in a chart V = V

x= V (x)

xwhere

the components of vector fields (and similarly for other tensor fields) are nowsmooth functions.

It is important to understand the behaviour of tensors at a point andtensor fields under maps of manifolds.

Definition: Let f : M N be smooth. If Xp TpM then thepushforward of Xp is the tangent vector to N at f(p), denoted fXf(p),defined by

fX : C(f(p)) R by

g Xp(g f).

That fXf(p) is indeed a tangent vector is easy to show using the proper-ties of Xp.

Definition: Let f : M N be smoooth. Then the (induced) tangentmap at p is the linear map f : TpM Tf(p)N by Xp fXf(p).

Theorem: Let f : M N be smoooth where M is m dimensional andN is n dimensional. Let {xi} be local coordinates on U, about p M , and

18

let {y} be local coordinates about f(p) in N

Let Xp = Ai

xi|p then

fXf(p) = Ai

xi|p (y

f)

y|f(p) .

The proof uses the chain rule.In a similar way contravariant tensors at p, that is tensors of type (r, 0),

push forward.Vectors of a vector field can be pushed forward point wise, but the result

is not necessarily a vector field. Similarly for contravariant tensor fields.Exercise 1d: Write down the analogue of the theorem given just above

for a contravariant tensor.In contrast to tangent vectors one-forms at p (and covariant tensors, that

is tensors of type (0, s)) pull back under manifold maps. We consider thedual linear map to the tangent map.

Definition: The linear map f : T f(p)N T

pM , by f, dual to

the tangent map above, is defined by

f (Xp) = f,Xp = , fXf(p).

Theorem: Let f : M N be smoooth where M is m dimensional andN is n dimensional. Let {y} be local coordinates on V N and let {xi}be local coordinates on f1(V ) U M . If

= dy |f(p)

then

f =

xi(y f)dxi |p .

The analogue for covariant tensors at a point is obvious. In contrast tocontravariant tensor fields the pullbacks of covariant tensor fields are againtensor fields.

Exercise 2d*: Let (u, v) and (x, y, z) be affine coordinates for the affinespaces A2 and A3 respectively. Let f : A2 A3 by

(u, v) (u2 + v2, u2 v2, 2uv),

and let h : A3 A2 by

(x, y, z) (ay + z,x+ z).

19

Let P = (y + z) x

+ x y

and = ydx xdy + zdz be a vector field and a

one form on A3..(i) Compute P at p = (2, 1,1) and hP at h(p). Find the value of the

real number a for which hP is a vector field on A2.

(ii) Compute , P and find the points at which this is zero.(iii) Compute f .

4.2 The commutator or Lie bracket of two vector fields

It is a straightforward matter to see that the product of two vector fieldsdefined in a natural way is not a vector field. For if X and Y are two vectorfields and the product XY is defined by XY (f) = X(Y (f)) the Leibniz ruleis not satisfed, that is it is not a derivation. However an important vectorfield, their commutator, can be constructed from X and Y.

Definition: The commutator of two (smooth) vector fields X and Y isthe (smooth) vector field [X,Y ] defined by [X,Y ](f) = X(Y (f)) Y (X(f))for any f CM .

It is a straight forward matter to see that this is a derivation, in particularit satisfies the Leibniz rule. Note that [X,Y ] = [Y,X].

Exercise 3d: What goes wrong if one attempts to define a vector andcommutator vector field by the product rule XY (f) = X(f)Y (f)?

Exercise 4d: (i) In the simple cases it is easiest to use linearity andcompute a commutator term by term. However there is a general expressionin terms of coordinates which you should check. If in a particular coordinatesystem {x}

X = X(x)

xand Y = Y (x)

xthen

[X,Y ] = (X

xY Y

xX)

x.

(ii) Let (x, y, z) denote affine coordinates on the affine space A3. Com-pute [V,W ] where

V = x

x z3

y+ y2

z,

W = 2

x+ x

y

z.

20

Theorem: With the product of two vector fields defined as the commu-tator the space of vector fields is an algebra.

Moreover it is a Lie algebra (infinite dimensional) and the Jacobi identityholds for any three vector fields on M , X,Y, Z viz

[[X,Y ], Z] + [[Y, Z], X] + [[Z,X], Y ] = 0.

4.3 Integral curves and flows

Given a vector field X on M we can construct (a family of) curves that havethe property that their tangent vector at any point p is equal to X at p. Inthis sense the curves integrate the vectors of the vector field.

Definition: Let X be a vector field on M and consider a point p M .An integral curve of X passing through p is a curve C(t) in M such that

C(0) = p and C(d

dt) = XC(t)

for all t in some open interval (, ) R.The coordinate presentation of the condition for a curve to be an integral

curve of a given vector field X is a system of first order ordinary differentialequations with given intial conditions. The curve is given by the uniquesolution of these equations which exits locally by the theory of ordinary dif-ferential equations. The geometry of integral curves is obtained by givinggeometrical interpretations to standard results from (autonomous) differen-tial equations theory. To see this introduce a chart and local coordinatesystem, (U, ) and {x}, about p U so that X = X(x)

x|1(x) and

C(ddt

) = dC

dt(t)

x|C(t). Here C

= C gives a coordinate presentationof the curve. The curve is an integral curve of the vector field if, for givenX, C satisfy the odes:

d

dtC(t) = X(C(t))

with intial conditions C(0) = x(p).Definition: A vector field X is said to be complete if for all p on M the

integral curve of X can be extended to a curve on the manifold for all valuesof t.

Definition: The flow generated by a vector field X is a differentiablemap : R M M such that

21

i) at each point p M the tangent to the curve Cp(t) = (t, p) at p is Xii) (0, p) = piii) (t+ s, p) = (t, (s, p))To find the flow we just solve the odes d

dt(t, p) = X((t, p)), with intial

conditions (0, p) = p, for the integral curves (t, p) passing through p att = 0. The solutions satisfies condition (iii) because the o.d.e. system isautonomous.

