has components vab...d = ( _ )*<n-a)M| ¿fafl^... S\, where s is the signature of g (so ^(n — s) is the number of negative eigenvalues of the matrix of metric components (gab)). Therefore these tensors satisfy the relations

Vab-%,...h = (— )&n~8>n! Wf... SdM. (2.37)

The Christoffel relations imply that the covariant derivatives of

Vab...d and rjab-d with respect to the connection defined by the metric vanish, i.e. b_d

Using the canonical n-form, one can define the volume (with respect to the metric g) of an n-dimensional submanifold % as — f t n'.J*

Thus can be regarded as a positive definite volume measure on Ji. We shall often use it in this sense, and shall denote it by dv. Note that d is not meant to represent the exterior differential operator here; d« is simply a measure on J(. If / is a function on J(, one can define its integral over % with respect to this volume measure as

With respect to local oriented coordinates {xa}, this can be expressed as the multiple integral

J /1<7|4 da;1 da;2... da;n, which is invariant under a change of coordinates.

If X is a vector field on its contraction with *j will be an (n — 1 )-form field X .ij, where

This (n-l)-form may be integrated over any (n— l)-dimensional compact orientable submanifold . We write this integral as

where the canonical form »3 is regarded as defining a measure-valued form d ora on the submanifold "V. If the orientation of "V is given by the direction of the normal form na, then dcra can be expressed as nad<r where da is a positive definite volume measure on the submanifold "V. The volume measure dor is not unique unless the normal na is normalized. If na is normalized to unit magnitude in a metric g on i.e. nanbgab = ± 1, then dais equal to the volume measure on V defined by the induced metric on "f (to see this, simply choose an orthonormal basis with nagab as one of the basis vectors).

Using the canonical form, one can derive Gauss' formula from Stokes' theorem: for any compact w-dimensional submanifold % of Jt,

= ( - )<n-»-i<»-s>l Tf -t" r,a de%s tX<>.tU = »-19«...

on using relation (2.37) twice. Therefore f Xad(ra= f X'.edv j o<% j«

holds for any vector field X; this is Gauss' theorem. Note that the orientation on for which this theorem is valid is that given by the normal form *j such that <n, X) is positive if X is a vector which points out of If the metric g is such that (7ai>raara6 is negative, the vector ff"6^ will point into

Some of the geometrical properties of a manifold can be most easily examined by constructing a manifold called a fibre bundle, which is locally a direct product of ^ and a suitable space. In this section we shall give the definition of a fibre bundle and shall consider four examples that will be used later: the tangent bundle the tensor bundle T^i^Ji), the bundle of linear frames or bases and the bundle of orthonormal frames

A Ck bundle over a C' (O fc) manifold is a Ck manifold ß and a Ck sur jective map n: ßThe manifold ß is called the total space, is called the base space and n the projection. Where no confusion can arise, we will denote the bundle simply by ß. In general, the inverse image 7r-1(y) of a point pe Ji need not be homeomorphic to 7T~x(g) for another point qe~4(. The simplest example of a bundle is a product bundle (?4t x si, it) where si is some manifold and the projection ir is defined by n{p, v) = p for ally eui', v esi. For example, if one chooses *4t as the circle S1 and si as the real line JR1, one constructs the cylinder C2 as a product bundle over S1.

A bundle which is locally a product bundle is called a fibre bundle. Thus a bundle is a fibre bundle with fibre S? if there exists a neighbourhood % of each point qoiJ( such that n-1^) is isomorphic with in the sense that for each point peW there is a diffeomorphism <j>p of n~HP) onto J5" such that the map r]/ defined by \]r(u) = {n{u), is a diffeomorphism i}r: ir1^/) x Since is paracompact, we can choose a locally finite covering of by such open sets <2fa. If <2fa and are two members of such a covering, the map

is a diffeomorphism of ^"onto itself for each p e n The inverse images 7r-1(y) of points p&Ji are therefore necessarily all diffeo-morphic to 3? (and so to each other). For example, the Möbius strip is a fibre bundle over S1 with fibre JR1; we need two open sets <2^, to give a covering by sets of the form x R1. This example shows that if a manifold is locally the direct product of two other manifolds, it is nevertheless not, in general, a product manifold; it is for this reason that the concept of a fibre bundle is so useful.

