Differential geometry

The space-time structure discussed in the next chapter, and assumed through the rest of this book, is that of a manifold with a Lorentz metric and associated affine connection.

In this chapter, we introduce in § 2.1 the concept of a manifold and in §2.2 vectors and tensors, which are the natural geometric objects defined on the manifold. A discussion of maps of manifolds in §2.3 leads to the definitions of the induced maps of tensors, and of sub-manifolds. The derivative of the induced maps defined by a vector field gives the Lie derivative defined in §2.4; another differential operation which depends only on the manifold structure is exterior differentiation, also defined in that section. This operation occurs in the generalized form of Stokes' theorem.

An extra structure, the connection, is introduced in §2.5; this defines the covariant derivative and the curvature tensor. The connection is related to the metric on the manifold in §2.6; the curvature tensor is decomposed into the Weyl tensor and Ricci tensor, which are related to each other by the Bianchi identities.

In the rest of the chapter, a number of other topics in differential geometry are discussed. The induced metric and connection on a hypersurface are discussed in §2.7, and the Gauss-Codacci relations are derived. The volume element defined by the metric is introduced in § 2.8, and used to prove Gauss' theorem. Finally, we give a brief discussion in §2.9 of fibre bundles, with particular emphasis on the tangent bundle and the bundles of linear and orthonormal frames. These enable many of the concepts introduced earlier to be reformulated in an elegant geometrical way. §2.7 and §2.9 are used only at one or two points later, and are not essential to the main body of the book.

2.1 Manifolds

A manifold is essentially a space which is locally similar to Euclidean space in that it can be covered by coordinate patches. This structure permits differentiation to be defined, but does not distinguish intrinsically between different coordinate systems. Thus the only concepts defined by the manifold structure are those which are independent of the choice of a coordinate system. We will give a precise formulation of the concept of a manifold, after some preliminary definitions.

Let Rn denote the Euclidean space of n dimensions, that is, the set of all n-tuples (a;1,a;®, ...,£") ( — co<xi<co) with the usual topology (open and closed sets are defined in the usual way), and let \Rn denote the 'lower half' of Rn, i.e. the region of Rn for which x1 < 0. A map 0 of an open set 0 c: Rn (respectively \Rn) to an open set 6' c: Rm (respectively \Rm) is said to be of class Cr if the coordinates (a;'1, x's,..., x'm) of the image point 4>{p) in 0' are r-times continuously differentiable functions (the rth derivatives exist and are continuous) of the coordinates (x1, xxn) of p in 6. If a map is Cr for all r ^ 0, then it is said to be C7°°. By a C° map, we mean a continuous map.

A function /on an open set <3 of Rn is said to be locally Lipschitz if for each open set <S with compact closure, there is some constant K such that for each pair of points p,qe%, \f(p) —f{q)| < K \p — q\, where by \p\ we mean

{{*1(P)?+(AP)?+- + (*n(i>))*}i. A map <f> will be said to be locally Lipschitz, denoted by C1-, if the coordinates of 4>{p) are locally Lipschitz functions of the coordinates of p. Similarly, we shall say that a map 0 is CT~ if it is Cr_1 and if the (r— l)th derivatives of the coordinates of §{p) are locally Lipschitz functions of the coordinates of p. In the following we shall usually only mention Cr, but similar definitions and results hold for CT~.

If SP is an arbitrary set in Rn (respectively \Rn), a map 0 from SP to a set SP' <=■ R™ (respectively ^Rm) is said to be a CT map if <f> is the restriction to SP and SP' of a Cr map from an open set 0 containing SP to an open set <S' containing SP'.