For each t the flow defines a map t : M M such that t+s = t s.For t = small it follows that, in local coordinates, x, about p

(p) = (, p) = x(p) + X(p) + O(2).

Hence for small enough t t(p) is injective and smooth. Hence it is a diffeo-morphism onto its image (at least for an open set in M).

Definition: A one parameter group of diffeomorphisms of M is a collec-tion of diffeomorphisms with t R such that

(i) 0 is the identity map(ii) t =

1t

(iii)t(s) = t+sDefinition: The one parameter group of diffeomorphisms generated by

a vector field X is, for each t, the map t : M M by p (t, p).Definition: The (infinitesimal) generator of a one parameter group of

diffeomorphisms t is the vector field given by X =tt |t=0..

Exercise 5d*: Let (x, y, z) denote affine coordinates on the affine spaceA3. Let

t : (x, y, z) (xe3t, ye2t, z 2ta)

be a one parameter family of maps of A3.(i) Verify that t is a one-parameter group of diffeomorphisms when the

number a equals one. Is this the unique value for a?(ii) When a = 1 show that the vector field V = 3x

x+ 2y

y 2

z

generates t.(iii) Compute the Lie bracket [V,W ] of V and W , where W = y

x+x

y.

(iv) Find the one-parameter group of diffeomorphisms generated by W .

5 Differential forms

Exterior pforms are totally anti-symmetric tensors of type (0, p). Theyadmit an (exterior) product of a pform and a qform, the result of which

22

is a(p + q)form. This product is, modulo conventional numerical factors,the totally skew symmetric part of the tensor product of the forms. Un-fortunately different authors choose different conventions (usually one of twostandard ones) for the numerical factor so care must be taken when readingthem. The resulting associative algebra is called the exterior algebra. Allthis can be defined over just a vector space or pointwise over a manifold.In the case of smooth manifolds and smooth tensor fields (say) there is alsoa derivative, the exterior derivative) mapping differential pform fields todifferential (p+1)form fields. In terms of components this is obtained (upto conventional numerical factors) by taking the totally anti-symmetric partof the partial derivative of the components. The exterior derivative and theLie derivative are two derivatives mapping tensors into tensors which exist,without the imposition of extra structure, on a manifold. Differential formsarise naturally in the theory of integration on a manifold as, for example, inStokes theorem and in gauge theories. They play a central role in the studyof topological properties of manifolds by cohomological techniques.

Definition: A (0, s) tensor is called anti- or skew-symmetric if

(XP (1), ..., XP (s)) = sgn(P )(X1, ..., Xs).

Such a tensor is called a sform and s is the degree of the form.Here P is a permutation and sgn(P ) is its sign. The permutation is a

bijection P : (1...s) (1....s) and can always be written in terms of an even(sgn(P ) = 1) or an odd (sgn(P ) = 1) number of interchanges (where 2neighbouring integers are permuted). If an sform has components (withrespect to some basis), 1.........s then 1.........s = [1.........s]. Here thesquare brackets indicate the totally anti-symmetric part of 1.........s , that is

[1.........s] =1

s!

P

(sgnP )P (1).........P (s) .

Example: [] =12( ) so = [] iff = .

Definition: A pform on a manifold M is a smooth anti-symmetric(0, p) tensor on M with a 0form being a function in CM .

Since (X1, ., Y, ...., Y, .., Xp) = 0 for a differential form all forms withp > n = dim M must vanish. Furthermore all nforms are proportional.

The space (module over the ring C(M)) of exterior forms on M isdenoted (M) and

(M) = 0(M) 1(M) ... n(M).

23

The exterior algebra on (M) is introduced by defining the exterior product.Definition: If p(M), q(M) then their exterior, or wedge,

product is the (p+ q)form

( )(X1, ..., Xp+q) =

P

sgn(P )( )(XP (1), ..., XP (p+q)).

Examples: Using a local coordinate system {x}

dx dx = dx dx dx dx,

dx dx dx = dx dx dx + dx dx dx + dx dx dx

dx dx dx dx dx dx dx dx dx.

Theorem: If p(M), q(M) then = (1)pq .Hence if either p or q is even or zero the wedge product is commutative; if

p and q are both odd it is anti-commutative. In particular, if is a pform,with p odd then = 0; for instance the wedge product of a one formwith itself is zero, a useful calculational result, particularly in the light of:

Theorem: A basis for p(M) at a point p M is

dx1 |p dx2 |p .... dx

p |p .

and, if dim M = n, the dimension Dp of the vector space p(M) at p is(np

)and the dimension of the vector space (M) at p is 2n. p(M) at p,

with the exterior product is an algebra.In terms of a local coordinate basis we write a pform (field) a

=1

p!1.........pdx

1 dx2 .... dxp ,

where the numerical factor takes care of double counting and 1.........p =[1.........p].

Example: When n = 3 and p = 2, = 12!dx

dx = 12dx1 dx2 +

23dx2 dx3 + 31dx

3 dx1.Example: When n = 3; D0 = 1, D1 = 3, D2 = 3,D3 = 1. In E3 and

rectangular Cartesian coordinates let

= xdx+ ydy + zdz,

= xdx+ ydy + zdz. Then

= xdy dz + ydz dx+ zdx dy,

24

where (x, y, z) is the vector or cross product of the two vectors given by(x, y, z) and (x, y, z). [cf Hodge duality in Euclidean three-space].

Theorem: If f : M N is a smooth map and and are smoothforms on N then

f ( ) = f f .

Definition: When p > 0, the interior product of a tangent vector V anda pform is the (p 1)form V , defined, pointwise, to be

(V )(X1, ...., Xp1) = (V,X1, ...., Xp1)

for any (p 1)vectors X1, ...., Xp1. A standard alternative notation is iV ,both will be used.