The tangent bundle is the fibre bundle over a Ck manifold Ji obtained by giving the set 8 = U Tp its natural manifold structure z>€ur and its natural projection into Thus the projection n maps each point of Tp into p. The manifold structure in S is defined by local coordinates {zA} in the following way. Let {a;1} be local coordinates in an open set of Ji. Then any vector VeTp (for any p&<%) can be expressed as V = Vi8j8xi\p. The coordinates {zA} are defined in 7r-1(<20 by {zA} = {a:4, Va}. On choosing a covering of Ji by coordinate neighbourhoods the corresponding charts define a Cfc_1 atlas on £ which turn it into a Ck~1 manifold (of dimension n2); to check this, one needs only note that in any overlap (<2fa n ^p) the coordinates {a;^} of a point are Ck functions of the coordinates of the point, and the components { Vaa} of a vector field are Cfc_1 functions of the components {Va/h of the vector field. Thus in 7r-1(<2fa n.®^), the coordinates {zAa} are Ck~1 functions of the coordinates {zA^\.

The fibre 7r—is Tp, and so is a vector space of dimension n. This vector space structure is preserved by the map Tp->Rn, which is given by <j>a,p(u) = Va(u), i.e. <j>a p maps a vector at p into its components with respect to the coordinates {xaa}. If {a^} are another set of local coordinates then the map {i>a,p)°{i>p,p~1)18 a linear map of Rn onto itself. Thus it is an element of the general linear group OL(n, R) (the group of all non-singular nxn matrices).

The bundle of tensors of type (r,s) over denoted by Trs{Jf), is defined in a very similar way. One forms the set S = U TrB(p), defines peur the projection n as mapping each point in Trs(p) into p, and, for any coordinate neighbourhood W in JK, assigns local coordinates {zA} to by {zA} = {x\ Ta-bce] where {a;4} are the coordinates of the point p and {Ta—bcdj are the coordinate components of T (that is, X - Ta-bc_d8/8xa®...®da;d|p). This turns S into a Ck~x manifold of dimension rar+8+1; any point u in Tre(Ji) corresponds to a unique tensor T of type (r, s) at n(u).

The bundle of linear frames (or bases) L(J() is a Cfc_1 fibre bundle defined as follows: the total space & consists of all bases at all points of JK, that is all sets of non-zero linearly independent «-tuples of vectors {E0}, E0 6 for each pe~4( (arunsfrom 1 tow). The projection n is the natural one which maps a basis at a point p to the point p. If {a-4} are local coordinates in an open set c then

are local coordinates in tt-1^),-where Ea} is the jth components of the vector E„ with respect to the coordinate bases 8/8xi. The general linear group OL(n, E) acts on in the following way: if {Ea} is a basis at p then A e OL(n, E) maps u — {p, Ea} to

When there is a metric g of signature s on one can define a sub-bundle of the bundle of orthonormcd frames which con sists of orthonormal bases (with respect to g) at all points of . 0{JK) is acted on by the subgroup 0(£(ra-M), l(n — s)) of OL(n,E). This consists of the non-singular real matrices Aab such that

where O^ is the matrix diag(+l, +1,..., +1, -1, -1,..., Tl). £(w-M) terms h(n — s) terms It maps (p, Ea) e O(^) to (p, Aab E6) e O(ui'). In the case of a Lorentz metric (i.e. s = n— 2), the group 0(n— 1,1) is called the n-dimensional Lorentz group.

