Differential Geometry 2015-16 Lecture Notes

Home , Directional derivative, Interior product

Imperial College QFFF MSc ——————– Diﬀerential Geometry 2015-16 Lecture Notes ——————–

Chris Hull Oct 2015

Contents

1 Introduction 5 1.1 Heuristic deﬁnition of (real) manifold ...... 5 1.2 Example: Stereographic projections of a 2-sphere; the manifold S2 ...... 8 1.3 Open sets in Rm in the ‘usual topology’ ...... 10 1.4 Topological space ...... 12 1.5 Topological structure of an m-dimensional real manifold M . 14 1.6 Example: circle S1 ...... 15 1.7 Topological invariants and compactness ...... 18 1.8 Formal deﬁnition of smooth m-dim’l real manifold ...... 21 2 1.9 Example: real projective space RP ...... 23 1.10 Product manifold ...... 27 1.11 Example: Torus ...... 28 1.12 Complex manifold ...... 29 1.13 Example: S2 as a complex manifold ...... 30 n 1.14 Example: CP complex projective space ...... 31 1.15 Examples of not being a manifold ...... 33 1.16 Manifolds with boundary ...... 34

1 2 Differential maps, tangent vectors and tensors 36 2.1 Differentiable maps ...... 36 2.2 Embedding and submanifolds ...... 38 2.3 An example: CY 3 ...... 39 2.4 Diffeomorphism ...... 40 2.5 Functions ...... 41 2.6 Curves ...... 42 2.7 Tangent vectors ...... 43 2.8 An aside: Dual vector spaces ...... 46 2.9 Cotangent vectors ...... 48 2.10 Tensors ...... 49 2.11 Contraction of tensors ...... 50 2.12 Tangent vectors act on functions ...... 51 2.13 Directional derivatives ...... 53 2.14 Differentials as Cotangent Vectors ...... 58 2.15 Tensors in a coordinates basis ...... 61

3 Induced maps, tensor fields and flows 62 3.1 Induced maps: push-forward ...... 62 3.2 Pull-back ...... 65 3.3 Vector, covector and tensor fields ...... 66 3.4 Tensor fields and induced maps ...... 69 3.5 Induced maps and diffeomorphisms ...... 72 3.6 Flows ...... 73 3.7 Lie Derivative of a vector field ...... 77 3.7.1 Components of the Lie Derivative of a vector field . . . 78 3.8 Lie Bracket of two vector fields ...... 81 3.9 Commuting flows ...... 83 3.10 Lie Derivative of a tensor field ...... 84 3.11 Example: Active coordinate transformations in GR and symmetry ...... 85

4 Diﬀerential forms 87 4.1 Cartan wedge product ...... 88 4.2 Exterior product ...... 91 4.3 Diﬀerential Form Fields ...... 92 4.4 Exterior derivative ...... 93 4.5 Pullback of forms ...... 94

2 4.6 3d vector calculus ...... 94 4.7 Coordinate free deﬁnition of the exterior derivative ...... 94 4.8 Interior product ...... 96 4.9 Lie derivative of forms ...... 96 4.10 Closed and Exact forms ...... 97 4.11 Physical application: Electromagnetism ...... 98 4.12 Orientation ...... 99 4.13 Top forms and volume forms ...... 100 4.14 Integrating a top form over a chart ...... 102 4.15 Integrating a top form over an orientable manifold ...... 104 4.16 Integration of r-forms over oriented r-dimensional submanifolds105

5 Stokes’ theorem and cohomology 106 5.1 Cycles and boundaries ...... 106 5.2 Stokes’ theorem ...... 107 5.3 Example. Gauss’s and Stokes’ law ...... 108 5.4 de Rham cohomology ...... 109 5.5 Example: Cohomology of R ...... 111 5.6 Example: Cohomology of S1 ...... 111 5.7 Cohomology and Topology ...... 113 5.8 Stoke’s Theorem and Cohomology ...... 113 5.9 Poincar´e’sLemma ...... 115 5.10 Example: Electromagnetism (again!) ...... 116 5.11 Poincar´eduality ...... 117

6 Riemannian Geometry I: The metric 118 6.1 The metric ...... 118 6.2 Metric inner product ...... 120 6.3 Volume element ...... 121 6.4 Epsilon symbol ...... 123 6.5 Hodge Star ...... 124 6.6 Inner product on r-forms ...... 125 6.7 Adjoint of d: d† ...... 126 6.8 3d vector calculus again...... 128 6.9 The Laplacian ...... 129 6.10 Ex. Electromagnetism again...... 130 6.11 Hodge theory ...... 131 6.12 Harmonic representatives for cohomology ...... 134

3 6.13 Maxwell’s Equations on a Compact Riemannian Manifold . . . 135

7 Riemannian Geometry II: Geometry 137 7.1 Induced metric and volume for a submanifold ...... 138 7.2 Length of a Curve and the Line Element ...... 139 7.3 Integration of n-forms over oriented n-dimensional submanifolds140 7.4 Hypersurfaces ...... 142 7.5 Electric and Magnetic Flux ...... 145 7.6 Stokes’s Theorem in 3D ...... 145 7.7 Gauss’s Theorem ...... 146 7.8 Connections ...... 147 7.9 Torsion ...... 150 7.10 Parallel transport and Geodesics ...... 152 7.11 Interpretation of Torsion ...... 153 7.12 Covariant derivative ...... 155 7.13 Metric connection ...... 158 7.14 Curvature ...... 160 7.15 Holonomy ...... 161 7.16 Non-coordinate basis ...... 164 7.17 Connections and curvature in non-coordinate bases ...... 169 7.18 Cartan’s structure equations ...... 171 7.19 Change of Basis and the Local Frame ...... 174 7.20 Tangent Bundle ...... 176

4 1 Introduction

• A manifold is a space which ‘locally’ looks like Rm (or Cm) • Calculus can be extended from Rm to the manifold. i.e. we have a notion of ‘diﬀerentiability’. For a complex manifold, complex analysis can be extended from Cm to the manifold, so we have a notion of ‘analyticity’.

1.1 Heuristic deﬁnition of (real) manifold We essentially require 2 things - an ‘atlas’ which is ‘smooth’

1. Atlas: m-dim’l manifold M is covered by patches, and there are maps which take patches of M into patches in Rm

Each patch of M, Ui, pairs with a map ψi to give a ‘chart’.

The maps ψi are invertible and

m ψi : M → R (1) p → ψi(p) (2)

A set of charts (Ui, ψi) covering M, so that ∪iUi = M, is called an ‘atlas’. −1 0 m The inverse ψi maps coordinates (x1, . . . , xm) in Ui ∈ R onto Ui ∈ M.

5 We explicitly give the maps ψi by giving

m ψi : M → R (3) p → x(p) = (x1(p), x2(p), . . . , xm(p)) (4)

Example of an altas: An altas! M = The World (i.e. a 2-sphere), 0 Ui =regions of the world, Ui =pages in altas, ψi =some appropriate coordinates eg. latitude and longitude.

6 2. Diﬀerentiability: transition from one patch to another is smooth.

If 2 patches Ui and Uj overlap, so Ui ∩ Uj 6= 0, then there is a map φji so

m 0 m 0 φji : R (Ui ) → R (Uj) (5) −1 x → y = φji(x) ≡ ψj · ψi (x) (6)

−1 for points such that p = ψi (x) ∈ Ui ∩ Uj. Recall that ψi is invertible.

φji is called the ‘transition function’.

Note that in the diagram, φij should be φji.

From the deﬁnition we see that φij is invertible since ψi, ψj are, and −1 furthermore φij = φji. Since this is a map from Rm → Rm, our usual notions of diﬀerentiability (smoothness) can be used.

Explicitly using coordinates where x = (x1, . . . , xm) and y = (y1, . . . , ym), we can write the transition function y = φji(x) as ya = ya(x1, . . . , xm) where a = 1, . . . , m. ∞ Smoothness requirement: Each ya(x1, . . . , xm) for a = 1, . . . , m is C in each argument xa.

7 1.2 Example: Stereographic projections of a 2-sphere; the manifold S2

Think of S2 as embedded in R3 as x2 + y2 + z2 = 1. Note that x, y, z provide a unique labelling of points but are not coordinates (there are obviously too many of them!).

Consider projecting a point (x, y, z) on the S2 into the plane z = 0 from the North pole. Let the intersection point be (X,Y, 0). Geometrically, x y X = Y = (7) 1 − z 1 − z

2 2 and these deﬁne a map ψN : S → R .

Now X,Y , which are functions of the point p = (x, y, z) look like good coordinates. BUT there is a problem at the North pole i.e. p = (0, 0, 1) where obviously something looks suspicious. The pole is mapped to inﬁnity in R2, but which inﬁnity? eg. is it at (0, ∞) or (∞, 0), or (0, −∞)? Hence the map ψN is not invertible due to the inclusion of the N-pole.

(technically the map fails to be a homeomorphism - see later...)

So take 2 charts: project onto plane z = 0 from N-pole for p ∈ UN , where UN excludes the N-pole, and project from S-pole onto plane z = 0 for p ∈ US, where US excludes the S pole.

8 x y X0 = Y 0 = (8) 1 + z 1 + z 2 2 We should make these charts cover the S and overlap eg. UN = S − 2 {Npole}, US = S − {Spole}

Diﬀerentiability?

Y 0 Y 1 + z 1 = X2 + Y 2 = = (9) X0 X 1 − z X02 + Y 02 Transition functions explicitly given as X Y X0 = X0(X,Y ) = Y 0 = Y 0(X,Y ) = (10) X2 + Y 2 X2 + Y 2 deﬁne the map

9 This is smooth, C∞, in overlap (note that this is true as both poles are excluded, where things are not smooth).

Remark: We thought of S2 as a subset of R3 in order to simply label the points. This is a common method of explicitly constructing manifolds, although it is not necessary to do this. A very important point is that an atlas provides an intrinsic deﬁnition of a manifold – eg. an Atlas is as good as a globe, and describes the same Earth 2-sphere.

1.3 Open sets in Rm in the ‘usual topology’ The canonical example of an open set in Rm is an open ball. m The open ball B(p0) of radius centred on a point p0 in R are the points p in m R whose distance from p0 is less than . If p has coordinates x(p) and p0 has coordinates x0, then the distance between p and p0 is the Euclidean distance 2 2 2 2 |x(p)−x0| (i.e. for a point y = (y1, . . . , ym), |y| = (y1) +(y1) +...+(ym) ) Then m B(x0) = {p ∈ R with |x(p) − x0| < } (11)

10 This is the interior of a sphere sm−1 in Rm. Example: For m = 1 this gives the familiar ‘open interval’ (x0 − , x0 + ).

Deﬁnition of an open set in Rm in the ‘usual’ or ‘metric’ topology: Set of points forms an ‘open set’ in Rm if for any point in the set we can ﬁnd an open ball of non-zero radius, centred on the point and also contained in the set. i.e. there are no ‘boundary points’.

Hence: all open sets in Rm can be formed by a union of open balls.

This notion of open set deﬁnes topology on Rm - called the ‘metric topology’ as it used a distance measure to deﬁne the open sets.

11 1.4 Topological space Topological space: A set of points X, together with a set of subsets J = {Oi|i ∈ I} so that,

1. 0 ∈ J and X ∈ J ( 0 - empty set )

2. ∪a∈AOa ∈ J for any A ⊂ I, ﬁnite or inﬁnite.

3. ∩b∈BOb ∈ J for any ﬁnite subset B ⊂ I.

Then the {Oi} are called ‘open subsets’ of X.

Examples: ‘trivial’ topology on any set of points X: J = {0,X} - not very useful ‘discrete’ topology on any set of points X: J = all subsets of points - also not very useful ‘metric’ topology on Rm: J =set of all possible unions of open balls. ‘metric’ topology on R: J =set of all possible unions of open intervals. An open interval (a, b) is the set of all points x ∈ R with a < x < b.

Continuous map: Given two topological spaces, a continuous map f : X → Y is a map from points in X to points in Y such that the inverse image of an open set in Y is an open set in X.

(Note: For a map g : A → B the inverse image of a subset C ⊂ B is the set {a ∈ A|g(a) ∈ C}. This is not related to the inverse map g−1.)

Homeomorphism: invertible continuous map between topological spaces X,Y such that the inverse map is also continuous. Then open sets in X are mapped to open sets in Y and vice versa.

If for topo spaces X,Y there exists a homeomorphism we say that X,Y are homeomorphic - then they have the same topology.

Intuitively: we can continuously deform one into the other.

Closed Set: A subset U of X is closed if the complement of U, X \ U, is open.

12 Examples: ‘metric’ topology on R: A closed interval is closed. A closed interval [a, b] is the set of all points x ∈ R with a ≤ x ≤ b. ‘metric’ topology on Rm: Closed balls are closed. A closed ball of radius m centred on point x0 in R is

m C(x0) = {p ∈ R with |x(p) − x0| ≤ } (12)

Hausdorﬀ (‘topological smoothness’) : 0 for any 2 distinct points x, x there exists open subsets Ux,Ux0 containing 0 x, x respectively s.t. Ux ∩ Uy = 0.

13 1.5 Topological structure of an m-dimensional real manifold M 1. M is a Hausdorﬀ topological space with open sets J

2. M has an ‘Altas’; a family of pairs (the charts) {(Ui, ψi)} where Ui ∈ J such that:

(a) the charts cover M so ∪iUi = M 0 m (b) the ψi are homeomorphisms from Ui into open subsets Ui of R .

The set of open sets {Ui} is called an open cover. This means it is a set of open sets that covers M, so that each point of M is in at least one of the open sets Ui in the cover. It is a subset of the set of all open sets J.

Thus locally our manifold is homeomorphic to (‘looks like’) Rm.

14 1.6 Example: circle S1

Can think of S1 as embedded in R2 so x2 + y2 = 1.