Theorem: The interior product is an anti-derivation (an example of agraded derivation) and nilpotent, viz

iV ( ) = iV + (1)p iV where

p(M);

i2V = 0.

Definition: If p(M) then the exterior derivative of , d p+1(M) and is defined by

d(X1, ...., Xp+1) =

i

(1)i+1Xi((X1, .., Xi1, Xi+1, ..., Xp+1))

+

ij

(1)i+j([Xi, Xj], X1, .., Xi1, Xi+1, .., Xj1, Xj+1, .., Xp+1).

If p = 0, and = f C(M)

df(X) = X(f).

Theorem: If in a local coordinate system

=1

p!1.........pdx

1 dx2 .... dxp ,

then,

d =1

p!1.........pdx

dx1 dx2 .... dxp

=1

p![1.........p]dx

dx1 dx2 .... dxp .

25

Theorem:(i) d2 = 0.

(ii) If p(M), q(M) then

d( ) = d + (1)p d,

that is d is nilpotent and an anti-derivation.Definition: If pM then is said to be closed if d = 0 and exact

if = d for some p1M .Every exact form is closed, since d2 = 0, but closed forms may not be

(globally) exact, there may be topological obstructions. The Poincare lemmasays that if M is contractible any smooth closed pform, p positive, definedon M is exact.

Theorem: Hamiltons equations of classical mechanics can be written inthe form

iV = dH,

where H is the Hamiltonian function on 2n dimensional phase space M, V isa (Hamiltonian) vector field onM and , the (canonical) symplectic two formon M, is non-degenerate (of maximal rank) and closed ( d = 0). Solutionsof Hamiltons equations correspond to integral curves of V .

By Darbouxs theorem local coordinates (q, p), = 1..n, can be intro-duced such that = dpdq

. In these canonical or symplectic coordinatesthe above equations lead directly to the equations found in elementary textson analytical mechanics

d

dtq =

H

p,dpdt

= H

q.

Theorem: If f : M N is a smooth map and is a smooth pformon N then the pull-back of , f , is a smooth pform on M . Furthermorethe exterior derivative and the pullback commute

f d = d(f ).

Definition: If p(M) and V is a vector field the Lie derivative of with respect to V is denoted V

p(M) and is given by

p = 0 : V = V d,

p 0 : V = V d + d(V ).

26

Theorem: The Lie derivative is a derivation.

If p(M), q(M) then V ( ) = (V ) + (V ).

Furthermore it satisfies (see texts) the H. Cartan formulae

V = d iV + iV d, (as above), and

V iW iW V = i[V,W ]

V W W V = [V,W ]

V d = d V .

The Lie derivative is the covariant generalization of the directionalpartial derivative which can be defined without adding extra structure to amanifold. Let be a smooth one-form and V a smooth vector field. If inlocal coordinates (x), = dx

and V = v x

then

V = (v

x +

xv)dx.

Now it is always possible to choose particular local coordinates so that V =

x1, that is V is tangent to the x1 coordinate lines. This is sometimes called

the straightening up lemma and, thinking geometrically, is obviously truealthough it needs an analytic proof. Then in these particular coordinatesthe Lie derivative takes the form of a simple directional derivative, V =(

x1)dx

. But this is a coordinate dependent result. It is straightforwardto obtain similar formulae when p 1.

Exercise 1e: Let M be a three dimensional manifold and let (x, y, z) belocal coordinates for U M . On U let three differential forms be

= xzdy dz z2dx dy,

= 2zdx xydy + x3dz,

= 3xdy 7yz3dz.

(i) Compute , and . Check, by explicit calculation, that = but = .

(ii) Compute d, d and d. Check, by explicit calculation, that d2 = 0and d( ) = d d.

27

(iii) Let V = y x

z y

+ x z

. Compute the interior product of V witheach of the three differential forms and with their exterior derivatives. Hencecompute the Lie derivative with respect to V of the three forms.

Exercise 2e*: Let (u, v, w) and (t, x, y, z) be affine coordinates for theaffine spaces A3 and A4 respectively. Let be a mapping from A3 to A4 by

: (u, v, w) (u, v 2w, v2, w + u3).

If is the 2-form on A4,

= xdt dy tdx dz

compute and d. Verify that d = d().Let P be the vector field on A4 given by

P = t + x + y + xz.

Compute P and evaluate the Lie derivative of with respect to P.Consider the 2-form, and the function f on A4,where

= dt dx+ dy dz ; f = t2 + x2 y2 2z2.

Find the vector field X which satisfies the equation X = df , and com-pute the Lie derivative of with repect to X.

Exercise 3e: Define the Lie derivative of a vector field X with respectto (or in the direction of) a vector field V to be the vector field VX by

,VX = V ,X V ,X,

for any one form 1(M).(i) Compute a local coordinate expression for VX and hence verify that

VX = [V,X].

(ii) Verify that

V (X) = (VX) +X(V ).

(iii) Read about the Lie derivative of tensors of type (r, s), in particulartensors of type (0, 2).

28

Definition: Let M be a (connected) manifold covered by charts (Ui, i).M is said to be orientable if for any charts (Ui, i) and (Uj, j) with Ui andUj overlapping there exist local coordinates, {x

} for Ui and {y} for Uj such

that J = det(x

y) is positive-definite.