A Cr cross-section of a bundle is a Cr map <J>: such that is the identity map on thus a cross-section is a Cr assignment to each point p of ^ of an element <b(p) of the fibre ir~\p). A cross-section of the tangent bundle T{J() is a vector field on a cross-section of is a tensor field of type (r, s) on a cross-section of is a set of n non-zero vector fields {Ea} which are linearly independent at each point, and a cross-section of O(ui') is a set of orthonormal vector fields on

Since the zero vectors and tensors define cross-sections in and these fibre bundles will always admit cross-sections. If is orientable and non-compact, or is compact with vanishing Euler number, there will exist nowhere zero vector fields, and hence cross-sections of T{J() which are nowhere zero. The bundles Li^Ji) and O(^) may or may not admit cross-sections; for example L(S2) does not, but L(En) does. If admits a cross-section, is said to be paraMelizable. R. P. Geroch has shown (1968 c) that a non-compact four-dimensional Lorentz manifold admits a spinor structure if and only if it is parallelizable.

One can describe a connection on Ji in an elegant geometrical way in terms of the fibre bundle A connection on may be regarded as a rule for parallelly transporting vectors along any curve y(t) in Thus if {En} is a basis at a point p = y(i0), i.e. {p, En} is a point u in L{*4(), one can obtain a unique basis at any other point y(t), i.e. a unique point y{t) in the fibre n~l{y{t)), by parallelly transporting {En} along y(t). Therefore there is a unique curve y(t) in called the lift of y(t), such that:

(3) the basis represented by the point y(t) is parallelly transported along the curve y(t) in . In terms of the local coordinates {zA}, the curve y(t) is given by

Consider the tangent space TU(L{*£)) to the fibre bundle L(JK) at the point u. This has a coordinate basis The «.-dimensional subspace spanned by the tangent vectors {{8/8t)^0|u} to the lifts of all curves y(t) through p is called the horizontal subspace Hu of TU(L(,.#)). In terms of local coordinates, so a coordinate basis of Hu is {d/Sx" — Em} r*aj d/fiE^}. Thus the connection in ^ determines the horizontal subspaces in the tangent spaces at each point of Conversely, a connection in may be defined by giving an «.-dimensional subspace of TU(L(JK)) for each with the properties:

(1) If AeGLfaR1), then the map A*: Tu(L(J?))-+TA{lù(L{^))

maps the horizontal subspace Hu into

(2) Hu contains no non-zero vector belonging to the vertical subspace Vu.

Here, the vertical subspace Vu is defined as the n2-dimensional subspace of TU(L{^#)) spanned by the vectors tangent to curves in the fibre 7t~1[7t(u)); in terms of local coordinates, Vu is spanned by the

vectors {djdE^}. Property (2) implies that Tu is the direct sum of Hu and Vu.

The projection map n: induces a surjective linear map

77*: Tv(L(Ji)) TM(Ji), such that tt*(VJ = 0 and n* restricted to Hu is 1-1 onto Thus the inverse n*-1 is a linear map of onto Hu. Therefore for any vector X e Tp(Jf) and point u e n-l{p), there is a unique vector X sHu, called the horizontal lift of X, such that 77*(X) = X. Given a curve y(t) in J(, and an initial point u in ?r_1(y(<0)), one can construct a unique curve y(t) in L(J(), where y(t) is the curve through u whose tangent vector is the horizontal lift of the tangent vector of y(<) in Jl. Thus knowing the horizontal subspaces at each point in one can define parallel propagation of bases along any curve y(t) in JK. One can then define the covariant derivative along y(t) of any tensor field T by taking the ordinary derivatives with respect to t, of the components of T with respect to a parallelly propagated basis.

If there is a metric g on whose covariant derivative is zero, then orthonormal frames are parallelly propagated into orthonormal frames. Thus the horizontal subspaces are tangent to 0{Ji) in and define a connection in 0(Ji).

Similarly a connection on Ji defines «.-dimensional horizontal subspaces in the tangent spaces to the bundles T(Jt) and Tre(Ji), by parallel propagation of vectors and tensors. These horizontal subspaces have coordinate bases

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