Take angular coordinate θ on circle, x = cos θ, y = sin θ. This gives a chart (U, ψ) where U is given by (x, y) for θ = (0, 2π), and the map ψ is,

1 ψ : S → R (13) p = (x, y) → θ (14)

BUT this misses the point on S1 corresponding to θ = 0. How annoying. (U, ψ) does not provide an altas as it doesn’t cover all S1.

15 Try U given by (x, y) for θ = (0, 2π + ) with > 0. This deﬁnitely now does cover the circle. However now ψ is not a homeomorphism!

θ = x and θ = 2π + x for x < are the same point on S1 so ψ clearly not invertible.

So even for a circle we actually need 2 charts to provide an atlas. eg.

U1 : θ = (0, 2π) (15)

ψ1 : p = (x = cos θ, y = sin θ) → θ (16) and

0 U2 : θ = (−π, +π) (17) 0 0 0 ψ2 : p = (x = cos θ , y = sin θ ) → θ (18)

16 This now covers S1, and on the intersections θ = θ0 (0 < θ0 < π) or θ = θ0+2π (−π < θ0 < 0) so both transition functions are clearly smooth.

17 1.7 Topological invariants and compactness Topological invariants are quantites that depends only on the topological structure of a manifold.

A famous example is the Euler character. If we think of a 2d surface in 3d space and triangulate it, then the quantity

χ = number(faces) − number(edges) + number(points) (19) is independent of the triangulation and measures the topology. eg. χ(S2) = 2, χ(T 2) = 0, and for a Riemann surface of genus G, χ(Riem) = 2−2 genus.

Another famous example is given by the Betti numbers br, r = 0, 1, . . . , m for Pm r an m-dimensional manifold. The Euler number in general is, χ = r=0(−1) br

A basic topological invariant is compactness. A manifold may be compact (eg. S1) or non-compact (eg. R).

Deﬁnition: A topological space is compact if for every open cover there exists a ﬁnite subset (possibly the whole set itself) which is also an open cover.

Example. S1 is compact. Some examples of covers: take cover {U1,U2}. then ﬁnite subset {U1,U2} is indeed a cover.

take cover {U1,U2,U3}. then ﬁnite subset {U1,U2} or {U1,U2,U3} are covers.

18 Example. R1 is not compact take cover {U1,U2}. then ﬁnite subset {U1,U2} is indeed a cover.

BUT for the following cover with unit length open sets Ui, there is no ﬁnite subset which covers R1, so it is not compact.

Example. An open interval (a, b) in R1 is not compact. It is covered by the open intervals

Un = (a, b − L/n), n = 2, 3, 4,...,L = b − a but no ﬁnite subset of these covers the interval. (Heine-Borel) Theorem: If a topological space, X, is a subset of Rm s.t. it is a closed subset (i.e. its complement is open) and is bounded in extent

19 in Rm then X is compact.

A subset of Rm is bounded in extent if the distance between any two points is less than some bound R, with the same bound for all pairs of points, i.e. a subset of X ⊆ Rm is bounded if there is some positive real number R such that |x − x0| < R for all points x, x0 ∈ X. A subset of Rm is closed if its complement in Rm is open, i.e. the set of points {x ∈ Rm : x∈ / X} is open.

The closed interval [a, b] is compact, as it is closed and bounded.

Example. S2 is compact. It can be embedded as a subset in R3 as x2 + y2 + z2 = 1, so all point in S2 lie in interval x, y, z ∈ [−1, 1] so extent of S2 is bounded.

20 1.8 Formal deﬁnition of smooth m-dim’l real manifold 1. M is a Hausdorﬀ topological space with open sets J

2. M has an ‘Altas’: a family of pairs (the charts) {(Ui, ψi)} where Ui ∈ J such that:

(a) the charts cover M so ∪iUi = M 0 m (b) the ψi are homeomorphisms from Ui into open subsets Ui of R .

−1 3. Given Ui ∩Uj 6= 0, then the map (the transition function) φij = ψi ·ψj from Rm → Rm is C∞, smooth.

The conditions 1), 2) imply that M can ‘locally topologically smoothly’ be thought of as Rm.

The condition 3) implies that M can ‘locally diﬀerentially smoothly’ be thought of as Rm.

m The transition function is a map φij from ψj(Ui ∩ Uj) ⊂ R to ψi(Ui ∩ Uj) ⊂ Rm: φij : ψj(Ui ∩ Uj) → ψi(Ui ∩ Uj)

A point p ∈ Ui ∩ Uj has coordinates x(i) = ψi(p) in the (Ui, ψi) chart and coordinates x(j) = ψj(p) in the (Uj, ψj) chart. The two coordinate systems are related by x(i) = φij(x(j)) m Note that x(i) ∈ R is an m-vector. Introducing a coordinate index µ = µ 1 2 m 1, 2, . . . , m, we have coordinates x(i) = (x(i), x(i), . . . , x(i)) for points in Ui, µ and similarly we have coordinates x(j) for points in Uj. Then for each µ µ there is a transition function φij, so that we have

µ µ ν x(i) = φij(x(j)) .

Note: any open set of a manifold is itself a manifold!

21 Remark: A manifold typically admits (infinitely) many different altases. This simply expresses the freedom in choosing a covering of open sets, and coordinates on those sets. 0 0 Two altases {(Ui, ψi)}, {(Ui , ψi)} for M are compatible if the union of these is also an atlas for M. The equivalence class of all compatible atlases for M constitute the differentiable structure of M.

Remark: In the definition of a smooth manifold, the transition functions are smooth, i.e. infinitely differentiable, C∞. This can relaxed to define a Cr manifold as one with transition functions that are Cr, i.e. they can be differentiated r times and the r’th derivative is continuous.

22 2 1.9 Example: real projective space RP ≡ set of undirected lines through the origin in R3

Geometrically:

2 A directed line is labeled by a point on S2. Hence RP is S2 with its antipo- dal points identiﬁed

3 Take point in R (x1, x2, x3) which labels a line provided 6= (0, 0, 0), i.e. take point in R3 − {0}.

However, (x1, x2, x3) labels the same line as (λx1, λx2, λx3) for any λ ∈ R.

23 2 3 Hence RP ≡ R −{0} with the identiﬁcation of points (x1, x2, x3) ∼ (λx1, λx2, λx3) for all λ ∈ R − {0}.

The labels (x1, x2, x3) are sometimes called the ‘homogenous coordinates’.

2 Now we want coordinate charts on RP .

Take chart:

2 U1 : all p ∈ RP s.t. x1 6= 0 (20) 2 2 ψ1 : RP → R (21) x2 x3 p = (x1, x2, x3) → , = (a1, a2) (22) x1 x1

This correctly encodes the identiﬁcation (x1, x2, x3) ∼ (λx1, λx2, λx3).

Note we must exclude all points with x1 = 0 or else ψ1 not a homeomorphism.

[If one tried to extend U1 to contain points with x1 = 0, then ψ1 would formally take such points to coordinates with (a1, a2) both inﬁnite. Then there 2 are no points in R with ﬁnite coordinates (a1, a2) that correspond to points 0 2 with x1 = 0. This means there is no region U ⊂ R for which ψ1 can extend 0 to a homeomorphism between a region containing points with x1 = 0 and U .]

So require further charts:

2 U2 : all p ∈ RP s.t. x2 6= 0 (23) 2 2 ψ2 : RP → R (24) x1 x3 p = (x1, x2, x3) → , = (b1, b2) (25) x2 x2 and

2 U3 : all p ∈ RP s.t. x3 6= 0 (26) 2 2 ψ3 : RP → R (27) x1 x2 p = (x1, x2, x3) → , = (c1, c2) (28) x3 x3

24 2 Now the charts {(U1, ψ1), (U2, ψ2), (U3, ψ3)} form an altas covering RP .

Check the transition functions are smooth: In overlap p ∈ U1 ∩ U2 we have,

2 2 φ21 : R → R (29) x2 x3 x1 x3 (a1, a2) = , → (b1(a1, a2), b2(a1, a2)) = , (30) x1 x1 x2 x2

1 a2 ∞ Hence b1 = and b2 = . This is indeed C smooth, since a1 6= 0 in U2 a1 a1 and hence in the overlap.

There are lots of overlaps!

n Generalisation to RP .

n Now RP ≡ set of undirected lines in Rn+1.

n+1 Label a line using ‘homogeneous coordinates’ (x1, x2, . . . xn+1) in R − {0} and identifying (x1, . . . , xn+1) ∼ (λx1, . . . , λxn+1) for all λ ∈ R − {0}.

Construct (n + 1) charts:

n Um : all p ∈ RP s.t. xm 6= 0 (31) n n ψm : RP → R (32)

25 x1 x2 xm−1 xm+1 xn+1 p = (x1, . . . , xn+1) → , ,..., , ..., (33) xm xm xm xm xm

0 0 Transition functions from Ui → Uj are again smooth. [Ex for reader!]

26 1.10 Product manifold

If X is an m-dim’l manifold with atlas {(Ui, ψi)} and Y is an n-dim’l manifold with atlas {(Vj, ξj)}, then X × Y is the ‘product manifold’, an (m + n) dim’l manifold with altas {(Wij, σij)} deﬁned as

• points: a point in X × Y is labelled as (p, q) with p ∈ X and q ∈ Y

• patches: (p, q) ∈ Wij iﬀ p ∈ Ui and q ∈ Vj • maps:

m+n σij : X × Y → R (34) (p, q) → (ψi(p), ξj(q)) (35)

m+n i.e. vector (x1, . . . , xm, y1, . . . , yn) ∈ R . Obviously satisﬁes manifold conditions.

27 1.11 Example: Torus Torus T n = S1 × ... × S1, product of n circles eg. T 2 is homeomorphic to the surface of a donut/bagel

28 1.12 Complex manifold An m-dim’l complex manifold, M, is deﬁned by:

1. (Hausdorﬀ) topological space with open sets J

2. Atlas of charts {(Ui, ψi)} where Ui ∈ J such that:

(a) the charts cover M so ∪iUi = M 0 m (b) the ψi are homeomorphisms from Ui into open subsets Ui of C .

−1 3. Given Ui ∩Uj 6= 0, then the map (the transition function) φij = ψi ·ψj from Cm → Cm are analytic (or holomorphic) complex functions.

0 0 Explicitly in coordinates: a point in Ui = (z1, . . . , zm), and a point in Uj = (u1, . . . , um). Then

m m φij : C → C (36) (z1, . . . , zm) → (u1(z), . . . un(z)) (37) so that all the ui(z1, . . . , zn) are analytic (holomorphic) functions in their arguments zi.

Recall: An analytic (holomorphic) function f(z1, . . . , zn) is a function that is complex diﬀerentiable i.e. any derivatives wrt some zi depend only on the position z1, . . . , zn that the derivative is evaluated at, and these derivatives ∗ are ﬁnite. A result of this is that the function only involves zi and not zi eg. f(z1, z2) = 1/z1 + 1/z2 is analytic away from z1 = 0 and z2 = 0, while 2 2 f(z1, z2) = |z1| + |z2| is not. Recall that an analytic function obeys the Cauchy-Riemann relations; write zi = xi + iyi with real xi, yi. Then an analytic function f(z1, . . . zn) = fr(z) + ifi(z) obeys, ∂f ∂f ∂f ∂f r = i and i = − r , ∀ i (38) ∂xi ∂yi ∂xi ∂yi

Note that we may always think of an m dim’l complex manifold as a 2m dim’l real manifold by deﬁning an altas using twice the number of real coordinates with zi = xi + iyi.

29 1.13 Example: S2 as a complex manifold

2 Recall: real manifold S is defined by atlas of 2 charts {(UN,S, ψN,S)}. Coor- 2 0 dinates in R are (XN,S,YN,S) with UN,S being defined by XN,S,YN,S being finite. Then the transition functions were;

XS,N YS,N XN,S = 2 2 ,YN,S = 2 2 (39) XS,N + YS,N XS,N + YS,N

Now take complex coordinates ZN,S = XN,S ±iYN,S which deﬁne two complex charts - again ZN,S is ﬁnite (i.e. the N and S pole are excluded in the N and S chart respectively). We then see the transition functions are simply, 1 ZN = (40) ZS which is analytic in the overlap of the 2 charts. (Note the overlap region excludes ZS = 0).

30 n 1.14 Example: CP complex projective space n CP ≡ the set of undirected complex lines through origin in Cn+1. [A complex line is z = a + λb, for a, b, λ ∈ C, so it is 2 real dimensional!]

n The deﬁnition is exactly as for RP except x0s → z0s.

n+1 Take ‘homogeneous coordinates’ (z1, . . . , zn+1) in C − {0} where we identify (z1, . . . , zn+1) ∼ (λz1, . . . , λzn+1) for any λ ∈ C − {0}.

Require (n + 1) charts:

n Um : all p ∈ CP s.t. zm 6= 0 (41) n n ψm : CP → C (42) z1 z2 zm−1 zm+1 zn+1 p = (z1, . . . , zn+1) → , ,..., , ..., (43) zm zm zm zm zm (m) (m) = Z1 ,...Zn (44)

Then e.g. (1) (1) z2 zr zn+1 Z1 ,...Zn = ,..., ,..., z1 z1 z1 (2) (2) z1 z3 zr zn+1 Z1 ,...Zn = , ..., ,..., z2 z2 z2 z2

Transition functions eg. for U1,U2 overlap:

(1) (1) (2) (1) (2) (1) φ21 : Z1 ,...,Zn → Z1 (Z ),...,Zn (Z ) (45) with, (1) (2) 1 (2) Zi Z1 = (1) ,Zi = (1) for i 6= 1 (46) Z1 Z1 Hence the transition functions are indeed analytic on the overlaps (where (1) Z1 6= 0).