If M is orientable and dim M = n, there exists a nowhere vanishingnform, , called the volume element. Two volume elements and arecalled equivalent if = h where h C(M) and h is positive-definite onM . If h is negative-definite and are inequivalent. An assignment ofa volume form to M (or more precisely an equivalence class of volume ele-ments) is the assignment of an orientation to an orientable M . An orientablemanifold admits two inequivalent orientations, often called right-handed andleft-handed or positive and negative. Let M be orientable, as above, and let = hidx

1 dx2 .... dxn on Ui,with hi C(Ui) positive-definite. Then

at any point p Ui Uj, = hiJdy1 dy2 .... dyn and hiJ is positive

definite. Hence can be extended as a volume element over M . Now letf C(M) and let M be oriented with volume element as above. Thendefine

Ui

f =

i(Ui)

f(1i (x))hi(1i (x))dx

1dx2...dxn,

where the integral on the right-hand side is the elementary multiple integralon i(Ui) R

n. This definition can be extended, on an oriented para-compact manifold, to the atlas independent definition of a function over M ,

Mf, by summing the contributions from each chart domain, as above, and

making use of a technical device, called a partition of unity subordinate tothe covering, to avoid double or repeated counting from overlaps. Furtherdetails can be found in the texts, for example on pages 166-167 at the end ofNakaharas discussion of this topic which has been used here. More gener-ally, a formulation of integration of pforms on manifolds is briefly sketchedin the next section.

6 Integration on oriented manifolds and Stokes

theorem

Integration on oriented manifolds intrinsically involves differential forms.Definition: Let Ip = [0, 1]

p = {{x1, ...., xp} Rp | 0 x 1} be a unitpcube in Rp. The 0cube is I0 = {0} R.

29

i) A pcell in M is a C map C : J M , where J is an open set in Rp

that contains Ip.ii) A 0cell is a map from {0} M , that is it just a point.iii) The support | C | of a pcell is the set C(Ip) M .Definition: A (cubical) pchain, p, in M is a finite formal linear com-

bination of pcells on M with real coefficients

p = r1C1 + ...rkCk,

where ri R and Ci are pcells.The support of the pchain is | p |= | Ci |.Clearly there is a vector space -and hence an abelian group - structure

on the space of pchains.The boundary of a pcube is a (p 1)chain constructed as the sum of

all the (oriented) sides with a plus sign for front sides and a minus sign forback sides.

Definition: Define the maps

()i : Ip1 Ip by

()i (t1, ....., tp1) = (t1, .., ti1, , ti, ..., tp1),

where = 0 or 1.These maps project a (p 1)-cube into a side of a pcube.Definition: If C is a pcell in M then the boundary of C is denoted C

and defined to be the (p 1)chain

C =

p

i=1

(1)i+1(C (1)i C

(0)i ).

The boundary of a pchain p = r1C1 + ...rkCk is

p = r1(C1) + ...rk(Ck).

Boundaries have no boundaries, that isTheorem:

2 = 0.

Definition: Let C be a pcell in M and a pform on M , then theintegral of over C is

C

=

Ip

C.

30

where if C = f(t1, ..., tp)dt1 ... dtp, the right hand side is understood tomean the usual integral of multi-variable calculus, i.e.

Ip

C =

1

0

....

1

0

f(t1, ..., tp)dt1...dtp,

Let p = r1C1 + ...rkCk be a pchain then

p

=k

i=1

ri

Ci

.

A central theorem in differential geometry is Stokes theorem. It incorpo-rates as special cases the Greens, Gausss and Stokes theorems of elementaryvector calculus.

Stokes Theorem: If p1M and is a pchain then

d =

.

Recall that pM then is said to be closed if d = 0 and exact if = d for some p1M . Every exact form is closed, since d2 = 0, butnot every closed form is exact. On Rn every closed form is exact but fortopologically non-trivial manifolds this may not be the case and is the basisof cohomology. From Stokes theorem if is closed then

= 0. This

result, plus boundary conditions, gives rise to conservation laws in physicalfield theories.

Exercise 1f: Suppose C is the spiral C : R R3, t (sin t, cos t, t) and and are the one forms on R3 with

= xdy ydx and = xydz + yzdx+ zxdy,

Evaluate

C and

C when t [0, 1].

Exercise 2f: Construct particular examples of Stokes theorem to showthat it implies

(a) Greens theorem(b) Stokes theorem for the integral of a one form along a closed curve in

R3

(c) Gausss (i.e. the divergence) theorem.Exercise 3f: (from the 2004 exam)

31

(a) State Stokes theorem for the integral of an exact (p + 1)form overa (p+ 1)chain on a manifold M .

(b) Suppose M is the subset of R3 consisting of points (x, y, z) such thatx2 + y2 > 2.

Let be the one-chain on M defined by 0 t 1

(t) = (3 sin 2t, 3 cos 2t, sin 2t).

Also let , , be one forms and a two form on M with

= (x+ y)dz + (x+ z)dy,

= yzdx+ xzdy + xydz,

=1

x2 + y2(ydx xdy),

= dx (dy + dz).

(i) Show that = 0.(ii) Show that d = and d = d = 0.(iii) Evaluate

,

and

.

(iv) Either find a function f such that df = or prove that no suchfunction exists. Repeat for and .

7 Affine connections (covariant derivative, par-

allelism, curvature)

Only affine connections are considered in this course and connection alwaysmeans affine connection.

In this section the covariant generalization of the partial derivative isconsidered. The way in which any smooth tensor field of type (r, s) can bedifferentiated to give another smooth (r, s) tensor field is exhibited. In orderto do this additional structure - called a connection - must be added to themanifold structure. In general connections are not unique. Connections canbe introduced via vector fields or one-forms. The former will be the startingpoint here but a differential form point of view will also be presented.

Up to this point bases used in applications and examples were usuallycoordinate bases. From this point just as much emphasis will be placedon general bases. In addition results are often presented from two points

32

of view, one which emphasises the role of vector fields and the other whichemphasises the dual differential form point of view. It is useful to keep inmind the following technical result.

Theorem: Consider a basis of vector fields {ea} tangent to M and dualbasis of one form fields {a}, a = 1.. dimM = n.

(i) If f C(M) thendf = ea(f)

a.