A Cr n-dimensional manifold ^ is a set ^ together with a Cr atlas <fia}, that is to say a collection of charts where the are subsets of ^ and the <j>a are one-one maps of the corresponding eila to open sets in Rn such that

is a C map of an open subset of-R" to an open subset ofi?n (see figure 4). Each is a local coordinate neighbourhood with the local coordinates (a — 1 ton) defined by the map <pa (i.e. iîp then the coordinates ofp are the coordinates of<pa(p) in Rn). Condition (2) is the requirement that in the overlap of two local coordinate neighbourhoods, the coordinates in one neighbourhood are Cr functions of the coordinates in the other neighbourhood, and vice versa.

Figure 4. In the overlap of coordinate neighbourhoods^ and coordinates are related by a Cr map <j>a o <j>0~1-

Another atlas is said to be compatible with a given Cr atlas if their union is a Cr atlas for all The atlas consisting of all atlases compatible with the given atlas is called the complete atlas of the manifold; the complete atlas is therefore the set of all possible coordinate systems covering

The topology of ^ is defined by stating that the open sets of consist of unions of sets of the form <%ia belonging to the complete atlas. This topology makes each map <f>a into a homeomorphism.

A Cr differentiable manifold with boundary is defined as above, on replacing 'R*' by '\Rn'. Then the boundary of denoted by is defined to be the set of all points of whose image under a map <f>a lies on the boundary of in Rn. ¿L^is an (n — 1 )-dimensional Cr manifold without boundary.

These definitions may seem more complicated than necessary. However simple examples show that one will in general need more than one coordinate neighbourhood to describe a space. The two-dimensional Euclidean plane R2 is clearly a manifold. Rectangular coordinates (x, y; — oo < x < oo, —co<y<oo) cover the whole plane in one coordinate neighbourhood, where 4> is the identity. Polar coordinates (r,6) cover the coordinate neighbourhood (r > 0, 0 < 6 < 2n); one needs at least two such coordinate neighbourhoods to cover R2. The two-dimensional cylinder C2 is the manifold obtained from R2 by identifying the points (x, y) and {x+ 2k, y). Then (x,y) are coordinates in a neighbourhood (0 < x < 2n, —co<y<co) and one needs two such coordinate neighbourhoods to cover C2. The Möbius strip is the manifold obtained in a similar way on identifying the points (x, y) and (x + 27t, —y). The unit two-sphere S2 can be characterized as the surface in R3 defined by the equation (a;1)2 + (a;2)2 + (a;3)2 = 1. Then

are coordinates in each of the regions x1 > 0, x1 < 0, and one needs six such coordinate neighbourhoods to cover the surface. In fact, it is not possible to cover S2 by a single coordinate neighbourhood. The n-sphere Sn can be similarly defined as the set of points

A manifold is said to be orientable if there is an atlas <fia} in the complete atlas such that in every non-empty intersection n <2^, the Jacobian \8xil8x'i\ is positive, where (a;1, ...,xn) and (x'1, ...,x'n) are coordinates in and ^ respectively. The Möbius strip is an example of a non-orientable manifold.

The definition of a manifold given so far is very general. For most purposes one will impose two further conditions, that is HausdorfF and that is paracompact, which will ensure reasonable local behaviour.

A topological space is said to be a Hausdorff space if it satisfies the HausdorfF separation axiom: whenever p, q are two distinct points in ^ there exist disjoint open sets "V in ^ such that pe%,qe 1r. One might think that a manifold is necessarily Hausdorff, but this is not so. Consider, for example, the situation in figure 5. We identify the points b, b' on the two lines if and only if xb = yb. < 0. Then each point is contained in a (coordinate) neighbourhood homeomorphic to an open subset of R1. However there are no disjoint open neighbourhoods b a

Figure 5. An example of a non-Hausdorff manifold. The two lines above are identical for x = y < 0. However the two points a (x = 0) and o' (y = 0) are not identified.

ffl, satisfying the conditions ae^o'e/, where a is the point x = 0 and a' is the point y — 0.