1 Note: Consider CP . This consists of lines through the origin in C2, speciﬁed by homogeneous coordinates (z1, z2). We have 2 patches. The patch U1 has

31 (1) (1) z1 6= 0, and we deﬁne a single complex coordinate Z = z2/z1, so Z 6= 0 in U1. The patch U2 has z2 6= 0, and we deﬁne a single complex coordinate (2) (2) Z = z1/z2, so Z 6= 0 in U2. Then we have a manifold with one complex dimension (corresponding to 2 real dimensions) with an atlas of two charts. (1) (2) In the overlap U1 ∩ U2, Z 6= 0 and Z 6= 0, and the transition function is (2) 1 2 Z = Z(1) . But this is exactly the same as our S example earlier, replacing (1) (2) 2 Z = 1/ZN and Z = 1/ZS. Hence the complex manifold S is simply 1 1 n CP . This is special for CP . For n > 1, CP is not s sphere.

32 1.15 Examples of not being a manifold • Any space with a boundary eg. [0, ∞) or a [0, ∞) × S1 etc....

• Some simple topological spaces

3 3 • Orbifolds: ex. R /Z2 deﬁned by taking R and identifying points by the Z2 action (x, y, z) ∼ (−x, −y, −z). This is not a manifold. The origin is ‘ﬁxed’ under this action and leads to an ‘orbifold’ singularity. [A space with manifold structure everwhere except at distinct points where there is a singularity which is locally of the type Rm/Γ for some discrete group Γ is called an orbifold.]

33 1.16 Manifolds with boundary We can encompass manifolds with boundary by modifying our deﬁnition of a m manifold so that the maps ψi are homeomorphisms into H = {(x1, . . . , xm) ∈ m R |xm ≥ 0}.

The boundary of Hm is the space Rm−1 with xm = 0, which we denote ∂Hm. The open sets of Hm are given by the intersection of open sets of Rm with Hm, so that if U is an open set of Rm, then V = U ∩Hm is an open set of Hm. An open ball lying entirely in Hm is then open in Hm. A half-ball with centre p m m lying in ∂H is the intersection B(p)∩H . This is the hemisphere given by m m the part of B(p) with x ≥ 0, with boundary B(p)∩∂H that is an m−1 dimensional closed ball in ∂Hm. For example if p is the point with co-ordinates (0, 0,... 0), then the half ball is the region {x ∈ Rm : |x| < and xm ≥ 0} with boundary the region {x ∈ Rm : |x| < and xm = 0}. For m = 3, the boundary of half of the n-dimensional ball is a disc, the 2-dimensional interior of a circle. General open sets are formed by taking arbitrary unions of open balls in Rm, and then taking the intersection with Hm.

The set of points in M that are mapped to the boundary of Hm form the

34 boundary of M, denoted ∂M. The boundary points of a manifold form a manifold themselves, or a disconnected sum of manifolds if the boundary has more than one component.

Consider an open set Ui of M, which contains part of the boundary, so that 0 m Ui ∩ ∂M= 6 0. The map ψi takes Ui to an open set Ui of H . The coordinates (x1, . . . , xm−1, xm) of a point p ∈ M are x(p) = ψi(p), and points in the boundary p ∈ ∂M ∩ Ui) will have coordinates (x1, . . . , xm−1, 0).

m The transition functions φij are required to be smooth maps from H to Hm.

35 2 Diﬀerential maps, tangent vectors and tensors

Manifolds allow calculus to be extended from Rm onto them - hence the ‘differential’ in ‘diﬀerential geometry’. We shall now explore how this happens.

2.1 Diﬀerentiable maps Deﬁne a map f;

f : M → N p → q = f(p) (47) where M is an m dim’l manifold with altas {(Ui, ψi)} and N is an n dim’l with altas {(Va, φa)}.

Let p ∈ Ui and q ∈ Va with q = f(p).

Now the map, −1 m m φa · f · ψi : R → R (48)

36 µ 0 and so we can use usual calculus. If we have coordinates {x } on Ui and α 0 {y } on Va then explicitly,

α µ −1 µ y (x ) = φa · f · ψi (x ) (49) for each α = 1, . . . , n.

−1 ∞ The map f is smooth at p if the (n-vector of) functions φa · f · ψi are C (smooth) in their arguments xµ. The map is said to be Cr if the functions −1 r φa · f · ψi are C . Sometimes (as in e.g. Nakahara’s book) smooth maps are referred to as diﬀerentiable.

−1 For complex manifolds, the map f is analytic at p if the functions φa · f · ψi are analytic in their arguments xµ.

Consider a point p in an overlap of charts p ∈ Ui ∩ Uj. Take coordinates µ 0 µ 0 {x } in Ui and {y } in Uj. For p ∈ Ui ∩ Uj, x = ψi(p) and y = ψj(p), with −1 x = φij(y) where φij = ψi · ψj .

−1 −1 −1 Now φa · f · ψi (x) = φa · f · ψi (φij(y)) = φa · f · ψj (y).

−1 If φa · f · ψi (x) is smooth, then since the transition functions are smooth, −1 then we are guaranteed that φa · f · ψi (y) is also smooth. i.e. the manifold structure ensures that if our map is smooth in one coordinate system, it will be in any other.

37 2.2 Embedding and submanifolds Consider a smooth map f : M → N where dimM ≤ dimN .

The map f deﬁnes an embedding of the manifold M into N if the map f is one-to-one.

Then the image f(M) is a submanifold of N that is diﬀeomorphic to M.

Here is an example where the map does not deﬁne an embedding as it is not one-to-one.

[ Technically this is called an immersion, and the map f∗ which we meet later is one-to-one.]

Whitney Embedding Theorem Any real smooth m-dimensional manifold can be smoothly embedded in R2m. Note that some manifolds can be embedded in Rn for some n < 2m, but 2m is the smallest dimension that will work for all manifolds.

38 2.3 An example: CY 3 Low energy string theory reduces to a theory of gravity and matter in 10 dimensions.

Simple ‘supersymmetric’ vacuum solutions giving ‘realistic’ phenomenology 4 have a spacetime manifold R × CY3 (i.e. a product manifold).

Here note that R4 refers to Minkowski spacetime, R1,3 [we will not mention metrics for a while, and as a manifold Minkowski spacetime is simply R4]

CY3 is a ‘Calabi-Yau’ manifold, a compact 3 complex dimensional manifold. In fact there are many of them, the only restriction being a topological one (vanishing ﬁrst chern class which implies you can always ﬁnd a metric with vanishing Ricci tensor).

The fact that CY3 is compact physically allows low energy observers to only ‘see’ 4 spacetime dimensions.

n All these CY3 manifolds can be thought of as submanifolds of CP for some (n > 3).

For example:

4 5 5 The ‘Quintic’ - studied a lot - is embedded in CP as z1 + . . . z5 = 0, where 4 zi are homogeneous coordinates for CP .

Ex. Have a think about what that looks like!

39 2.4 Diﬀeomorphism If a smooth map f : M → N is invertible and the inverse map f −1 is also smooth then f is a ‘diﬀeomorphism’.

For such a diﬀeomorphism f we say M and N are ‘diﬀeomorphic’, meaning they are the same manifold i.e. M ≡ N as manifolds. [Recall for topo spaces X ≡ Y if there exists a homeomorphism between them.]

We denote the set of diffeomorphisms of a manifold M as diff(M), so any f ∈ diff(M) takes,

f : M → M p → p0 = f(p) (50)

We may think of these diﬀeomorphisms as smooth relabelings of points.

Obviously a diffeomorphism also defines a homeomorphism at the level of the topological spaces M and N . But two manifolds with the same topology are not necessarily diffeomorphic! Only seem to be highly non-trivial examples - eg R4 due to Donaldson. The 7-sphere with the usual topology allows 28 different differentiable structures, while the 11-sphere allows 992 different differentiable structures. This is rather unusual – in most cases, there is only one differentiable structure.

40 2.5 Functions The simplest smooth maps. A function f is a smooth map such that,

f : M → R p → f(p) (51)

µ 0 Using a chart (Ui, ψi) with coordinates x in Ui we have a coordinate repre- sentation of a function; for a point p ∈ Ui,

−1 m f · ψi : R → R −1 x = ψi(p) → f · ψi (x) (52)

Denote set of functions on M as F(M).

This inherits the usual ring structure from R. Namely we may add/multiply two functions to get a third simply by adding/multiplying their values at all points; • (f + g)(p) = f(p) + g(p) • (f · g)(p) = f(p)g(p)

A constant function is one where f(p) = c for all p ∈ M, for some c ∈ R.

41 2.6 Curves Essentially the opposite of a function! An open curve C is a smooth map such that;

C : R → M λ ∈ (a, b) → p = C(λ) (53)

µ 0 In a chart (Ui, ψi) with coordinates x in Ui we may write,

m ψi ·C : R → R λ → x = ψi ·C(λ) (54)

An open curve has ‘ends’. One can also consider closed curves as maps S1 → M.

42 2.7 Tangent vectors

Consider an m-dim manifold M and a point p ∈ M and two curves C1, C2 passing through p, so that,

p = C1(0) = C2(0) (55)

We can use the fact that our manifold locally looks like Rm to say whether or not the two curves have the same ‘direction and speed’ at p. Taking a chart covering the neighbourhood of p,(U, ψ), with coordinates xµ then we say that the curves are tangent at p iﬀ;

d µ d µ x (C1(λ)) = x (C2(λ)) (56) dλ λ=0 dλ λ=0 d µ i.e. the curves are tangent to each other at p. We note that dλ x (C1,2(λ)) λ=0 ∈ Rm, and hence the ‘direction and speed’ are governed by a point in Rm.

We define the equivalence at p, denoted C1 ∼ C2, iff C1, C2 are tangent at p. Note that if two curves are tangent in one coordinate chart, they are tangent in any other coordinates – i.e. our definition of this equivalence is coordinate independent.

Tangent vector: Geometrically we then deﬁne a tangent vector, Vp, at the point p to be the class of all curves passing through p which are tangent to

43 each other. Given a curve C through p, we denote this equivalence as [C]. We say that C is a representative of the class.

Tangent space: We then deﬁne the tangent space at p as the set of all tangent vectors as p. We denote the tangent space Tp(M).

This tangent space has a linear/vector structure, again inherited from the fact that the manifold is locally Rm. Consider a point p in a chart (U, ψ), where w.l.o.g. we choose the homeomorphism, ψ, so that ψ(p) = 0 ∈ Rm. Then given a, b ∈ R, and two tangent vectors Vp,1,Vp,2 ∈ Tp(M) with representatives C1,2(λ) so that Vp,1 = [C1],Vp,2 = [C2], we deﬁne,

Vp,3 = a Vp,1 + b Vp,2 ∈ Tp(M) (57) where Vp,3 = [C3(λ)] and,

−1 C3(λ) ≡ ψ [a (ψ ·C1(λ)) + b (ψ ·C2(λ))] (58) with the righthand side being defined from the usual linearity of Rm. Whilst this definition uses a particular chart, one obtains the same curve C3 whichever chart one uses, and hence the definition is independent of the local choice of coordinates.

More explicity we could write this relation as

d µ d µ d µ x (C3(λ)) = a x (C1(λ)) + b x (C2(λ)) . (59) dλ λ=0 dλ λ=0 dλ λ=0

Hence we see that the tangent space Tp(M) is actually isomorphic to the vector space Rm. Thus we can write any element as

µ Vp = v ep,(µ) (60) where ep,(µ) ∈ Tp(M) are a set of m basis tangent vectors (each with repre- µ m sentative curves C(µ)), and v ∈ R are the components of the tangent vector in this basis.

As for any vector space, the components in one basis are related to those in another by an element of the general linear group. If we have two bases 0 0 a {ea} and {ei}, then these are related as ei = Ai ea, where the matrix

44 a Ai ∈ GL(m, R) (i.e. is an invertible m × m matrix - since both are bases 0i 0 a 0i a a this must be invertible). Then since Vp = V ei = V ea then V Ai = V relates the components.

45 2.8 An aside: Dual vector spaces

i Take a vector space V , with basis {ei} so v = v ei ∈ V .

Consider a linear function, f, on V ;

f : V → R X i v → f(v) = v f(ei) (61) i since it is linear. Hence f is determined by knowing the set {f(ei)}.

The space of linear functions is also a vector space due to the linear structure;

f3(v) = (a1f1 + a2f2)(v) = a1f1(v) + a2f2(v) (62) for a1, a2 ∈ R. Hence given linear maps deﬁned by {f1(ei)} and {f2(ei)} we can construct a third, {a1f1(ei) + a2f2(ei)}.

This is called the dual vector space to V , denoted V ∗. Note that since to ∗ deﬁne linear map we require {f(ei)}, we must have dim(V ) = dim(V ).

i ∗ ∗ i Take basis {e } for V so f ∈ V is f = fie (note the position of indices)

i i i Make the choice that basis {e } is dual to the basis {ei}: i.e. e (ej) = δ j. One can always choose this basis.

0 a Note then that under a basis transformation, ei = Ai ea for A ∈ GL, then 0i −1 i a we require a corresponding transform of the dual basis, e = (A )a e .

46 We have an inner product:

∗ < ·, · >: V × V → R (f, v) → < f, v >≡ f(v) (63) where, i j j i i f(v) = fie (v ej) = fiv e (ej) = fiv (64) and the inner product is bilinear in its arguments.

From the bilinearity of this inner product we see that we may also think of v ∈ V as being a linear function acting on f ∈ V ∗, giving the value v(f) ≡< f, v >. Hence we see that (V ∗)∗ = V (at least for ﬁnite dim’l vector spaces).

47 2.9 Cotangent vectors

∗ TpM is a vector space. Tp M is the dual vector space - the co-tangent space

∗ Elements of Tp M are cotangent vectors - linear functions on vectors (also known as one-forms, as we will see later!)

∗ Take ωp ∈ Tp M. Then we can think of ωp as the map,

ωp : TpM → R vp → ωp(vp) (65)

The bilinear inner product < ·, · > is the map;

∗ < ·, · >: Tp M × TpM → R (ωp, vp) → < ωp, vp >≡ ωp(vp) (66)

µ ∗ α Take a basis {eµ} for TpM. and a dual basis {e } in Tp M. Then ωp = ωαe µ α and vp = v eµ, and ωp(vp) = ωαv .

