Because of (2.7), any Lie derivative commutes with d, i.e. for any p-form field co, d^io) = Z^dio).

From these formulae, as well as from the geometrical interpretation, it follows that the Lie derivative L^T|p of a tensor field T of type [r, s) depends not only on the direction of the vector field X at the point p, but also on the direction of X at neighbouring points. Thus the two differential operators defined by the manifold structure are too limited to serve as the generalization of the concept of a partial derivative one needs in order to set up field equations for physical quantities on the manifold; d operates only on forms, while the ordinary partial derivative is a directional derivative depending only on a direction at the point in question, unlike the Lie derivative. One obtains such a generalized derivative, the covariant derivative, by introducing extra structure on the manifold. We do this in the next section.

2.5 Covariant differentiation and the curvature tensor

The extra structure we introduce is a (affine) connection on A connection V at a point p of ^ is a rule which assigns to each vector field X at p a differential operator Vx which maps an arbitrary CT[r > 1) vector field Y into a vector field VXY, where:

(1) VXY is a tensor in the argument X, i.e. for any functions/, g, and C1 vector fields X, Y, Z,

(this is equivalent to the requirement that the derivative Vx at p depends only on the direction of X at p)\

(2) Vx Y is linear in Y, i.e. for any CP- vector fields Y, Z and a., fie R1,

(3) for any C1 function / and C1 vector field Y,

Then Vx Y is the covariant derivative (with respect to V) of Y in the direction X at p. By (1), we can define VY, the covariant derivative of Y, as that tensor field of type (1,1) which, when contracted with X, produces the vector Vx Y. Then we have

A CT connection V on a Ck manifold ^ {k > r + 2) is a rule which assigns a connection V to each point such that if Y is a Cr+1 vector field on then VY is a Cr tensor field.

Given any Cr+1 vector basis {Ea} and dual one-form basis {E°} on a neighbourhood we shall write the components of VY as Ya. b, so

The connection is determined on °U by n? Cr functions defined by

For any C1 vector field Y,

Thus the components of VY with respect to coordinate bases {d/dx0}, {da;6} are Ya.b = 8Yaj8xb + r*^ Yc.

The transformation properties of the functions are determined by connection properties (1), (2), (3); for

TV = <E"',VEi.Ec.> = <<D«'aE",Vw(<D/Ec)>

if E„. = <!>„." E„, E°' = C>a'„ E". One can rewrite this as r°w = ®a'a(Eb.(<t>c.a)+<v 0/ r^).

In particular, if the bases are coordinate bases defined by coordinates {x"'}, the transformation law is

. _ 8xa' / 82xa dx^fa^ „ \ 6'c' ~ 8xa \8xb'8x°'+ 8xvdF to)'

Because of the term Eb\G>c.a), the r^ do not transform as the components of a tensor. However if VY and VY are covariant derivatives obtained from two different connections, then

VY—^Y = (rv-fv) FcE6®Ea will be a tensor. Thus the difference terms (P^— f0^) will be the components of a tensor.

The definition of a covariant derivative can be extended to any Cr tensor field if r > 1 by the rules (cf. the Lie derivative rules):

(1) if T is a C tensor field of type (q, s), then VT is a Cr"1 tensor field of type (g,i+l);

(2) V is linear and commutes with contractions;

(3) for arbitrary tensor fields S, T, Liebniz' rule holds, i.e.

We write the components of VT as [VKhT)a-de_g = Ta-deg.h. As a consequence of (2) and (3), where {Ea} is the dual basis to {EJ, and methods similar to those used in deriving (2.12) show that the coordinate components of VT are

+ (all upper indices) - T^heTab-djf ^-(all lower indices). (2.13)

As a particular example, the unit tensor Ea®E°, which has components Sab, has vanishing covariant derivative, and so the generalized unit tensors with components • • • Sa'\, S[%iSa>bt...Sap\p

(p < n) also have vanishing covariant derivatives.