An atlas is said to be locally finite if every point p e has an open neighbourhood which intersects only a finite number of the sets <2f0. Jt is said to be paracompact if for every atlas there exists a locally finite atlas rjfpj with each contained in some A connected HausdorfF manifold is paracompact if and only if it has a countable basis, i.e. there is a countable collection of open sets such that any open set can be expressed as the union of members of this collection (Kobayashi and Nomizu (1963), p. 271).

Unless otherwise stated, all manifolds considered mil be paracompact, connected Cm Hausdorff manifolds without boundary. It will turn out later that when we have imposed some additional structure on (the existence of an affine connection, see §2.4) the requirement of para-compactness will be automatically satisfied because of the other restrictions.

A function/on a Ck manifold J( is a map from ^to i?1. It is said to be of class Cr (r < k) at a point p of if the expression fo of / on any local coordinate neighbourhood <?Ua is a CT function of the local coordinates at p\ and / is said to be a Cr function on a set V of ^ if /is a CT function at each point pe'V.

A property of paracompact manifolds we will use later, is the following: given any locally finite atlas {<%a, on a paracompact Ck manifold, one can always (see e.g. Kobayashi and Noinizu (1963), p. 272) find a set of Ck functions ga such that

(2) the support of ga, i.e. the closure of the set {p e ga(p) 4= 0}, is contained in the corresponding %a\

Such a set of functions will be called & partition of unity. The result is in particular true for functions, but is clearly not true for analytic functions (an analytic function can be expressed as a convergent power series in some neighbourhood of each point p e and so is zero everywhere if it is zero on any open neighbourhood).

Finally, the Cartesian product si x 88 of manifolds si, 88 is a manifold with a natural structure defined by the manifold structures of si, 88 \ for arbitrary pointsp esi, qe8S, there exist coordinate neighbourhoods V containingp, q respectively, so the point (p, q) e si x 88 is contained in the coordinate neighbourhood tfl x y in si x 88 which assigns to it the coordinates (a;1,*/*), where xi are the coordinates of p in and yi are the coordinates of q in

2.2 Vectors and tensors

Tensor fields are the set of geometric objects on a manifold defined in a natural way by the manifold structure. A tensor field is equivalent to a tensor defined at each point of the manifold, so we first define tensors at a point of the manifold, starting from the basic concept of a vector at a point.

A Ck curve A(i) in ^is a Ck map of an interval of the real line R1 into The vector (contravariant vector) (d/di)*! i0 tangent to the C1 curve A(i) at the point A(i0) is the operator which maps each C1 function/at A(l0) into the number (dfl&t)K| ,0; that is, (dfldt)K is the derivative of/in the direction of A(i) with respect to the parameter t. Explicitly,

The curve parameter t clearly obeys the relation (d/dt)kt = 1. If (x1 xn) are local coordinates in a neighbourhood of p,

(Here and throughout this book, we adopt the summation convention whereby a repeated index implies summation over all values of that index.) Thus every tangent vector at a point p can be expressed as a linear combination of the coordinate derivatives

Conversely, given a linear combination Vs(djdx})\p of these operators, where the Vs are any numbers, consider the curve A(J) defined by

2/*(A(i)) = x}(p) + t V*, for t in some interval [—e, e]; the tangent vector to this curve at p is V^d/dx*)^. Thus the tangent vectors at p form a vector space over jR1 spanned by the coordinate derivatives (8ldxs)\p, where the vector space structure is defined by the relation

which is to hold for all vectors X, Y, numbers a, /? and functions /. The vectors (d/cte*),, are independent (for if they were not, there would exist numbers Vs such that V}(dldx})\p = 0 with at least one Vi non-zero; applying this relation to each coordinate xk shows

Vidxkldxi = F* = 0, a contradiction), so the space ol"all tangent vectors to f utp, denoted by Tp(yi() or simply Tp, is an w-dimensional vector space. This space, representing the set of all directions at p, is called the tangent vector space to at p. One may think of a vector \eTp as an arrow at p, pointing in the direction of a curve A(t) with tangent vector V at p, the 'length' of V being determined by the curve parameter t through the relation V(t) = 1. (As V is an operator, we print it in bold type; its components Vj, and the number F(/) obtained by V acting on a function/, are numbers, and so are printed in italics.)