∗ ∗ Now (Tp M) = TpM, and hence a vector acts as a linear function on the cotangent space. So given vp ∈ TpM we have,

∗ vp : Tp M → R ωp → vp(ωp) ≡< ωp, vp > (67)

48 2.10 Tensors Tensors are the natural generalization of these maps. A (q, r) tensor is a linear function on q cotangent vectors and r tangent vectors:

∗ Ap : Tp M × ... × TpM × ... → R (ω(1), . . . , ω(q), v(1), . . . , v(r)) → Ap(ω(1), . . . , ω(q), v(1), . . . , v(r)) (68)

µ µ Let us take a basis {eµ} and dual {e } at p. Then ω(i) = ω(i)µe |p and µ v(i) = v(i)eµ. Since the tensor map is a linear map, we have,

β1 βr α1 αq Ap(ω(1), . . . , ω(q), v(1), . . . , v(r)) = ω(1)α1 . . . ω(q)αq v(1) . . . v(r)Ap(e , . . . , e , eβ1 , . . . , eβr ) (69) α1 αq Hence the map Ap is simply deﬁned by the collection {Ap(e , . . . , e , eβ1 , . . . , eβr )}. These are the ‘components’ of the tensor. Denote them,

α ...α A 1 q ≡ A (eα1 , . . . , eαq , e , . . . , e ) (70) β1,...,βr p β1 βr so that for any set of covectors ω(i) and vectors v(i) we can write the results of the action of A on them simply by the linear combination of the components,

A (ω , . . . , ω , v , . . . , v ) = ω . . . ω vβ1 . . . vβr Aα1...αq (71) p (1) (q) (1) (r) (1)α1 (q)αq (1) (r) β1,...,βr

Hence the space of (q, r) tensors is a vector space with dimension mq+r.

q The set of (q, r) tensors at a point p is denoted as Jr,p(M). For convenience we will drop the ‘M’ here.

We may deﬁne a tensor product (outer product), denoted ⊗, with action q1 q2 q1+q2 ⊗ : Jr1,p × Jr2,p → Jr1+r2,p in the following way;

q1 q2 q1+q2 Let A ∈ Jr1,p and B ∈ Jr2,p. Then C = A ⊗ B ∈ Jr1+r2,p is deﬁned by

C(ω(1), . . . , ω(q1+q2), v(1), . . . , v(r1+r2)) ≡ A(ω(1), . . . , ω(q1), v(1), . . . , v(r1)) ×

B(ω(q1+1), . . . , ω(q1+q2), v(r1+1), . . . , v(r1+r2))(72)

∗ for any ω(i) ∈ Tp M and v(i) ∈ TpM. Note that this is compatible with the linear structure of the tensor spaces.

49 q1+q2 q1 q2 1 0 ∗ Then Jr1+r2,p = Jr1,p ⊗ Jr2,p. Since J0,p ≡ TpM and J1,p ≡ Tp M we may write in general;

q ∗ ∗ Jr,p = TpM ⊗ ... ⊗ TpM ⊗ Tp M ⊗ ... ⊗ Tp M (73)

Notation varies; typically mathematicians write a tensor as T (·, ·,..., ·) em- phasizing it is a map. Physicists would write it as

α ...α T = T 1 q e ⊗ ... ⊗ e ⊗ eβ1 ⊗ ... ⊗ eβr (74) p β1,...βr α1 αq giving the tensor components explicitly.

2.11 Contraction of tensors

q+1 q Given A ∈ Jr+1,p we may form a new tensor B ∈ Jr,p via the contraction of a specified covector ‘slot’ and a specified vector slot. Given a basis {ei} and dual basis {ei} we define the contraction map by

µ B(ω(1) . . . ω(q), v(1), . . . , v(r)) ≡ A(ω(1) . . . , e , . . . ω(q), v(1), . . . , eµ, . . . , v(r)) (75) Notationally in physics we would write,

α ...µ...α B = A 1 q e ⊗ ... ⊗ e ⊗ eβ1 ⊗ ... ⊗ eβr (76) β1,...µ...βr α1 αq

µ In maths one would write B = A(·, . . . , e ,..., ·, eµ,... ·).

50 2.12 Tangent vectors act on functions

Recall that a tangent vector, Vp, at the point p is the class of all curves passing through p which are tangent to each other. For an m-dimensional manifold M and a point p ∈ M with two curves C, C0 passing through p, so that, 0 p = C(λ0) = C (λ1) (77) for some λ0, λ1, the curves are tangent at p iﬀ;

d µ d µ 0 x (C(λ)) = x (C (λ)) (78) dλ dλ λ=λ0 λ=λ1 This allows us to deﬁne the directional derivative of a function. Take a curve C through p and a function f ∈ F(M). Then f ·C : R → R.

The rate of change of f along the curve deﬁnes the directional derivative at at p = C(λ ) 0 d (f ·C(λ)) dλ λ=λ0 Using a chart (U, ψ) we can break up f ·C = (f · ψ−1) · (ψ ·C) where

−1 m m f · ψ : R → R ψ ·C : R → R (79)

Then ψ ·C deﬁnes a curve in Rm which we write as xµ(λ) ≡ xµ(C(λ)) = ψ ·C(λ)

51 while f · ψ−1 deﬁnes a function on Rm which we write as f(x) ≡ f(ψ−1(x)) = f · ψ−1(x)

Then we may write the rate of change at p = C(λ0) as

d d µ ∂ f ·C(λ) = x (C(λ)) f(x) (80) dλ dλ ∂xµ p=C(λ0) λ=λ0 p using the chain rule.

An important property of the directional derivative is that any other curve C0 such that C0 ∼ C at p gives the same rate of change. Thus the deﬁnition only depends on the equivalence class of curves that are tangent to one another, i.e. depends only on the tangent vector. To see this, consider a point p ∈ M with two curves C(λ), C0(λ) passing through p, so that,

0 p = C(λ0) = C (λ1) (81) for some λ0, λ1, with the curves are tangent at p, so that

d µ d µ 0 x (C(λ)) = x (C (λ)) (82) dλ dλ λ=λ0 λ=λ1 Then the directional derivative for the two curves is the same:

d d µ ∂ f ·C(λ) = x (C(λ)) f(x) dλ dλ ∂xµ p=C(λ0) λ=λ0 p

d µ 0 ∂ = x (C (λ)) f(x) dλ ∂xµ λ=λ1 p

d 0 = f ·C (λ) (83) dλ 0 p=C (λ1) Thus the directional derivative at p for two curves C0 ∼ C that are tangent at p are the same, so that the directional derivative depends only on the equivalence class of curves at p,[C]. That is, for any tangent vector Vp ∈ TpM at p, we can deﬁne the directional derivative by

d Vp[f] = (f ·C(λ)) dλ λ=λ0 for any curve C(λ) in the class of curves corresponding to Vp.

52 2.13 Directional derivatives

For a tangent vector Xp ∈ TpM at p, we have deﬁned the directional derivative Xp[f] of a function f. Then for each tangent vector Xp ∈ TpM we have a directional derivative which is a map from the functions to the reals, ˆ Xp : F(M) → R (84) f → Xp[f] (85) which gives the rate of change of the function at p along any representative curve. This obeys the properties

• For a constant function c, Xp[c] = 0.

• Xp[f + g] = Xp[f] + Xp[g]

• ‘Leibnitz’ rule: Xp[fg] = Xp[f]g + fXp[g]

This has the natural linear structure of TpM, so that for a, b ∈ R and Xp,Yp ∈ TpM we have (aXp + bYp)[f] = a Xp[f] + b Yp[f] (86) for any function f ∈ F(M). Then writing Zp = aXp + bYp, we have ˆ ˆ ˆ Zp = aXp + bYp ˆ so that the vector space of maps Xp is isomorphic to the vector space of tangent vectors Xp, TpM. Then each tangent vector Xp corresponds to a ˆ unique directional derivative Xp, and vice versa.

The coordinate basis

A tangent vector Vp ∈ TpM at p is an equivalence class of curves [C] through p that deﬁnes a directional derivative at p of any function f, Vp[f]. These constructions are independent of any choice of coordinates. However, choosing a chart gives useful expressions. First, a tangent vector Vp ∈ TpM deﬁnes a vector in Rm for any curve in the corresponding class of curves [C] with p = C(λ0) by

µ d µ v = x (C(λ)) (87) dλ λ=λ0

53 As we have seen, vµ ∈ Rm is independent of the choice of curve in [C]. Then we have

d d µ ∂ f ·C(λ) = x (C(λ)) f(x) dλ dλ ∂xµ p=C(λ0) λ=λ0 p

µ ∂ = v µ f(x) (88) ∂x p

Next, we deﬁne a set of vectors eµ ∈ TpM (µ = 1, . . . , m) by the corresponding mapse ˆµ : f → eµ[f] where

∂ −1 eµ[f] ≡ µ (f · ψ (x)) µ = 1, . . . , m (89) ∂x ψ(p) and we can write ∂ eˆ = (90) µ ∂xµ Then

d µ f ·C(λ) = v eµ[f] (91) dλ p=C(λ0) so that µ Vp[f] = v eµ[f]

The set of vectors eµ ∈ TpM (µ = 1, . . . , m) form a basis for TpM and the vector V has components vµ in this basis:

µ Vp = v eµ The ∂ eˆ = (92) µ ∂xµ form a basis for directional derivatives in this coordinate chart. This basis is called the coordinate basis since it is based on a particular chart. The basis of tangent vectors eµ are often written as ! ∂ eµ = µ (93) ∂x p

54 but strictly speaking this is an abuse of of notation: the tangent vectors eµ are equivalence classes of curves and it is the corresponding mapse ˆµ that are derivatives.

For a chart (U, ψ) with p ∈ U, consider the line in Rm through ψ(p) in the xν direction for some ﬁxed ν given by

µ µ µ x(ν)(λ) = ψ (p) + δ ν(λ − λ0) so that dxµ (ν) = δµ dλ ν and x(λ0) = ψ(p). We will take λ to be in an interval (a, b) so that (x(ν)(λ)) ∈ U 0 = ψ(U) for all λ ∈ (a, b). For each ν, this gives a curve in U ⊆ M through p with C(ν)(λ) given by −1 C(ν)(λ) = ψ (x(ν)(λ)) with p = C(ν)(λ0). Then the class of curves [C(ν)] in M deﬁnes a tangent vector at p in TpM, and the directional derivative of a function f with respect to this tangent vector is

d d µ ∂ f ·C(ν)(λ) = x (C(ν)(λ)) f(x) dλ dλ ∂xµ p=C(λ0) λ=λ0 p

d µ ∂ = x (λ) f(x) dλ (ν) ∂xµ λ=λ0 p

µ ∂ = δ ν µ f(x) ∂x p

∂ = ν f(x) (94) ∂x p

= eν[f] (95)

Thus we see that the basis tangent vector eν is the class of curves [C(ν)] through p, and deﬁnes the directional derivative ∂ eˆ = (96) µ ∂xµ

55 Changing from one coordinate chart to another

The action Vp[f] of a tangent vector Vp on a function f is defined without reference to any coordinate system. Choosing coordinates xµ gives compo- µ µ nents v of Vp in that coordinates basis. Choosing different coordinates y defines a different coordinates basis, and the components v0µ will be different in the new coordinate system. The fact that they must define the same µ Vp[f] allows us to calculate the relation between the components v in one coordinate system to the components v0µ in another.

If p is contained in an overlap of two charts, p ∈ Ui ∩ Uj with coordinates µ 0 α 0 {x } in Ui and {y } in Uj, then the components of a directional derivative in the two corresponding coordinate bases transform to ensure it is invariant. In Ui we have ∂ eˆ = (97) µ ∂xµ while in Uj we have ∂ eˆ0 = (98) µ ∂yµ

Consider Vp acting on any function f ∈ F(M). Then we may write this action as

µ µ ∂f(x) Vp[f] = v eµ[f] = v µ (99) ∂x x(p) or equivalently as

0µ 0 0µ ∂f(y) Vp[f] = v eµ[f] = v µ (100) ∂y y(p) Since the partial derivative chain rule gives µ 0α ∂f(x(y)) 0α ∂x (y) ∂f(x) v α = v α µ (101) ∂y y(p) ∂y y(p) ∂x x(p) we see that on the overlap we see that for the two expressions for Vp[f] to agree for all functions f, we must have the relation between components µ µ 0α ∂x (y) v = v α (102) ∂y y(p)

56 While it is elegant to geometrically consider tangent vectors as equivalence classes of curves, it turns out to be very powerful to consider them from the isomorphic point of view, as directional derivatives. In particular the ‘diﬀer- ential’ in diﬀerential geometry is related to the important role that tangent vectors have acting on functions. Whilst this is obvious when taking the view of directional derivatives, it is far from obvious from the viewpoint of equivalence classes of curves that it would be a useful observation. This then allows one to extend ideas of calculus to manifolds.

Hence from now on, we will take tangent vectors to be directional derivatives and we will not distinguish between tangent vectors Xp and directional ˆ derivatives Xp.

• Thus we may write a general tangent vector in a coordinate basis as µ ∂ Vp = V ∂xµ p ∈ Tp(M).