If T is a C {r ^ 1) tensor field defined along a Cr curve A[t), one can define DT¡8t, the covariant derivative of T along A (t), as Vglet T where T is any Cr tensor field extending T onto an open neighbourhood of A. DT/dt is a Cr_1 tensor field defined along A(i), and is independent of the extension T. In terms of components, if X is the tangent vector to A(i), then T)Ta - de J8t = Ta "dt g.tiXh. In particular one can choose local coordinates so that A(i) has the coordinates xa(t), Xa = da^/di, and then for a vector field Y

The tensor T is said to beparallelly transported along A if DTf8t = 0. Given a curve A(i) with endpoints p, q, the theory of solutions of ordinary differential equations shows that if the connection V is at least C1- one obtains a unique tensor at q by parallelly transferring any given tensor from p along A. Thus parallel transfer along A is a linear map from Tr„(p) to Trs[q) which preserves all tensor products and tensor contractions, so in particular if one parallelly transfers a basis of vectors along a given curve from p to q, this determines an isomorphism of Tp to Tq. (If there are self-intersections in the curve, p and q could be the same point.)

A particular case is obtained by considering the covariant derivative of the tangent vector itself along A. The curve A(t) is said to be a geodesic curve if D / d \

is parallel to (S/dt)x, i.e. if there is a function / (perhaps zero) such that Xa.bXb = fXa. For such a curve, one can find a new parameter v(t) along the curve such that dv\dv)k- ;

such a parameter is called an afine parameter. The associated tangent vector V = (d/dv)A is parallel to X but has its scale determined by V(v) = 1; it obeys the equations the second expression being the local coordinate expression obtainable from (2.14) applied to the vector V. The affine parameter of a geodesic curve is determined up to an additive and a multiplicative constant, i.e. up to transformations v' =av + b where a, b are constants; the freedom of choice of 6 corresponds to the freedom to choose a new initial point A(0), the freedom of choice in a corresponding to the freedom to renormalize the vector V by a constant scale factor, V' = (1/a) V. The curve parametrized by any of these affine parameters is said to be a geodesic.

Given a C' (r ^ 0) connection, the standard existence theorems for ordinary differential equations applied to (2.15) show that for any pointy of and any vector Xp atp, there exists a maximal geodesic Ax(v) in with starting pointy and initial direction Xp, i.e. such that Ax(0) = p and (d/dv)A|„=0 = Xp. If r ^ 1 —, this geodesic is unique and depends continuously on p and Xp. If r ^ 1, it depends differentiably on p and Xp. This means that if r ^ 1, one can define a Cr map exp: Tp -> JK, where for each X e Tp, exp (X) is the point in JK a unit parameter distance along the geodesic Ax from p. This map may not be defined for all XeTp, since the geodesic Ax(v) may not be defined for all v. If v does take all values, the geodesic A(t>) will be said to be a complete geodesic. The manifold is said to be geodesically complete if all geodesies on are complete, that is if exp is defined on all Tp for every point p of

Whether is complete or not, the map expp is of rank n at p. Therefore by the implicit function theorem (Spivak (1965)) there exists an open neighbourhood of the origin in Tp and an open neighbourhood Jfp of j> in JK such that the map exp is a CT diffeomorphism of onto Jfp. Such a neighbourhood Jfp is called a normal neighbourhood of p. Further, one can choose Jfp to be convex, i.e. to be such that any point q of Jfp can be joined to any other point r in Jfp by a unique geodesic starting at q and totally contained in Jfp. Within a convex normal neighbourhoods one can define coordinates (a;1,...,xn) by choosing any point qe^V, choosing a basis {E0} oiTq, and defining the coordinates of the point r inS by the relation r = exp (xaEa) (i.e. one assigns to r the coordinates, with respect to the basis {E0}, of the point exp"1 (r) in Tg.) Then (8/8x% = Ef and (by (2.15)) = 0. Such coordinates will be called normal coordinates based on q. The existence of normal neighbourhoods has been used by Geroch (1968c) to prove that a connected C3 Hausdorff manifold with a C1 connection has a countable basis. Thus one may infer the property of paracompactness of a C® manifold from the existence of a C1 connection on the manifold. The 'normal' local behaviour of geodesies in these neighbourhoods is in contrast to the behaviour of geodesies in the large in a general space, where on the one hand two arbitrary points cannot in general be joined by any geodesic, and on the other hand some of the geodesies through one point may converge to ' focus' at some other point. We shall later encounter examples of both types of behaviour.