(ii) If[ea, eb] = C

dabed

where Cdab = Cdba are the structure functions for the basis, then dually

da +1

2Cabc

b c = 0.

(iii) [ea, eb] = 0 if and only if the basis is a coordinate basis for some localcoordinate system.

Comment: Since the C s are functions in general, the Jacobi identityapplied to the basis vector fields,

[[ea, eb], ec] + [[eb, ec], ea] + [[ec, ea], eb] = 0

implies that

CdabCfcd + C

dbcC

fad + C

dcaC

fbd + ea(C

fbc) + eb(C

fca) + ec(C

fab) = 0.

This identity can also be obtained (a useful exercise) by taking the exteriorderivative of the dual equation da + 1

2Cabc

b c = 0. In the special casewhere the C s are constants (e.g. when they are the structure constants of aLie algebra) the Jacobi identity above reduces to

CdabCfcd + C

dbcC

fad + C

dcaC

fbd = 0.

Exercise 1g: (a) Prove parts (i)-(ii) of the theorem. Is the converse of(ii) true?

(b) If = 12!ab

a b is a smooth two-form field on M show that

d =1

3!abc

a b c, where

abc = 3(e[cab] f [cCfab]).

33

7.1 Affine connections and covariant derivatives

Definition: An (affine) connection D at a point p M is a differentialoperator, D, which assigns to each vector field X on M a mapping DX :TM TM such that for all Y, Z TM and f CM

i) DX(Y + Z) = DXY +DXZii) DX+YZ = DXZ +DYZiii) DfXY = fDXYiv) DX(fY ) = X(f)Y + fDXY, and by definition DXf = X(f).DXY is called the covariant derivative of Y along X.Definition: Da = Dea . In particular, for dual coordinate bases {

x

, dx}with respect to local coordinates {x}, = 1...n, D = D

x.

DX can be specified completely by specifying its action on basis vectorfields and thus specifying the connection components or coefficients.

Definition: Let D be an affine connection on M . Then the componentsof D with respect to a basis of tangent vector fields {ea}, as above, are

abc = a, Deceb

so thatDeceb =

abcea.

Theorem: Let X,Y TM and let X = Xaea and Y = Yaea. Then

DXY = [Xc(ecY

a + abcYb)]ea,

Hence the connection and covariant derivative are defined by specifying theconnection coefficients (or components with respect to the basis) abc.

Definition: The covariant derivative of a vector field Y TM is definedto be the (1, 1) tensor field DY where

DY (,X) = ,DXY ,

for any one-form and vector field X.It follows that

DY = DcYaea

c

where, using a standard notation the components of DY are written DcYa

andDcY

a = ecYa + abcY

b.

34

HenceDXY = (X

cDcYa)ea.

Caution: Many authors (including Neil Lambert in his notes) write acbwhere I write abc. I prefer to put the differentiation subscript last.

Definition: The (n n matrix-valued) connection one forms are givenby ab =

abc

c.Exercise 2g: Let

abc =

a, Dece

b be the connection components of aconnection D with respect to another set of dual bases {ea,

a}, where

ea = Sbaeb;

a = (S1)abb,

and (corresponding to multiplying a n n matrix-valued function by itsmatrix inverse)

(S1)abSbc =

ac .

Let ab =

abc

c be the corresponding connection one-forms of D.(a) Prove that

ab = (S

1)acdScb + (S

1)accdS

db .

(b) Consider the special case of two local coordinate systems, and thecorresponding coordinate bases on M . Deduce the transformation law forthe connection components with respect to the coordinate bases.

Hence the connection components are not the components of a (1, 2)tensor field on M but the difference of the connection components of twoconnections are clearly the components of a (1, 2) tensor field on M . Hencetwo affine connections differ by an tensor field of type (1, 2).

The torsion tensor of a connection is a geometrically important part of aconnection (see N. Lamberts notes).

Definition: The torsion tensor of a connection D is the (1, 2) tensor Tdefined by

T (,X, Y ) = , T (X,Y ),

for any 1(M) and X, Y T (M) where the vector field T (X,Y ) is

T (X,Y ) = DXY DYX [X,Y ].

Note that the torsion of D is zero if and only if

DXY DYX = [X,Y ].

35

Theorem: If {ea} and {a} are dual bases, as above, the the components

of T with respect to these bases, T (a, eb, ec) are

T abc = acb

abc C

abc,

so thatT = T abcea

b c.

Hence, if (x) are local coordinates, then the components of T with re-spect to the local dual coordinate bases are

T =

.

Note that T abc = Ta

cb.Definition: An affine connection is called symmetric when its torsion is

zero (for then = in a coordinate basis).

Definition: Let {a} be a basis of one forms and ab a connection oneform as above. The the torsion two form of the connection, a, is

a =1

2T abc

b c,

where, as above, T abc = acb

abc C

abc.

Just as the connection one form can be thought of as a matrix-valued oneform (a matrix with on forms as entries) the torsion two form can be thoughtof as a vector valued (column matrix) two form.

The connection can be extended to define a covariant derivative on anytensor field, essentially by linearity and by demanding that the covariantderivative satisfies the Leibniz rule and commutes with contraction. Firstconsider one forms.

Definition: Let 1(M) and X T (M). Then the covariant deriva-tive of along X is the one form DX where

DX(Y ) = X((Y )) (DXY )

for any Y T (M).In terms of components with respect to dual bases {ea} and {

a} as abovethis implies that

DX = Xa(ea(b) c

cba)

b, and hence

Dec(b) = bac

a.

36

Following the same lines as above the covariant derivative of a one-form canbe defined.

Definition: The covariant derivative of a one-form is defined to be the(0, 2) tensor field D where

D(X,Y ) = DX, Y ,

for any vector fields X and Y .It follows that

D = Daba b

where, using a standard notation the components of D are written Daband

Dab = eab cbac.