µ ∂f(x) • Vp acts on functions as Vp[f] = V ∂xµ ∈ R for f ∈ F(M). x(p) • In an overlap of two charts, with coordinates {xµ} and {yα}, we have,

! ! µ ∂ 0α ∂ Vp = V µ = V α (103) ∂x p ∂y p

with (the familar transformation of vector components from GR),

µ µ 0α ∂x (y) V = V α (104) ∂y y(p)

• Given a curve C(λ) with parameter λ, we denote the tangent vector to d the curve at the point p = C(λ) as dλ . Explicitly in a chart we have,

µ d dx (λ) ∂ = µ (105) dλ dλ ∂x p=C(λ)

57 2.14 Diﬀerentials as Cotangent Vectors

Given a function f : M → R, there exists a natural cotangent vector ω at a point p, i.e. a linear map ω : TpM → R which is given by its action on a vector V ∈ TpM by

ω : V → ω(V ) = V [f] (106)

µ ∂ In a coordinate basis V = V eµ ∈ Tp(M) where eµ = ∂xµ p and

µ ∂f(x) V [f] = V µ ∂x x(p)

µ µ for f ∈ F(M). This can be written as ω(V ) = ωµV where ω = ωµe is the cotangent vector at p whose components in the coordinate basis are

∂f(x) ωµ = µ ∂x x(p)

These are also the components of the diﬀerential df. Given a function f : M → R, the diﬀerential df at a point p is

∂f(x) µ df|p = µ (dx )|p (107) ∂x p

µ so that df = ωµdx .

We have defined a cotangent vector at p as an element of the dual of the tangent space TpM. Geometrically, we could instead define a cotangent vector at p as the equivalence class of functions that have the same gradient ∂f(x) { ∂xµ } at p. (This is easily shown to be coordinate independent). This is analogous to the definition of a tangent vector as an equivalence class of ∂f(x) curves. Any f in the equivalence class has the same gradient { ∂xµ } at p and so defines the same map ω : V → ω(V ) = V [f], so that there is a one-to-one correspondence between linear maps from the tangent space TpM to R and equivalence classes of functions, so that the two definitions agree.

58 The diﬀerential df is an example of a one-form at p, i.e. an expression of µ the form w = wµ (dx |p). There is a one-to-one correspondence between µ µ cotangent vectors w = wµe and one-forms w = wµdx , so that there is an ∗ isomorphism between the cotangent space Tp M and the space of one-forms at p. The isomorphism between cotangent vectors and one-forms is similar to that between tangent vectors and directional derivatives. Just as we can regard vectors as directional derivatives, we can regard cotangent vectors as one-forms. Then it is natural to identify the dual coordinate basis eµ with the space of diﬀerentials at p

µ µ e ≡ (dx |p) (108)

∂ µ The coordinate basis { ∂xµ |p} for TpM and the dual coordinate basis {dx |p} ∗ for Tp M are dual, i.e. they satisfy ∂ < dxµ, >= δµ ∂xα α

∗ α A general co-vector w ∈ Tp M can then be expressed as wp = wα (dx |p). Then the cotangent vector ω discussed above corresponding to a function f is precisely the differential ω = df, so that the differential df is the one-form defining the map df : TpM → R which is given by its action on a vector V ∈ TpM by df : V → df(V ) = V [f] (109)

d Given a curve C(λ) through p and a function f, then the tangent vector dλ |p d df(λ) and the covector df|p have inner product: < df|p, dλ |p >= dλ . Hence the notation df!

To see this explicitly, use µ d dx (λ) ∂ = µ (110) dλ dλ ∂x p=C(λ) and

∂f(x) µ df|p = µ (dx )|p (111) ∂x p

59 together with ∂ < dxµ, >= δµ ∂xα α to ﬁnd µ d dx (λ) ∂f(x) df(λ) < df|p, |p >= µ = dλ dλ ∂x p dλ p Diﬀerentials and change of coordinates ∂ Consider a coordinate basis, so that eµ = ∂xµ p . Recall under a change of coordinates y = y(x), then

! α 0 ∂ ∂x (y) eµ → eµ = µ = µ eα (112) ∂y p ∂y y(p)

Recall that for a dual basis to this coordinate basis we require

µ 0µ 0 µ < e , eα >=< e , eα >= δ α

From the chain rule property of partial derivatives we find the dual coordinate basis transform as µ µ 0µ ∂y (x) α e → e = α e (113) ∂x x(p) Differentials also transform this way, ∂yµ dyµ = dxα (114) ∂xα so that it is consistent to identify the dual coordinate basis with the space of differentials at p µ µ e ≡ (dx |p) (115) ∂ µ The coordinate basis { ∂xµ |p} and the dual coordinate basis {dx |p} satisfy ∂ < dxµ, >= δµ ∂xα α

60 2.15 Tensors in a coordinates basis We may now write a general (q, r) tensor at p as

∂ ∂ α1...αq β1 βr Tp = T ⊗ ... ⊗ ⊗ dx ⊗ ... ⊗ dx (116) β1,...βr α1 αq ∂x ∂x p

Under a change of coordinates x → y = y(x) one can easily see the components of the tensor T α1...αq in the x coordinate basis are related to the β1,...βr components T 0α1...αq in the y coordinate basis as is familar from GR; β1,...βr

∂yα1 ∂yαq ∂xν1 ∂xνr 0α1...αq µ1...µq Tβ ,...β = ...... Tν ,...ν (117) 1 r ∂xµ1 ∂xµq ∂yβ1 ∂yβr 1 r

61 3 Induced maps, tensor ﬁelds and ﬂows

3.1 Induced maps: push-forward Suppose we have a map f : M → N . What happens to the tangent space at a point TpM? A map is induced by f on TpM at point p called the push-forward, denoted f∗, such that;

f∗ : TpM → Tf(p)N

V → f∗V (118)

To deﬁne this push forward map f∗ let us ﬁrst consider a function on N , g ∈ F(N ). Then we can pull-back the function g onto a function on M by

g : N → R (119)

g · f : M → R (120) The pull-back of the function g by f is sometimes written as f ∗(g), so ∗ f (g) ≡ g · f. Recall that a vector V ∈ TpM can then act on the function g · f to give its rate of change V [g · f]. Likewise the vector f∗V ∈ Tf(p)N can act on g as f∗V [g] to give its rate of change.

We deﬁne the push-forward map f∗, by equating these rates of change

f∗V [g] ≡ V [g · f] (121) for any function g ∈ F(M). This has the important property that it is a linear map: f∗(aV + bW ) = af∗(V ) + bf∗(W ) (122)

62 for V,W ∈ TpM and a, b ∈ R. The map f∗ : TpM → Tf(p)N between tangent spaces can be thought of as a linear approximation to the map f.

Using a chart (U, ψ) on M containing the point p with coordinates xµ, and a chart (V, ξ) on N containing the point f(p) with coordinates yα we may µ ∂ α ∂ write V = v ∂xµ |p and f∗V = w ∂yα |f(p). Then

−1 −1 f∗V [g · ξ (y)] ≡ V [g · f · ψ (x)] (123)

Next we deﬁne y(x) = ξ · f · ψ−1(x) (124) so we can write

g · f · ψ−1 = (g · ξ−1) · (ξ · f · ψ−1) = (g · ξ−1)(y(x)) so that

α ∂ −1 µ ∂ −1 w α g · ξ (y) = v µ g · f · ψ (x) (125) ∂y y=ξ·f(p) ∂x x=ψ(p)

µ ∂ −1 = v µ (g · ξ )(y(x)) ∂x x=ψ(p) µ ∂ α ∂ −1 = v µ y (x) α g · ξ (y) ∂x ∂y y=ξ·f(p) using the chain rule. Hence

α α ∂y µ w = µ v (126) ∂x p are the components of the push forward f∗V .

The push-forward f∗ generalizes to tensor products, so that

f∗ : TpM ⊗ TpM ⊗ ... → Tf(p)N ⊗ Tf(p)N ⊗ ...

V ⊗ W ⊗ ... → f∗V ⊗ f∗W ⊗ ... (127)

This then deﬁnes a push forward map of tensors of type (q, 0), so

q q f∗ : J0,p(M) → J0,f(p)(N ) (128)

63 Using coordinates as above, we write

∂ ∂ A = aµ1,...,µq ⊗ ... ⊗ (129) µ1 µq ∂x ∂x p and ∂ ∂ α1,...,αq f∗A = b ⊗ ... ⊗ (130) α1 αq ∂y ∂y f(p) so that the push forward is given by

∂yα1 ∂yαq bα1,...,αq = ... aµ1,...,µq (131) µ1 µq ∂x ∂x p

Given two maps f : M → N and g : N → O we can compose to form g · f : M → O. Associated with this is the push forward (g · f)∗ which, for some vector V ∈ TpM is given by

(g · f)∗V = g∗ · f∗V (132)

64 3.2 Pull-back We may deﬁne the pull-back f ∗ map. This ‘pulls’ back the cotangent space from N to M as

f : M → N ∗ ∗ ∗ f : Tp M ← Tf(p)N f ∗w ← w (133)

The pull-back is deﬁned as follows. Take a vector v ∈ TpM, so that f∗v ∈ ∗ ∗ Tf(p)N . Take a covector w ∈ Tf(p)N . Then for any v, w we require,

∗ < f w, v >=< w, f∗v > (134) which deﬁnes the pull back map f ∗.

This is a linear map:

f ∗(aw + bv) = af ∗(w) + bf ∗(v) (135)

∗ ∗ ∗ ∗ for w, v ∈ Tf(p)M and a, b ∈ R. The map f : Tf(p)N → Tp M between tangent spaces can again be thought of as a linear approximation to the map f.

Take coordinates xµ on a chart containing p in M, and yα in a chart con- µ ∂ taining f(p) in N , so that y = y(x). Then we may express v = v ∂xµ ∈ TpM α ∗ and w = wαdy ∈ Tf(p)N . Since

∂yα ∂ f v = vµ (136) ∗ ∂xµ ∂yα we see that for the above to hold we must have,

∂yα f ∗w = w dxµ (137) α ∂xµ

As for the push-forward map, the pull-back map f ∗ naturally extends to ten- 0 sor products of covectors and hence to Jr,p tensors.

65 3.3 Vector, covector and tensor fields We will define a tensor field as a map that smoothly assigns a tensor to each point in M. That is, we have a map

q A : M → Jr,p

p → Ap (138)

We will need to define what it means for a tensor field to depend smoothly on the point p. We will sometimes write this map as A(p) to emphasise that the tensor field depends on the point p. With a coordinate chart, this becomes a tensor A(x) depending on the coordinates x = ψ(p).

q We denote the space of (q, r) tensor ﬁelds as Jr .

1 0 Example: Vector ﬁelds are in J0 , covector ﬁelds are in J1 .

Consider first vector fields. A vector field assigns to each point p ∈ M a vector Vp ∈ TpM. Then for any function g : M → R we have the directional derivative Vp[g] which gives a real number for each point p ∈ M. This defines a function on M by

V [g]: M → R p → Vp[g] (139)

Then the action of a vector on functions generalizes to vector ﬁelds on functions as

1 J0 × F → F V , g → V [g] (140)

We can now deﬁne a vector ﬁeld V as being smooth if V [g] is a smooth function on M for all smooth functions g : M → R.

µ ∂ For a vector ﬁeld in a chart we may write V = v (x) ∂xµ . Then the condition µ ∂ that V be a smooth vector ﬁeld is that V [g](x) = v (x) ∂xµ g(x) is a smooth function of x for all smooth functions g(x). This requires that each component vµ(x) is itself a smooth function.

66 ∗ A cotangent vector ﬁeld assigns to each point p ∈ M a covector ωp ∈ Tp M. Then for any vector ﬁeld V on M, ωp(Vp) gives a real number for each point p ∈ M, so that ω(V ) is a real function

ω(V ): M → R p → ωp(Vp) (141)

Then a cotangent vector ﬁeld ω is said to be smooth if ω(V ): M → R is a smooth function for all smooth vector ﬁelds V . Introducing coordinates, µ µ ∂ µ ω = ωµ(x)dx and V = v (x) ∂xµ and ω(V ) = ωµ(x)v (x). Then ω(V ) will be a smooth function of x for all smooth functions vµ(x) if and only if each component ωµ(x) is itself a smooth function.

A(q, r) tensor at a point p was deﬁned as a linear function on q cotangent vectors and r tangent vectors:

∗ Ap : Tp M × ... × TpM × ... → R (142) (ω(1), . . . , ω(q), v(1), . . . , v(r)) → Ap(ω(1), . . . , ω(q), v(1), . . . , v(r))

For a (q, r) tensor field, we require that this map from a point p to R be smooth for all smooth covector fields ω(1), . . . , ω(q) and all smooth vector µ fields v(1), . . . , v(r). This requires that in some chart with coordinates {x }, each of the components of the tensor in the coordinate basis are smooth functions on M.

The smoothness requirement for a tensor ﬁeld can then be expressed by requiring that when we express the tensor at points in some chart with coordinates {xµ} the components of the tensor in the coordinate basis are smooth functions on M.

The properties of a manifold then imply that if the components are smooth in one coordinate system, then the components will be smooth in any other overlapping coordinates.

0 Example: The metric tensor ﬁeld which we will encounter later is g ∈ J2 , µ ν and we can explicitly write locally in coordinates as g = gµν(x)dx ⊗ dx .

67 Note that tensor fields inherit the linearity of the tangent space. Given q two tensor fields A, B ∈ Jr (M), and two functions a, b ∈ F(M), we define addition of tensor fields and their multiplication by functions to give a tensor q field C = aA + bB ∈ Jr (M), defined so that C(p) = a(p)A(p) + b(p)B(p) at each point p. This will be smooth if a, b ∈ F(M) are smooth functions and A, B are smooth tensor fields. In some chart the components of C are given by a linear combination of the components of A, B:

Cα1...αq (x) = a(x)Aα1...αq (x) + b(x)Bα1...αq (x) (143) β1...βr β1...βr β1...βr

68 3.4 Tensor ﬁelds and induced maps We have seen how a map between manifolds induces push-forwards and pull- backs of certain tensors at a point p. In this section, we discuss extending these to tensor ﬁelds.

A map f : M → N induces the push-forward map

f∗ : TpM → Tf(p)N

V → f∗V (144) on the tangent space at a point p. For a point p the vector f∗Yf(p) is deﬁned by

f∗Y [g] = Y [g · f] (145) f(p) p for any function g on N . We can attempt to extend this to deﬁne the push- forward of a vector ﬁeld Y (p) by requiring this to hold for all p ∈ M.