Given a Cr connection V, one can define a Cr-1 tensor field T of type (1, 2) by the relation

T(X, Y) = Vx Y — VYX — [X, Y], where X, Y are arbitrary Cr vector fields. This tensor is called the torsion tensor. Using a coordinate basis, its components are mi _ p» _ -pi ¡k — 1 jk 1 ki-

We shall deal only with torsion-free connections, i.e. we shall assume T = 0. In this case, the coordinate components of the connection obey = rfw, so such a connection is often called a symmetric connection. A connection is torsion-free if and only if f.i} —f-ji for all functions /. From the geodesic equation (2.15) it follows that a torsion-free connection is completely determined by a knowledge of the geodesies on JK.

When the torsion vanishes, the covariant derivatives of arbitrary C1 vector fields X, Y are related to their Lie derivative by

and for any C1 tensor field T of type (r, s) one finds

- (all upper indices) + Tab~dj/_gXi.e+(all lower indices). (2.17)

One can also easily verify that the exterior derivative is related to the covariant derivative by dA = -4a...c;aA da^A ... A AxPo(AA)a_cd = (- )pAla„x.

where A is any y-form. Thus equations involving the exterior derivative or Lie derivative can always be expressed in terms of the co-variant derivative. However, because of their definitions, the Lie derivative and exterior derivative are independent of the connection.

If one starts from a given point p and parallelly transfers a vector Xp along a curve y that ends at p again, one will obtain a vector K'p which is in general different from Xp; if one chooses a different curve Y» the new vector one obtains at p will in general be different from Xp and X'p. This non-integrability of parallel transfer corresponds to the fact that the covariant derivatives do not generally commute. The Riemann (curvature) tensor gives a measure of this non-commutation. Given C+1 vector fields X, Y, Z, a C*-1 vector field R(X, Y) Z is defined by a Cr connection V by

Then R(X, Y) Z is linear in X, Y, Z and it may be verified that the value of R(X, Y) Z at p depends only on the values of X, Y, Z at p, i.e. it is a Cr-1 tensor field of type (3,1). To write (2.18) in component form, we define the second covariant derivative VVZ of the vector Z as the covariant derivative V(VZ) of VZ; it has components

Then (2.18) can be written

B>MX'Y*& = (Z".idY*).cXc-(Z".dX\cYc

= (Z°:dc-Z°;cd)X°Y*, where the Riemann tensor components Rabcd with respect to dual bases {E0}, {E°} are defined by R"^ = <E°, R(EC, Ed) E6>. As X, Y are arbitrary vectors, Z°;dc-Z°.cd = RabcdZ" (2.19)

expresses the non-commutation of second covariant derivatives of Z in terms of the Riemann tensor.

Since

=> <iJ,VxVxZ> = -£((»), VyZ))— (VXT), VrZ)

holds for any C2 one-form field T) and vector fields X, Y, Z, (2.18) implies

<E°, R(EC, Ed) E6) = Ec( <E°, VBrf E6>) — Ed( (E°, VBc E6>)

- <VEc VBd E„> + <VBd Ea, VEe E6> - (E°, E„>. Choosing the bases as coordinate bases, one finds the expression

for the coordinate components of the Riemann tensor, in terms of the coordinate components of the connection.

It can be verified from these definitions that in addition to the symmetry = = <> (2.21a)

the curvature tensor has the symmetry

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