HenceDX = (X

cDcb)b.

The extension of all this to an (r, s) tensor field is straightforward and detailsare left as a reading (e.g. N. Lamberts notes) exercise. It can be summarizedin the following definition which is calculationally useful.

Definition: The covariant derivative of a (r, s) tensor T is defined to bethe (r, s+ 1) tensor field DT where

DT = DcTa1...arb1......bs

ea1 ... ear c b1 ... bs ; &

DcTa1...arb1......bs

= ecTa1...arb1......bs

+a1pcTpa2...arb1......bs

+ ...+ arpcTa1...ar1pb1......bs

pb1cTa1...arpb2......bs

... pbscTa1...arb1......bs1p

.

Exercise 3g: (i) Let D be a connection and let {ea, a} be the usual

dual bases. If f CM then (Daf)a and (DaDbf)

a b are, respectivelya one-form and a tensor of type (0, 2). Show that DaDbf = DbDaf for anyf if and only if the torsion of D is zero.

(ii) Verify Cartans first structure equations for an affine connection D

a = da + ab b.

The terms on the right-hand side are often called the covariant exteriorderivative (or exterior covariant derivative) of the basis co-frame with re-spect to the affine connection. This formulation can be used an an alterna-tive starting point for the theory of connections.

37

7.2 Parallelism and curvature

Given a connection vectors (and tensors) at different points can be comparedand hte notion of parallel objects at different points can be introduced.

Definition: Let C : I R M be a curve with tangent vector X. Atensor T0 at p = C(0) is said to be parallely transported along the curve toa parallel tensor field T along the curve if

DXT = 0,

Tp = T0.

This is a system of first order ordinary differential equations, with giveninitial conditions at t = 0, so it admits a unique local solution.

Definition: Let C : I R M be a curve with tangent vector X. TheC is called an auto-parallel (or affinely parametrized geodesic) if DXX = 0.

The parallel transport of a vector Y from a point p to a point q is usuallypath (im C) dependent. When this is not the case the connection is saidto be integrable or flat. The curvature of a connection is a measure of theextent to which it is not flat, and the curvature is is zero if and only if theconnection is flat. In components with respect to a basis of vector fields{ea}, as above, the condition for D to be locally flat is that the differentialequations

DcYa = 0,

admit, locally, a unique solution with given initial condition Y a = Y a0 at p M. The integrability conditions for this equation impy that the quantities

Rabcd = ec(abd) ed(

abc) +

afc

fbd

afd

fbc C

fcd

abf

are zero. Note that Rabcd = Rabdc.

It turns out that these quantities are the components of a (1, 3) tensorfield on M , the curvature tensor of the connection, conventionally defined ina two stage process as follows.

Definition: Let X,Y be smooth vector fields on M . Then R(X,Y ) :TM TM is the linear map, Z R(X,Y )Z, where

R(X,Y )Z = DXDYZ DYDXZ D[X,Y ]Z.

A straightforward calculation using a basis, with and X = Xcec, Y = Yded,

gives

R(X,Y ) = [R(X,Y )]abea b, where

[R(X,Y )]ab = RabcdX

cY d.

38

By the linearity properties explicitly exhibited above the quantities Rabcd canbe identified as the components of a (1, 3) tensor, the curvature tensor R ofD with

R(,X, Y, Z) = ,R(X,Y )Z,

for any one form and vector fields X,Y, Z.In terms of components with respect to the dual bases above the curvature

tensor of the connection D is given by

R(,X, Y, Z) = RabcdaZbXcY d,

R=Rabcdea b c d.

Finally, exhibiting all this in terms of basis vectors and components, it followsthat

R(ec, ed) : TM TM by eb R(ec, ed)eb=Rabcdea, and

Rabcdea = R(ec, ed)eb = DecDedeb DedDeceb D[ec,ed]eb

= {ec(abd) ed(

abc) +

afc

fbd

afd

fbc C

fcd

abf}ea.

Definition: A vector field X is called a parallel vector field, with respectto a connection D, if DX = 0.

For a general connection there are no parallel vector fields.Theorem: A connection D is flat if and only if there exists a basis of

parallel vector fields.The result follows from the fact that a parallel basis, by definition, satisfies

Dea = 0.Theorem: Let D be a flat connection with zero torsion. Then there

exists a coordinate system and corresponding coordinate basis relative towhich the components of the connection are zero.

The second theorem follows because a parallel basis for a flat connectionin the zero torsion case satisfies T abc =

acb = 0. Therefore C

abc = [eb, ec] = 0.

The latter condition is the necessary and sufficient condition for there to bea local coordinate system with the basis vector fields as the coordinate basisvector fields. (The proof of the latter, involving Frobenius theorem, is notpart of this course.)

Definition: The Ricci curvature of an affine connection is the (0, 2)tensor with components Rbd = R

abad.

The curvature definition is neatly summed up in Cartans second struc-ture equation for an affine connection.

39

Definition: The curvature two-form of an (affine-) connection one formab is given by Cartans second structure equations for an affine connectionD,

ab = dab +

ac

cb.

A straightforward calculation shows that, expanded in the basis a, the cur-vature two form ab (a n n matrix-valued two form) is

ab =1

2Rabcd

c d.

Theorem: The first Bianchi identity is given by

da + abb = ab

b.

The second Bianchi identity is given by

dab + ac

cb

ac

cb = 0.

These identities are computed by taking the exterior derivatives of the firstand second Cartan structure equations for D - a straightforward exercise.

Exercise 4g*: The Lie brackets of a basis of vector fields on a manifoldM are given by

[ea, eb] = Ccabec = (J

cakb J

cbka)ec

where ka and Jab are constants and

kaJab = kb and J

ab J

ba = J

aa .

A connection D on M is defined by

Debea = Jcbkaec.