There are two problems with this definition. First, it is not well-defined if f is not one-to-one. To see this, suppose that there are two points p, p0 ∈ M that map to the same point q ∈ N , so that f(p) = f(p0) = q. Then in 0 general f∗ will map Y (p) and Y (p ) to two different vectors in TqN , one with f∗Y [g]|q defined by Y [g · f]|p and one by Y [g · f]|p0 . Thus for this to be a good definition, the map f must be one-to-one, i.e. an embedding.

Embeddings For an embedding f, the image f(M) is a submanifold of N .A vector field on M can be pushed forward to a vector field on f(M), but this only defines a vector field on the submanifold f(M), not a vector field on the whole N .

For a vector field Y , Y [g · f] is a function on M (mapping a point p ∈ M to the directional derivative wrt Y of g · f at p, which is a real number) while f∗Y [g] is a function on a subspace of N . Then for a one-to-one map f, the definition (145) of the push-forward of the vector field Y can be stated as the requirement that for any g,

(f∗Y [g])(f(p)) = Y [g · f](p) (146)

69 This can then be written as

(f∗Y [g]) · f = Y [g · f] (147) where now both sides are viewed as functions on M. If Y is a smooth vector field, then Y [g · f] is a smooth function on M for all smooth f, g, so that (f∗Y [g]) · f is smooth. This requires that (f∗Y [g]) be smooth on f(M), so that f∗Y is a smooth vector field on f(M). This uses the fact that M and f(M) are diffeomorphic, so that a smooth inverse for f is defined on f(M), although the definition of the inverse cannot be extended to the whole of N .

The pull-back, however, can be extended to covector ﬁelds without such problems. The pull-back of f

f : M → N ∗ ∗ ∗ f : Tp M ← Tf(p)N f ∗w ← w (148) was deﬁned by requiring

∗ < f w, v > =< w, f∗v > (149) p f(p)

∗ for all v ∈ TpM and w ∈ Tf(p)N . This can be extended to define the pull- back of covector fields w(q) on N , by requiring this to be true for all vector fields v(p) on M and at all points p ∈ M. Note that this gives a well-defined covector field on M, even if f is not one-to-one. The covector field on M is smooth if the covector field w(q) on N is smooth.

If f is a diffeomorphism, then we may push-forward a vector field defined over the whole of M to obtain a vector field on the whole of N , or pull-back a covector field defined over the whole of N to obtain a covector field defined over the whole of M. Recall that f(M) is diffeomorphic to M for embeddings, so that for embeddings one can relate vector and covector fields on M to ones on f(M).

Consider a coordinate basis, with coordinates x for an open set U in M containing p and coordinates y for an open set V in N containing f(p), as set up in section 3.1. The map f : M → N deﬁnes a coordinate map x → y(x)

70 α ∗ where y(x) is defined by equation (124). A covector w = wαdy ∈ Tf(p)N pulls back to ∂yα f ∗w = w dxµ ∈ T ∗M (150) α ∂xµ p α For a covector field w = wα(y)dy on N , this defines a covector field

∗ ∗ µ f w = (f w)µ(x)dx on M with components ∂yα (f ∗w) (x) = w (y(x)) µ α ∂xµ

These components are smooth functions of x as the components wα(y) are smooth functions of y (as we are assuming w is a smooth covector ﬁeld) and y is a smooth function of x, as f is a smooth map. Then f ∗w has components that are smooth functions of x, so it is a smooth covector ﬁeld on M.

µ ∂ Consider now the push-forward. The push-forward of a vector V = v ∂xµ |p α ∂ at p is a vector f∗V = w ∂yα |f(p) at f(p). with

α α ∂y µ w = µ v (151) ∂x p

α ∂ We can now try to use this to construct a vector field w (y) ∂yα on N from µ ∂ a vector field v (x) ∂xµ on M. This requires regarding both sides of (151) as functions of y. To regard the components vµ(x) as functions of y would require inverting the y(x) to obtain a function x(y). This is not possible for general f. If f is not 1-1, then there is no well-defined inverse, as more than one x can give the same y. If f is 1-1, then for a point q ∈ f(M), there is a point p in M with f(p) = q. Then with f 1-1, a point in V ∩ f(U) with coordinates y will be the image of a unique point in U with coordinates x, so that there is a smooth function x(y) defined on V ∩f(U). Then on V ∩f(U), µ ∂yα v (x) defines a function v(x(y)) and similarly ∂xµ (x) becomes a function of y, so that (151) defines wα(y) as a smooth function of y in V ∩ f(U). Of course, this cannot be extended to points in N outside f(M).

71 3.5 Induced maps and diﬀeomorphisms In general;

0 • We cannot push forward a covector (or Jr,p tensor), nor pull back a q vector (or J0,p tensor). This is because the map f is not invertible in general.

• We can push/pull vectors/covectors at points. We cannot push vector fields unless the map is 1-1. If it is 1-1, it only defines vector fields on the image space.

However, for diﬀeomorphism maps we may do much more, because the inverse map exists and is also smooth. If the map f is a diﬀeomorphism, we can use −1 −1 ∗ (f )∗ to pull back a vector or (f ) to push forward a covector. In this q 1 case any tensor ∈ Jr,p can be pushed or pulled. For example A ∈ J1,p is pushed forward as

∂ ∂yα ∂xν ∂ A = Aµ ⊗ dxν → (f A) = f ⊗ (f −1)∗ A = Aµ ⊗ dyβ ν ∂xµ ∗ ∗ ν ∂xµ ∂yβ ∂yα (152) Furthermore, we can push-forward not just a tensor at a point, but an entire tensor field - simply by pushing forward the tensor at each point p ∈ M. Conversely we may pullback an entire tensor field. If M and N are diffeomorphic, any tensor field on M can be mapped to a tensor field on N , and vice versa.

For an embedding, i.e. a 1-1 map f : M → N , the image f(M) is diffeomorphic to M, so tensor fields on M can be mapped to ones on f(M). If a tensor field on M is smooth, then so is the tensor field on f(M) defined by the induced map.

72 3.6 Flows Consider a vector ﬁeld V . This generates integral curves whose tangents at any point p are given by Vp.

Consider a curve,

C : R → M λ → p(λ) (153)

d The tangent vector at p(λ) is dλ |p. Hence for the curve to be generated by a d vector ﬁeld V we require Vp = dλ |p.

Take a chart containing p with coordinates xµ. Then we may write the curve as xµ(λ) and our vector ﬁeld in the chart as

∂ V = V µ(x) (154) ∂xµ and the tangent to the curve through p(λ) as

µ d dx (λ) ∂ = µ (155) dλ dλ ∂x p(λ)

Then if C is an integral curve of V we have,

dxµ(λ) = V µ(x(λ)) (156) dλ We can use this to ﬁnd the integral curve explicitly. Assume that at λ = 0 µ µ the curve passes through the point p0 with coordinates x(0) = x (p0) so that

73 µ µ µ x (0) = x(0). Then given we know V (x) the above equation provides a ODE to integrate to ﬁnd xµ(λ) and hence C.

Recall that a smooth ODE has a unique solution for some finite interval about λ = 0. If the integral curve through p extends an integral curve for the vector field V that is defined for all λ ∈ R, we say it is complete. A vector field V is complete if, at every point p, the integral curve for V through p is complete. In fact on a compact manifold it is proven that one can extend λ → ±∞, so that the integral curve is complete – there is a theorem saying that every vector field on a compact manifold is complete.

Consider this solution which we denote as σV (λ, p0). In coordinates we write,

µ µ x (λ) = σV (λ, x(0)) (157) so that σV (0, x(0)) = x(0).

Then taking integral curves starting at all points in M we may think of σV for a complete vector ﬁeld V as the map

σV : R × M → M 0 (λ, p) → p = σV (λ, p) (158)

This map gives the flow defined by the vector field V . We require the map σV : R × M → M to be smooth.

Remarks: From the ﬂow equation (156) we see

• For a real c ∈ R, the vector field cV gives the flow σcV (λ) = σV (cλ) i.e. scaling the vector field by a constant doesn’t change the integral curves, only the parameterization of those curves.

• The ﬂow obeys σV (λ + s, p) = σV (λ, σV (s, p)). (For a proof, see Naka- hara.)

74 For a fixed λ, the flow σV gives a diffeomorphism σV (λ): M → M (in- ducing an ‘active’ coordinate transform).

This diﬀeomorphism then has the Abelian group structure of an identity

σV (0) = IdM (159) an inverse −1 (σV (λ)) = σV (−λ) (160) and multiplication σV (λ) · σV (s) = σV (λ + s) (161) This is the group of (active) coordinate transformations generated by V .

75 Consider the active transformation for an inﬁnitesimal λ = , so that p0 = 0 σV (, p), so that the inverse transformation gives p = σV (−, p ). In coordinates, 0µ µ µ µ µ 2 x = σV (, x ) = x + V (x) + O (162) where we have Taylor expanded the solution to (156). This result is familiar from GR.

The inverse transformation gives

µ µ 0 0µ µ 0 2 x = σV (−, x ) = x − V (x ) + O (163)

Note that this implies for any function f,

∂ f(x0) = f(x) + O () = f(x) + V α(x) f(x) + O 2 (164) ∂xα

76 3.7 Lie Derivative of a vector field Given a vector field V , we can differentiate a function f by taking its rate of change at a point along an integral curve of V passing through that point. The tangent vector to the curve is Vp at the point p, and the derivative of f at p is given by 0 df f(p ) − f(p) Vp[f] = ≡ lim (165) →0 dλ p 0 where the integral curve through p is given by σV (λ)p, and hence p = σV ()p.

This derivative of a function wrt the vector ﬁeld V gives a new function which we denote V [f] so

V [f]: M → R p → Vp[f] (166)

Likewise we may differentiate a vector field Y wrt the vector field V to obtain another vector field using the Lie derivative, denoted LV . This is similarly defined as σV (−)∗Y |p0 − Y |p LV [Y ]| ≡ lim ∈ TpM (167) p →0 0 where again p = σV ()p.

77 The crucial addition to the derivative above is pulling-back the vector using the push-forward σV (−)∗. Since Y |p0 ∈ Tp0 M and Y |p ∈ TpM live in different tangent spaces , at p0 and p, they cannot be subtracted to give a vector.

However, by pulling Yp0 back as σV (−)∗Y |p0 ∈ TpM then we can use the linearity of the vector space TpM to take the diﬀerence σV (−)∗Y |p0 and Y |p giving another vector in TpM.

3.7.1 Components of the Lie Derivative of a vector ﬁeld

µ ∂ µ ∂ Let us use coordinates so V = V (x) ∂xµ and Y = Y (x) ∂xµ and xµ(p0) = xµ(p) + V µ(p) + O(2) (168)

Now we may evaluate the Lie derivative explicitly. Take the vector ﬁeld Y 0 evaluated at p = σV ()p, so ! µ 0 ∂ 0 Yp = Y (x(p )) µ (169) ∂x p0

78 −1 and we can now push-forward by σV () = σV (−) to obtain the vector we want at p,

ν ! µ 0 ∂x (p) ∂ 0 σV (−)∗Yp = Y (x(p )) µ 0 ν ∈ TpM. (170) ∂x (p ) ∂x p

Now using the expression for xµ(p0) above, Y µ(x(p0)) can simply be expanded in the inﬁnitesimal as ! µ 0 µ α ∂ µ 2 Y (x(p )) = Y (x(p)) + V (x(p)) α Y (x) + O( ) (171) ∂x x=x(p)

This can be thought of as using (164) for the components Y µ(x0). From (163) we have

xµ(p) = xµ(p0) − V µ(x(p0)) + O(2) (172)

This can also be seen from

xµ(p) = xµ(p0) − V µ(x(p)) + O(2) (173) using V µ(x(p)) = V µ xµ(p0) − V µ(x(p)) + O(2) = V µ(xµ(p0)) + O(2) (174)

We can now use (172) to compute

ν ! ∂x (p) ν ∂ ν 2 µ 0 = δµ − µ V (x) + O( ) ∂x (p ) ∂x x=x(p0) (175)

But using (164), we can move from x(p0) to x(p) up to a term of order : ! ! ∂ ν ∂ ν µ V (x) = µ V (x) + O() (176) ∂x x=x(p0) ∂x x=x(p) so that

ν ! ∂x (p) ν ∂ ν 2 µ 0 = δµ − µ V (x) + O( ) (177) ∂x (p ) ∂x x=x(p)

79 Then we ﬁnd µ ν ν α ∂ µ µ ∂ ν 2 ∂ 0 σV (−)∗Yp = Y (x)δµ + δµV (x) α Y (x) − Y (x) µ V (x) + O( ) ν ∂x ∂x ∂x x=x(p) ν µ ∂ ν µ ∂ ν 2 ∂ = Y (x) + V (x) µ Y (x) − Y (x) µ V (x) + O( ) ν (178). ∂x ∂x ∂x x=x(p) where we see everything in the component of the vector is computed at p. We may now subtract Yp from this, divide by , and take the vanishing epsilon limit to give the Lie derivative in components at the point p, µ ∂ ν µ ∂ ν ∂ LV Y |p = V (x) µ Y (x) − Y (x) µ V (x) ν . (179) ∂x ∂x ∂x x=x(p)

80 3.8 Lie Bracket of two vector fields The Lie Bracket [·, ·] is a map taking two vector fields into another vector field,

1 1 1 [·, ·]: J0 × J0 → J0 X,Y → [X,Y ] (180) defined by requiring that for all functions g ∈ F(M), the vector field [X,Y ] satisfies, [X,Y ][g] ≡ X[Y [g]] − Y [X[g]]. (181) µ ∂ µ ∂ In components, so X = X (x) ∂xµ , Y = Y (x) ∂xµ one can easily see that, ∂ ∂ ∂ [X,Y ] = Xα(x) Y µ(x) − Y α(x) Xµ(x) . (182) ∂xα ∂xα ∂xµ

Then we see that the Lie derivative is simply expressed in terms of the bracket as LV Y = [V,Y ]. (183) The Lie bracket is bilinear. It is also skew-symmetric as

[X,Y ] = −[Y,X] (184) and satisﬁes the Jacobi identity,

[[X,Y ],Z] + [[Y,Z],X] + [[Z,X],Y ] = 0. (185)

81 Something we will require later is how the Lie bracket transforms under a diffeomorphism. Consider a diffeomorphism f ∈ Diff(G). Then we may show that for two vector fields X,Y and their push forwards f∗X, f∗Y the following holds:

f∗ ([X,Y ]|p) = [f∗X, f∗Y ] |f(p) (186)

Proof: Recall, for a point p the vector f∗X is deﬁned by

f∗X[g] = X[g · f] (187) f(p) p for any function g on N . As

f∗X[g] = (f∗X[g] · f) (188) f(p) p we have that, for a map f : M → N and any function g ∈ F(N ),

(f∗X[g]) · f = X[g · f] (189) where both sides are viewed as functions on M. Then

X [Y [g · f]] = X [(f∗Y )[g] · f] = {(f∗X) [(f∗Y )[g]]}· f (190) and hence

f∗([X,Y ])[g] · f = [X,Y ][g · f] = X [Y [g · f]] − Y [X[g · f]]

= {(f∗X) [(f∗Y )[g]]}· f − {(f∗Y ) [(f∗X)[g]]}· f

= {[f∗X, f∗Y ][g]}· f (191)

This then implies

f∗ ([X,Y ]) = [f∗X, f∗Y ] (192)

82 3.9 Commuting flows For 2 vector fields X,Y , the Lie bracket [X,Y ] (or Lie derivative) measures the closure of the flows generated by X and Y . If [X,Y ] 6= 0 then starting at a point p and moving along one flow by λ then the other by parameter gives a different point to taking the other order of flows, i.e.