If {Ea} (a = 1 to n) are any set of n vectors at p which are linearly independent, then any vector YeTp can be written V = F°Ea where the numbers {Fa} are the components of V with respect to the basis {Ea} of vectors at p. In particular one can choose the E„ as the coordinate basis (dldx1)^; then the components F{ = Vix1) — (da^/dt)^ are the derivatives of the coordinate functions xi in the direction V.

A one-form, (covariant vector) u> at p is a real valued linear function on the space Tp of vectors at p. If X is a vector at p, the number into which u> maps X will be written (u>, X); then the linearity implies that

<w, aX+fiY> = a<u>, X>+/?<w, Y>

holds for all a.fteR1 and X, Y eTp. The subspace of Tp defined by (ta, X) = (constant) for a given one-form u>, is linear. One may therefore think of a one-form at p as a pair of planes in Tp such that if (u>, X) = 0 the arrow X lies in the first plane, and if (u>, X) = 1 it touches the second plane.

Given a basis {Ea} of vectors at p, one can define a unique set of n one-forms {E°} by the condition: E{ maps any vector X to the number Xi (the ith component of X with respect to the basis {Ea}).

Then in particular, (E°, Eb> = 6%. Defining linear combinations of one-forms by the rules

<au) + yffTj,X> = a<u),X>+yff<Tj,X>

for any one-forms u>, tj and any a, fieR1, XeTp, one can regard {E°} as a basis of one-forms since any one-form u> &tp can be expressed as w = wiE< where the numbers o^ are defined by fc^ = (w, E{). Thus the set of all one forms at p forms an n-dimensional vector space at p, the dual space T*p of the tangent space Tp. The basis {E°} of one-forms is the dual basis to the basis {Ea} of vectors. For any we?1 X & Tp one can express the number (to, X) in terms of the components tait X{ of to, X with respect to dual bases {E°}, {Ea} by the relations <<•>, X> - <,.., E', XI - u,t X>.

Each function f on defines a one-form df at p by the rule: for each vector X, X) = Xf.

df is called the differential of/. If (x1, ...,xn) are local coordinates, the set of differentials (da;1, da;2,..., da;n) at p form the basis of one-forms dual to the basis (8/dx1, dfdx2,..., dj8xn) of vectors at p, since

(da;', dldxi} = dx^dxi = In terms of this basis, the differential df of an arbitrary function / is given by df = (dfjdx*) da;1.

If df is non-zero, the surfaces {/ = constant} are (n — 1 )-dimensional manifolds. The subspace of Tp consisting of all vectors X such that <d/,X> = 0 consists of all vectors tangent to curves lying in the surface {/ = constant} through p. Thus one may think of df as a normal to the surface {/ = constant} at p. If a + 0, a df will also be a normal to this surface.

From the space Tp of vectors at p and the space T*p of one-forms at p, we can form the Cartesian product n? = T*pxT*px...xT*sx TpxTpx...xTr r factors s factors i.e. the ordered set of vectors and one-forms (tj1, ...,tjr,Yj,...,ys) where the Ys and tjs are arbitrary vectors and one-forms respectively.

A tensor of type (r, s) at p is a function on 11® which is linear in each argument. If T is a tensor of type (r, s) at p, we write the number into which T maps the element (tj1, ..., if, Yj,..., Yg) of n® as

Then the linearity implies that, for example, Tft1,..., if, ccX+fiY, Y„ ..., Y.) = «. Ttf,..., if, X, Y„..., Y.)

holds for all a,yBeR1 and X,YeTp.