(a) Write down the definition of the torsion of a connection and verifythat the torsion of D is zero.

(b) Show thatDecDedeb = kbkdJ

ac ea.

(c) The components of the curvature tensor of D are given by

Rabcdea = DecDedeb DedDeceb D[ec,ed]eb.

Compute the Ricci tensor.

40

Exercise 5g: (i) Use Cartans second structure equations to show thatunder the change of basis given in Exercise 2(g) ab

ab where

ab = (S1)ac

cdS

db .

(ii) Verify the first set of Bianchi identities above.(iii) Verify the second set of Bianchi identities above.(Optional: Read about the tensor calculus forms of these identities and

the Ricci identities for a general tensor field. The Ricci identities for a vectorfield X = Xaea are (DcDd DdDc)X

a = RabcdXb + T fcdDfX

a.

8 Metrics on manifolds

The second additional structure on a smooth manifold M that will be con-sidered is a smooth metric g, a generalization of the familiar inner, scalar ordot product on Euclidean three-space. We shall consider non-degenerate,symmetric metric tensors of different signatures. Pure mathematicians mostcommonly deal with Riemannian metrics but physicists are also interested inmetrics of other signatures - pseudo-Riemannian metrics, most notably therelativistic metrics of Lorentzian signatures. A metric tensor provides anisomorphism between the cotangent and tangent spaces which reflects itselfin the tensor calculus notion of raising and lowering of the indices of tensorcomponents. The covariant derivative of a metric is zero with respect to aunique torsion free affine connection - the Levi-Civita connection and this isthe one that is always used (unless otherwise explicitly stated) when metricdifferential geometry is considered.

Definition: A metric g on M is a smooth symmetric and non-degeneratetensor field of type (0, 2).

Since g is symmetric g(X,Y ) = g(Y,X) for any X,Y TM ; since g isnon-degenerate g(Y,X) = 0 for any given vector field X and all vector fieldsY if and only if X = 0.

Definition: A metric g on M is called Riemannian when g(X,X) > 0for all non-zero X TM , otherwise it is called pseudo-Riemannian.

In terms of the usual dual bases ea and a:

g = gaba b, gab = gba, a, b = 1.. dimM = n,

and the determinant of the matrix (gab) is non-zero. As a consequence of non-degeneracy a metric also defines a symmetric (2, 0) tensor g1 = gabea eb

41

where gab = gba andgabgbc =

ac .

A metric defines an inner product on the tangent space at each point p M,and similarly the metric inverse defines an inner product on the cotangentspace at each point. The metric defines an isomorphism between the tangentspace and cotangent space at each point given by X TpM T

pMwhere, for any Y TpM

(Y ) = g(X,Y ),

and inversely T pM X TpM by

,X = g1(, ),

for any T pM . This corresponds to the lowering and raising of indiceswith the metric and its inverse of tensor calculus; that is, if X = Xaea and = a

a thengabX

b = a and gaba = X

a.

Conventionally one writes gabXb = Xa etc. These isomorphisms, and index

raising and lowering, extend in the obvious way to tensors of general type.Definition: Let g be a Riemannian metric on M, that is at each p M andfor any X TpM , the quadratic form g(X,X) is positive definite.

Let X,Y TpM , then(a) The norm of X is [g(X,X)]1/2. X is of unit length or norm if

g(X,X) = 1. A non-zero vector, X, can be normalized by rescaling X X

[g(X,X)]1/2.

The cosine of the angle between X and Y is g(X,Y )[g(X,X)]1/2[g(Y,Y )]1/2

and X and

Y are orthogonal if and only if g(X,Y ) = 0.Definition: A basis,ea, of vector fields is said to be an orthonormal basis

for a metric g if the corresponding metric components are

g(ea, eb) = ab = diag(1, ...1,1, ... 1).

The metric is said to have signature (p, q) when there are p plus ones andq = np minus ones. The metric is Riemannian (or of Euclidean signature)if p = n and pseudo-Riemannian otherwise, e.g it is called Lorentzian if p = 1,q = n1 (or p = n1, q = 1), etc. An orthonormal frame or basis of vectorfields always exists locally, essentially by elementary linear algebra, and the

42

signature is an invariant. The dual basis (co-frame) to an orthonormal basissatisfies

g1(a, b) = ab = diag(1, ...1,1, ... 1).

Every manifold admits a global Riemannian metric. A manifold M admitsa Lorentzian metric globally if and only if it admits a no-where vanishingvector field.

Orthonormal bases are not unique.Theorem: Let ea be an orthonormal frame for a metric g = ab

a b.Then ea = L

baeb is also an orthonormal basis if and only if the matrix-valued

functions L = (Lab ) take values in the group O(p, q)

LabLcdac = bd.

The notions of the norm and the length given above (but not alwaysangle or normalizability) can be extended to pseudo-Riemannian manifoldsby replacing g(X,X) by | g(X,X) |. The notion of orthogonality extendswithout change.

Definition: Let C : [t1, t2] M be a curve with end points, and letp = C(t1) and q = C(t2). Then the length of the path (ImC) from p to q iss where

s =

t2

t1

| g(X,X) |1/2 dt,

where X = dC(t)dt

is the tangent vector to the curve at C(t). This motivatesthe often used expression of a metric in terms of its line-element:

ds2 = gaba b = gab

ab,

as, for example, in terms of a coordinate basis, ds2 = g(x)dxdx or in

terms of an orthonormal basis ds2 = abab.

Definition: Let g and D be, respectively, a metric and a connection onM , then D is said to be a metric connection (for g) when Dg = 0.

When D is a metric connection for D and X and Y are two vector fields,with X and Y parallely transported along a curve C, then the norm of thesevectors and, when defined, the angle between them are constant along C.

Theorem: (The fundamental theorem of metric differential geometry).Let g be a metric on a manifold M , then there is a unique torsion-free, metricconnection, D, called the Levi-Civita connection.