σX (λ) · σY ()p 6= σY () · σX (λ)p (193)

2 vector ﬁelds are said to be ‘commuting’ if [X,Y ] = 0, and then the order by which one ﬂow doesn’t matter.

Theorem: The two diﬀeomorphisms σX (λ) · σY () and σY () · σX (λ) are equal iﬀ [X,Y ] = 0. See example sheet 3.

An example of commuting vector ﬁelds are those given by a coordinate basis ∂ { ∂xµ }.

83 3.10 Lie Derivative of a tensor field Notation; let us define the Lie derivative of a function as the usual derivative so LV f ≡ V [f] (194) We have just defined the Lie derivative on a vector field. The Lie derivative q may be extended to allow derivation of any (q, r) tensor field A ∈ Jr by a vector field V as ∗ σ(+) A|p0 − A|p q LV [A]| ≡ lim ∈ Jr,p (195) p →0

q Recall that since σ() is a diffeomorphism we may pull-back any Jr,p0 tensor q to Jr,p. Now the Lie derivative wrt a vector field V defines a map,

q q LV : Jr → Jr

A → LV A (196)

Exercise: using coordinates xµ, show that the Lie derivative of a covector µ ﬁeld w = wµ(x)dx is given explicitly in components as

∂ ∂ L A = V α w + w V α dxµ (197) V ∂xα µ α ∂xµ

Show that the same result for LV A can be obtained by demanding that

LV < A, Y >=< LV A, Y > + < A, LV Y > (198) for any vector ﬁeld Y and covector ﬁeld A.

Check that the Lie derivative as deﬁned obeys the ‘Leibnitz rules’,

LV (fA) = V [f]A + fLV A

LV (A1 ⊗ A2) = (LV A1) ⊗ A2 + A1 ⊗ (LV A2) (199) for a function f and tensor ﬁelds A, A1,A2. This can be used to derive the form of the Lie derivative of any (q, r) tensor in a coordinate basis, using the results for vectors and covectors.

84 3.11 Example: Active coordinate transformations in GR and symmetry

0 Given a vector field V , the flow σV () takes a point p to a point p = σV (, p). 0 If p and p are in the same coordinate patch Ui, then the coordinates x = ψ(p) will transform to the coordinates of the new point p0, x0 = ψ(p0). For infinitesimal , the diffeomorphism σV () gives the infinitesimal ‘active’ coordinate transformation xµ → x0µ = xµ + V µ + O(2) (200) Such transformations play an important part in GR, particularly in per- turbation theory. These transformations are diffeomorphisms, which induce corresponding coordinate transformations.

Such active coordinate transformations are to be distinguished from passive coordinate transformations. In an active transformation, the point p is trans- formed to a different point p0, with corresponding changes of coordinates. For a passive transformation, the point remains unchanged, but a different coordinate system is chosen. Consider a coordinate patch U, and two different charts ψ : U → Rm and ψ0 : U → Rm. One leads to coordinates x = ψ(p) and the other to coordinates x0 = ψ0(p). The two coordinates are related by the smooth function x0(x) = ψ0 · ψ−1(x). Transforming from coordinates x to coordinates x0 is a passive coordinate transformation – the point p remains unchanged, but one changes from coordinates x for U to coordinates x0.

Given a tensor field T defined on M, the Lie derivative tells us how the components of the tensor transform under a diffeomorphism. If T has components ν1...νr µ 0ν1...νr 0µ Tµ1...µp in the x coordinates and T µ1...µp in the x coordinates (related by an active coordinate transformation as above) then the difference in these components ν1...νr 0ν1...νr ν1...νr δTµ1...µp = T µ1...µp − Tµ1...µp (201) defines another tensor δT , where one finds;

2 δT = − LV T + O( ) (202)

Hence the Lie derivative can be used to compute the change in components of a tensor at a point, under an inﬁnitesimal ‘active’ coordinate transform.

85 If LV T = 0 then the diffeomorphism σV is said to be a symmetry transformation of the tensor T . We see from above that the active coordinate transform generated by V leaves the components of the tensor unchanged. For the metric tensor g, the diffeomorphisms that are symmetries of g are isometries and the vector fields satisfying LV g = 0, so that they generate isometries, are Killing vectors.

86 4 Diﬀerential forms

These objects generalize the notion of diﬀerentials found in calculus.

A form of order 1, or a one-form, is a cotangent vector. A covector at p 0 ∗ µ V ∈ J1,p = Tp M is, in a coordinate basis, V = Vµdx .

0 A diﬀerential form of order r is a totally antisymmetric tensor in Jr,p. Take 1 r vectors V1,...,Vr ∈ J0,p. Then a form, T , satisﬁes,

T (V1,...,Vi+1,Vi,...,Vr) = −T (V1,...,Vi,Vi+1,...Vr) (203) for any i. i.e. it is antisymmetric under any neighbouring vector pair interchange.

For a permutation of r objects, P : (1, 2, . . . , r) → (P1,P2,...Pr), then the sign of the permutation, sign(P ), is +1 for an even permutation (i.e. requires an even number of neighbour pair interchanges) or −1 for an odd permutation.

Following from the antisymmetry of T above, for any permutation, P , of the vector arguments of T we see,

T (V1,V2,...Vr) = sign(P ) T (VP1 ,VP2 ,...,VPr ) (204)

87 4.1 Cartan wedge product Deﬁne the Cartan Wedge product ,∧, by

µ µ µ X µ µ µ dx 1 ∧ dx 2 ∧ ... ∧ dx r ≡ sign(P ) dx P1 ⊗ dx P2 ⊗ ... ⊗ dx Pr (205) P where the sum is over all possible permutations P : 1, 2, . . . , r → P1,P2,...,Pr. For example, this gives,

dxµ1 ∧ dxµ2 = dxµ1 ⊗ dxµ2 − dxµ2 ⊗ dxµ1 (206)

From the deﬁnition of ∧ we see this is antisymmetric under interchange of neighbouring indices. Hence we see that,

µ µ µ µ µ µ dx 1 ∧ dx 2 ∧ ... ∧ dx r = sign(P )dx P1 ∧ dx P2 ∧ ... ∧ dx Pr (207) and therefore this product forms a coordinate basis for the antisymmetric tensor r-forms. Thus we may write an r-form w in a coordinate basis, 1 w = w dxµ1 ∧ dxµ2 ∧ ... ∧ dxµr (208) r! µ1µ2...µr where the components of the antisymmetric tensor w are given by ∂ ∂ ∂ wµ µ ...µr = w( , ,..., ) (209) 1 2 ∂xµ1 ∂xµ2 ∂xµr and are antisymmetric; w = sign(P )w . Note the conven- µ1µ2...µr µP1 µP2 ...µPr tional 1/r! above is designed to make the equation above for the components simple.

Example:

µ ν Take a general (0, 2) tensor, say A = Aµνdx ⊗ dx . We may decompose this into its symmetric and antisymmetric parts as

µ ν µ ν A = A(µν)dx ⊗ dx + A[µν]dx ⊗ dx (210)

1 1 0 where A(µν) = 2 (Aµν + Aνµ) and A[µν] = 2 (Aµν − Aνµ). A 2-form, A , is deﬁned by the antisymmetric part, 1 A0 = A dxµ ⊗ dxν = A dxµ ∧ dxν (211) [µν] 2 [µν] 88 µ1 µr 0 In general a (0, r) tensor T = Tµ1,...,µr dx ⊗ ... ⊗ dx deﬁnes an r-form T by 1 T 0 = T dxµ1 ⊗ ... ⊗ dxµr = T dxµ1 ∧ ... ∧ dxµr (212) [µ1,...,µr] r! [µ1,...,µr] where our basis projects onto only the antisymmetric part of Tµ1,...,µr .

89 Comments

Let M be an m-dimensional manifold.

r 0 • Denote r-forms at p as Ωp(M). These inherit the linearity of Jr,p and hence again form a vector space.

1 ∗ • Ωp(M) = Tp M. Hence covectors are one-forms. • We cannot have an r-form with r > m due to their antisymmetry. r Hence Ωp(M) vanishes for r > m.

• The dimension of the vector space T ∗p is m. The dimension of the ∗ ∗ vector space formed by the (0, r) tensors at p given by Tp ⊗ ...Tp is mr i.e. a (0, r) tensor in general has mr components. The dimension r of the vector space Ωp(M) is given by the number of components of an r-form, m! dim Ωr(M) = (213) p (m − r)! r! the no. ways of choosing r from m.

So we have; dim Ω1 = m, basis dxµ 2 1 µ ν dim Ω = 2 m(m − 1), basis dx ∧ dx , components are antisymmetric matrices . . dim Ωm = 1, basis dx1 ∧ dx2 ∧ ... ∧ dxm

We note that dim Ωr = dim Ωm−r for r > 0.

0 We can include the case r = 0 above by simply deﬁning Ωp = R. This will be convenient later.

90 4.2 Exterior product We may use our wedge product on the basis covectors to construct the ‘exterior product’;

r1 r2 r1+r2 ∧ :Ωp (M) × Ωp (M) → Ωp (M) ω, ξ → ω ∧ ξ (214)

1 α1 αr 1 α1 αr where for ω = ωα ...α dx ∧...∧dx 1 and ξ = ξα ...α dx ∧...∧dx 2 r1! 1 r1 r2! 1 r2 we deﬁne the product as

1 α α β β ω ∧ ξ ≡ ω ξ dx 1 ∧ ... ∧ dx r1 ∧ dx 1 ∧ ... ∧ dx r2 (215) α1...αr1 β1...βr2 r1!r2! Note that this is similar to the tensor product, but it automatically antisym- metrises the resulting tensor to give the r1 + r2-form.

If r1 + r2 > dim M then the wedge product vanishes.

The wedge product is obviously bilinear. Take ξ ∈ Ωq, η ∈ Ωr and ω ∈ Ωs. Some other properties of the product are

• ‘Graded commutativity’ ξ ∧ η = (−1)qrη ∧ ξ, eg. dx ∧ dy = −dy ∧ dx

• This implies ξ ∧ ξ = 0 if q is odd. e.g. dx ∧ dx = 0.

• Associativity (ξ ∧ η) ∧ ω = ξ ∧ (η ∧ ω).

91 4.3 Differential Form Fields An r-form field smoothly assigns an r-form to every point. It is a smooth 0 tensor field in Jr whose components are completely antisymmetric. Denote the set of r-form fields as Ωr(M).

0 Recall earlier we defined Ωp = R. Compatible with this we may define a 0-form field to be a function, i.e. Ω0(M) = F(M).

92 4.4 Exterior derivative Perhaps the most important feature of forms is that there is a very natural notion of diﬀerentiation of a form ﬁeld - the exterior derivative denoted d:

d :Ωr(M) → Ωr+1(M) ω → dω (216)

1 α1 αr where for ω = r! ωα1...αr dx ∧ ... ∧ dx we deﬁne, 1 ∂ dω = ω dxµ ∧ dxα1 ∧ ... ∧ dxαr (217) r! ∂xµ α1...αr

(Note that the resulting tensor is indeed antisymmetric.)

This is an interesting derivative. Previously when we diﬀerentiated a function we required a vector i.e. a directional derivative. Here we have a derivative operator that has no ‘direction’ built in. d enjoys a graded Liebnitz rule (which you should check!). For ξ ∈ Ωq and ω ∈ Ωr, d(ξ ∧ ω) = (dξ) ∧ ω + (−1)qξ ∧ (dω) (218) A very important property of the exterior derivative is that it is nilpotent, meaning, d2 = 0 (219) which follows simply from symmetry,

1 ∂2 d2ω = ω (dxν ∧ dxµ) ∧ dxα1 ∧ ... ∧ dxαr = 0 (220) r! ∂xν∂xµ α1...αr as the partial derivative is symmetric under µ ↔ ν.

Note that since Ωr vanishes for r > m, then for η ∈ Ωm, dη = 0.

Note we have included the case r = 0; d :Ω0 → Ω1. Then f ∈ F(M) = Ω0 ∂f(x) µ gives df = ∂xµ dx .