The space of all such tensors is called the tensor product

Trs(p) = Tv®...®Tp ®T*v®...®T*r r factors s factors

Addition of tensors of type (r, s) is defined by the rule: (T + T') is the tensor of type (r, s) at p such that for all Y{eTp, itf eT*p,

Similarly, multiplication of a tensor by a scalar ae R1 is defined by the rule: (aT) is the tensor such that for all YteTp, i^eT*p,

(«?) (»I1 if, Ylf ■ • •, Y.) = a.Ttf fl'.Y, Ys).

With these rules of addition and scalar multiplication, the tensor product Trs(p) is a vector space of dimension nr+* over J?1.

Let XtsTp (i = 1 to r) and <*ieT*p (j = 1 to s). Then we shall denote by Xx ® ... ® Xr ® u»1 ® ... ® u>s that element of TrB(p) which maps the element (rj1, ...,if, Ylf..., Ys) of FI® into

Xj) (»j2, X2>... X,) (to1, Yx>... <W, Ys>.

Similarly, if Re?%>) and Se2%>), we shall denote by R ® S that element of i'S+cfjp) which maps the element (tj1, ...,rf+v, Y1(..., Ys+i) of into the number

R(ij1, ...,T)S, Yj Yr)S(if+1 T)s+4, Yr+1,..., Yr+p).

With the product ®, the tensor spaces at p form an algebra over R.

If {E„}, {E°} are dual bases of Tp, T*p respectively, then

{Eai ® ... ® E^ ® E6i ® ... ® E6«}, (ait bj run from 1 to n), will be a basis for Tre(p). An arbitrary tensor T e TTs(p) can be expressed in terms of this basis as

T = Tai-arbi...bEai ® ... ® E^ ® E6» ® ... ® E6«

2.2] vectors and tensors 19

where {T10! - ^ 6i bj} are the components of T with respect to the dual bases {Ea}, {E°} and are given by

Relations in the tensor algebra at p can be expressed in terms of the components of tensors. Thus

Because of its convenience, we shall usually represent tensor relations in this way.

If {En.} and {E°'} are another pair of dual bases for Tp and T*p, they can be represented in terms of {E„} and {E°} by

where <3>a° is an n x n non-singular matrix. Similarly

where <3>a'a is another nxn non-singular matrix. Since {Ea.}, {Ea'} are dual bases,

- <E6',En.) = <4»'6E»,<VE0> = <V<I>V«6 = <t>a-a <t>b'a, i.e. 0a.a, 0a'a are inverse matrices, and 6"b — <&ab. <$b'b.

The components Tai 'arb,,...b-S °f & tensor T with respect to the dual bases {Ea.}, {Ea'} are given by

They are related to the components i1ai "%1...6, of T with respect to the bases {EJ, {E°} by

Jia\...a'r _ Jial --<h. . 0a'i ... (Jp'r 6i d> (2 4)

The contraction of a tensor T of type (r,s), with components Tab -dtf 0 with respect to bases {E„}, {E°}, on the first contravariant and first covariant indices is defined to be the tensor CJ(T) of type {r— l,s—l) whose components with respect to the same basis are

CftT) = T<*>~daf_0Eb ® ... ® Ed ® E>® ... ® E".

If {Ea,}, {Ea'} are another pair of dual bases, the contraction C}(T) defined by them is

C'J(T) = T""'-d-aT_B.Eb. ® ...®Ed,®W® ... ®E?

= Tab-daf...0^b ® ... ® Ed ® E/® ... ® E" = C}(T), so the contraction C\ of a tensor is independent of the basis used in its definition. Similarly, one could contract T over any pair of contra-variant and covariant indices. (If we were to contract over two contra-variant or covariant indices, the resultant tensor would depend on the basis used.)

The symmetric part of a tensor T of type (2,0) is the tensor <S(T) defined by

for all rh.The27**- We shall denote the components S(T)06 of &(T) by T(oW; then

Similarly, the components of the skew-symmetric part of T will be denoted by

In general, the components of the symmetric or antisymmetric part of a tensor on a given set of covariant or contravariant indices will be denoted by placing round or square brackets around the indices. Thus

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