43

If (x) are local coordinates, then the components of the Lev-Civita con-nection, the Christoffel symbols, with respect to the corresponding coordinatebases are

=1

2g(g, + g, g,).

The comma denotes partial differentiation e.g. g, =

xg.

Killing vector fields (excluded from Manifolds exam): On a manifoldwith metric g, a vector field V which satisfies the differential equations V g =0 is called a Killing vector field. If D is the Levi-Civita connection then inlocal coordinates, where V = V

xand V = gV

, Killings equations canbe written in the form DV +DV = 0, . On an ndimensional manifoldthere are at most 1

2n(n + 1) Killing vector fields, usually there are none.

Altogether they form a (representation of a) Lie algebra with commutatorthe Lie bracket. Killing vector fields are the infinitesimal generators of the(identity connected component) of the symmetry or isometry group of g. Iflocal coordinates are adapted to a particular Killing vector field V , so thatV =

x1say, then in these adapted coordinates Killings equations take the

form x1g = 0, and hence are satisfied if and only if x

1 is an ignorablecoordinate. If W is a second Killing vector field local coordinates can besimultaneously adapted to both V and W if and only if their Lie bracket iszero.

Definition: If the components, with respect to dual bases, as above, ofthe curvature tensor of the Levi-Civita connection D of a metric g are givenby Rabcd then the Riemann (curvature) tensor is the (0, 4) tensor Rabcd

a b c d where Rabcd = gaeR

ebcd, and the Ricci scalar is R = g

abRab, whereRab are the components of the Ricci tensor of D.

When Rab =Rngab the n dimensional manifold with metric is called an

Einstein space.Theorem: The Levi-Civita connection of a metric is flat, and hence

has zero curvature tensor, if and only if there is a local coordinate system,x, such that the coordinate basis of vector fields is orthonormal so thatds2 = dx

dx.Theorem: The components of the Riemann tensor satisfy the following

algebraic symmetries

Rabcd = R[ab]cd = Rab[cd] = Rcdab;

Ra[bcd] = 0,

44

and consequently has 112n2(n2 1) algebraically independent components.

These results hold, of course, in any basis.These tensor calculus results are often proven, using normal coordinates,

in relativity courses (not here). The following result is also included herefor completeness.

Theorem: The Ricci tensor of a Levi-Civita connection is symmetricand the corresponding symmetric Einstein tensor G = Gab

a b where

Gab = Rab 1

2gabR,

satisfies the contracted Bianchi identity

DaGab = 0.

The basic metric connection equations above can be summed up in Car-tans structure equations for a metric.

Theorem: Let a metric be given by ds2 = gabab and let ab be the con-

nection one-form of the Levi-Civita connection D. Then Cartans structureequations are given by

da + ab b = 0,

dgab gcbca gac

cb = 0,

dab + ab

ab =

1

2Rabcd

c d,

Rabcd = gaeRebcd, R = g

abRab.

The first equation says that the torsion of D is zero. The second says thatD is metric and the third and fourth define the curvature tensors and theircontractions. In the special case of an orthonormal basis the second equationreduces to

ab = ba.

A side comment for those doing Lie algebras: these connection one-forms( and the curvature two-forms) take values in the Lie algebra of so(p, q).

Note that a connection can be metric and have non-zero torsion and aconnection can have zero torsion without being metric - but the only metricconnection which also has zero torsion is the Levi-Civita connection.

45

Exercise 1h: Show that when an orthonormal basis is used, that isds2 = ab

ab, and da = 12Cabc

b c, then the Levi-Civita connection isab =

abc

c where

abc =1

2(Ccab Cabc Cbca).

Compare this result with the expression given earlier for the componentsof the Levi-Civita connection with respect to a coordinate basis, dx

,where

=1

2g(g, + g, g,).

Exercise 2h: A metric g, on a three dimensional manifold M is givenin local coordinates (x, y, z) by ds2 = 2e2ydx2 + 2dxdy dz2.

(i) Verify, by finding an orthonormal basis of one forms, or otherwise,that the signature of g is (1, 2).

(ii) Let f C(M) and suppose that the level set of this function, f = 1defines a two dimensional manifold S (a sub-manifold of M). Let X TM .Explain (or read) why X is called tangent to S if and only if X(f) = 0.

Let f be given by f(x, y, z) = 2xe2y. Find a non-trivial vector fieldY TM orthogonal to S (that is orthogonal to any tangent vector to S).

Exercise 3h: Let H be a two dimensional manifold and : H R3 begiven by (u, v) (t, x, y) where

t = sinh v

x = cosu cosh v,

y = sinu cosh v,

Show that the line element of the metric g induced on the sub-manifoldH from the Lorentzian three-metric g (with line element dx2 + dy2 dt2) onE

3 is the Lorentzian two-metric

ds2 = (cosh2 v)du2 dv2.

Compute the Christoffel symbols of the Levi-Civita connection of this two-metric. Use this connection to compute the covariant derivative of the oneform coshudu in the direction of the vector field

u.

Exercise 4h*: (a) Let a Riemannian metric on an n dimensional mani-fold be given by ds2 = ab

a b. Write down the first and second Cartan

46

structural equations satisfied by the the basis of one-forms a, the Levi-Civita connection one forms ab and the curvature two forms. Explain whyab = ba.

(b) When n = 2 show that there is only one independent Levi-Civita oneform and only one independent component of the curvature. Write out fullythe three Cartan structure equations for a Riemannian two-metric.

(c) Consider the metric g on R2 given in coordinates (x, y) by

ds2 = e2ydx2 + (1 + x2)2dy2.

Write down an orthonormal co-frame {1, 2} for g. Use Cartans structureequations, and this co-frame, to compute the components of the Levi-Civitaconnection.

47