93 4.5 Pullback of forms Suppose f : M → N . Since forms are (0, r) tensors they can be pulled back using f ∗ in the usual way. Additionally one can show

f ∗ (ξ ∧ ω) = (f ∗ξ) ∧ (f ∗ω) d (f ∗ω) = f ∗ (dω) (221)

4.6 3d vector calculus See the example sheet

A beautiful example of the power of forms is that they generalize 3d vector calculus. Recall for a function f(x) we may construct a vector field using grad, ∇f. For a vector field v(x), we may take the curl to give another vector field ∇ × v, or take the divergence to get a function ∇ · v.

Take M = R3; • d of a function f ∈ Ω0, df gives a one-form with components that are those of the gradient of f.

• A one-form has 3 component, and hence can be associated to a 3-vector ﬁeld. d of a one-form gives a 2 form, again with 3 components, where those components give the curl of the vector ﬁeld.

• A two-form is associated to a vector ﬁeld. d of it gives a 3-form with one component which is given by the divergence of the vector ﬁeld.

4.7 Coordinate free deﬁnition of the exterior derivative For ω ∈ Ωr(M) we may deﬁne the exterior derivative without reference to coordinates as

r+1 X i+1 (dω)(X1,...,Xr+1) ≡ (−1) Xi [ω(X1,...,Xi−1,Xi+1,...,Xr+1)] i=1 X i+j + (−1) ω([Xi,Xj],X1,...,Xi−1,Xi+1,...,Xj−1,Xj+1,...,Xr+1)(222) i

94 for any vector ﬁelds Xi.

For example, take ω ∈ Ω1, and vector ﬁelds X,Y . Then

dω(X,Y ) = X [ω(Y )] − Y [ω(X)] − ω ([X,Y ]) (223)

Exercise: show that the components this gives for dω in a coordinate basis (with ω ∈ Ω1) are the same as those given earlier.

95 4.8 Interior product The exterior product and derivative increase the degree of a form. A vector ﬁeld X provides a natural map reducing the degree of a form. This is denoted iX ,

r r−1 iX :Ω → Ω

ω → iX ω (224) and is deﬁned by

(iX ω)(V1,...Vr−1) ≡ ω (X,V1,...,Vr−1) (225)

1 µ1 µr Using coordinates we have, ω = r! ωµ1,...,µr dx ∧ ... ∧ dx , and then 1 i ω = Xνω dxµ1 ∧ ... ∧ dxµr−1 (226) X (r − 1)! νµ1...µr−1

2 It follows from the deﬁnition that iX is nilpotent, so that iX = 0.

4.9 Lie derivative of forms The Lie derivative of a form can conveniently be written using the interior and exterior products as

LX ω = (d iX + iX d) ω (227)

Exercise: show that the components this gives for LX ω in a coordinate basis (with ω ∈ Ω1) are the same as those given earlier.

96 4.10 Closed and Exact forms • An r-form ω ∈ Ωr is closed if dω = 0.

• An r-form ω ∈ Ωr is exact if ω = dσ where σ ∈ Ωr−1 is some r − 1 form.

Since d2 = 0, we immediately see that any exact form is also closed.

We are aware of the importance of exact diﬀerentials in physics - for example µ 1 in thermodynamics - and given a 1-form ﬁeld, ω = ωµdx ∈ Ω , an important ∂f(x) µ question is whether it is exact, i.e. can it be written as ω = df = ∂xµ dx for a function f?

We now see that a necessary condition is that dω = 0. However, we also see that this is not suﬃcient, since ω might be closed, but not exact.

A natural question then arises which is the starting point for ‘cohomology’ which we will discuss later; which forms are closed but not exact?

97 4.11 Physical application: Electromagnetism Recall the Gauss law ∇ · B = 0 and Faraday law ∇ × E + B˙ = 0. Using special relativity, electromagnetism is formulated in terms of an antisymmetric field strength tensor Fµν. Then these two laws are neatly packaged into the ‘Bianchi’ identity, σρµν ∂ρFµν = 0 (228) µ ν Since the field strength tensor F = Fµνdx ⊗ dx is antisymmetric, i.e. F (X,Y ) = −F (Y,X) for any vectors X,Y , then F is a 2-form, 1 F = F dxµ ⊗ dxν = F dxµ ∧ dxν (229) µν 2 µν Thinking of F as a 2-form we find that the Bianchi identity is precisely the statement that, dF = 0 (230) i.e. F is a closed form!

So using diﬀerential forms we have reduced 2 of the 4 Maxwell equations to a simple statement that the ﬁeld strength is a closed 2 form.

Note that we often formulate the relativistic theory in terms of a 4-vector potential Aµ so that Fµν = ∂µAν − ∂νAµ. In fact this is nothing other than µ the statement F = dA where A = Aµdx . If this were true, F would in fact be an exact 2-form. It is a very important point that Maxwell’s equations only imply F is closed. As we will see later locally (i.e. in a contractible coordinate patch) any closed form is exact. However this is not true globally (i.e. over the whole manifold). Hence in a chart we may write F = dA but not necessarily globally. We will return to this example a number of times ...

98 4.12 Orientation

∂ ∂ ∂ Consider a chart with coordinate basis vectors { ∂x1 , ∂x2 ,..., ∂xm }, where we are now interested in the order of the list of these basis vectors. Locally this order deﬁnes an orientation (analogous to left or right handed bases in R3).

Taking a permutation P : {1, 2, . . . , m} → {i1, i2, . . . , im} then the ordered ∂ ∂ ∂ set of basis vectors, { ∂xi1 , ∂xi2 ,..., ∂xim } deﬁnes the same/opposite orientation if P is even/odd.

∂ ∂ Consider an overlap between two charts with coordinate bases; { ∂x1 ,..., ∂xm } ∂ ∂ and { ∂y1 ,..., ∂ym }. The Jacobian matrix induced by changing coordinates locally from x → y, is given by ∂yµ(x)/∂xν. The determinant of this matrix tells us how the inﬁnitesimal volume element transforms. However, the sign of the determinant tells us how the orientations of the bases transform. A positive sign indicates x and y are oriented similarly. A negative sign means they are oppositely oriented.

A manifold is orientable if there exists an Atlas with charts (Ui, ψi) and µ coordinates x(i)(p) = ψi(p) such that on all overlaps Ui ∩ Uj 6= 0, then

µ ν Jij = det ∂x(i)/∂x(j) > 0 ∀ p ∈ Ui ∩ Uj. (231)

Then this Altas defines an orientation consistently over the whole manifold. The positive definite condition on all the Jacobians over the overlaps ensures that the orientation defined by one chart is consistent with that defined by its neighbours. For a non-orientable manifold, such an Altas cannot be found.

99 4.13 Top forms and volume forms Take an m-form, V on an m-dimensional manifold M. Since it has the high- est rank possible it is called a ‘top form’.

A top form has only one independent component, and in some coordinates we may write it as 1 V = v(x)dx1 ∧ dx2 ∧ ... ∧ dxm = v(x) dxµ1 ∧ ... ∧ dxµm (232) r! µ1µ2...µm where the ‘epsilon symbol’ µ1...µm = sign(P ) for permutation P : (1, 2, . . . , m) → (µ1, µ2, . . . , µm). Consider how the component of V transforms as we change coordinates from a chart Ui to chart Uj: ∂x1 ! ∂xm ! (i) µ1 (i) µm V = v(i)(x(i)) µ1 dx(j) ∧ ... ∧ µm dx(j) ∂x(j) ∂x(j) ∂x1 ∂xm ! (i) (i) 1 m = v(i)(x(i)) µ1...µm µ1 ... µm dx(j) ∧ ... ∧ dx(j) ∂x(j) ∂x(j) α β 1 m = v(i)(x(i)) det ∂x(i)/∂x(j) dx(j) ∧ ... ∧ dx(j) 1 m = v(j)(x(j)) dx(j) ∧ ... ∧ dx(j) (233)

α β We see that the Jacobian Jij = det ∂x(i)/∂x(j) of the coordinate transformation x(i) → x(j) naturally emerges.

Take an orientable manifold M, with an Altas such that Jij > 0. A volume form, V , is a top form on M, such that for any point p ∈ M, when we write V in the coordinate basis of a chart containing p, say p ∈ Ui, then 1 m V = v(x(i))dx(i) ∧ ... ∧ dx(i) with v(x(i)) > 0 (234) Remarks: 1. From the above discussion we note that one may only ﬁnd volume forms on manifolds that are orientable. 2. A top form is generally not a volume form. 3. On an orientable manifold (inﬁnitely) many volume forms can be found. 4. A volume element vanishes nowhere.

100 A volume form, V , defines an orientation on M. If we are interested in the orientation in some open set U, we simply find a chart (U, ψ) such that the coordinates x = ψ(p) are such that the component of V is positive definite over U. This then smoothly defines an orientation in the tangent space at each point in that chart. We may then continue to extend this to other charts. Eventually one will build up an Altas with Jij > 0, oriented in the sense of V .

Note that if V is a volume form, then −V is also a volume form but with the opposite orientation. For an Altas with Jij > 0 oriented in the sense of V , then one can ﬁnd an Altas oriented with V 0 = −V simply by taking the same charts, but performing an odd permutation of the coordinate labels i.e. 1 m 01 0m 0P1 0Pm {x(i), . . . , x(i)} → {x(i), . . . , x(i)} = {x(i) , . . . , x(i) } such that signP = −1.

Example: dx1 ∧ dx2 ∧ ... ∧ dxm deﬁnes the usual orientation on Rm.

101 4.14 Integrating a top form over a chart Consider an m-dimensional orientable manifold with a top form V ∈ Ωm, α β and choose an oriented Atlas so that Jij = det ∂x(i)/∂x(j) > 0. Then we deﬁne the integral V over a chart Ui (with coordinates x(i)) to be, Z Z m V ≡ d x(i) v(i)(x(i)) (235) 0 Ui Ui

1 m where V = v(i)(x(i))dx ∧ ... ∧ dx . The RHS of this is simply the usual multidimensional integration on Rm from Calculus.

We have used a particular chart to make this deﬁnition, and we should check that the deﬁnition is independent of this choice of chart. Consider the integration over an overlap region between two charts R = Ui ∩ Uj. Then we must show that Z Z m V = d x(i)v(i)(x(i)) R ψi(R) Z m = d x(j)v(j)(x(j)) (236) ψj (R) Now as discussed above, an m-form transforms as

1 m V = v(i)(x(i))dx(i) ∧ ... ∧ dx(i) α β 1 m = v(i)(x(i)) det ∂x(i)/∂x(j) dx(j) ∧ ... ∧ dx(j) 1 m = v(j)(x(j)) dx(j) ∧ ... ∧ dx(j) (237) Hence we see that Z Z m V = d x(j)v(j)(x(j)) R ψj (R) Z m α β = d x(j)v(i)(x(i)) det ∂x(i)/∂x(j) (238) ψj (R) and then using the standard transformation of the integration measure in Calculus, so that, Z Z dmx = dmx det ∂xα /∂xβ (239) (j) (i) (j) (i) ψj (R) ψi(R)

102 and then Z Z V = dmx det ∂xα /∂xβ v (x ) det ∂xα /∂xβ (i) (j) (i) (i) (i) (i) (j) R ψi(R) Z Z m m = d x(i)(x(i)))v(i)(x(i)) = d x(i)v(i)(x(i)) ψi(R) ψi(R)

−1 since det ∂xα /∂xβ = det ∂xα /∂xβ and det ∂xα /∂xβ = (i) (j) (j) (i) (j) (i) α β det ∂x(j)/∂x(i) since we have chosen our Altas so Jij > 0.

Let us now consider the top form V to be a volume form, rather than just an m-form so that its component is positive definite, and hence the volume form is nowhere vanishing. This then defines a positive definite integration R measure so that fV > 0 for any positive definite function f ∈ F(Ui) so Ui that f(p) > 0 for all p ∈ Ui.

103 4.15 Integrating a top form over an orientable manifold We may integrate a top form over an orientable manifold by using a ‘Parti- tion of unity’ on M. Let us take an oriented altas {(Ui, ψi)} on M so that any point p is found in a ﬁnite number of the Ui’s.

[This is always possible for a compact manifold, and technically the suﬃcient condition is ‘paracompactness’]

Take smooth functions i(p) on M such that;

• 0 ≤ i(p) ≤ 1

• i(p) = 0 if p∈ / Ui. P • i i(p) = 1 for all p ∈ M

For example, any smooth function f ∈ F(M) can be decomposed as X X f(p) = i(p)f(p) = fi(p) (240) i i where fi(p) = i(p)f(p) are new smooth functions that are only non-zero in the open set Ui.

Using this partition of unity we can integrate a top form V on M, by taking,

Z X Z V ≡ iV (241) M i Ui where we already know how to perform the integral of the top form iV over a chart Ui.

104 4.16 Integration of r-forms over oriented r-dimensional submanifolds Suppose we have an m-dimensional manifold M, and a one-to-one map f deﬁning an embedding of the r-dimensional manifold N into M, where r ≤ m. Thus,

f : N → M (242) and the image f(N ) is a submanifold of M (which is diﬀeomorphic to M).

Now suppose that N is orientable. Then a volume form on N may be pushed forward to give a volume form on the image submanifold f(M). Hence, given an orientation on N , the map f induces an orientation on the submanifold f(M).

Given such an n-dimensional oriented submanifold of M, we may naturally integrate an r-form ω ∈ Ωr(M) over it using the fact that we may always pull back a an r-form; Z Z ω ≡ (f ∗ω) (243) f(N ) N

Comment: Note that if the manifold N has a boundary ∂N , then that boundary is embedded in M as the (r −1)-dimensional submaniﬂold f(∂N ). Then one can integrate an (r − 1)-form over this boundary.

105 5 Stokes’ theorem and cohomology

5.1 Cycles and boundaries We will refer to a closed oriented submanifold V of dimension r as an r- submanifold.

[ Recall: a closed submanifold is one whose complement is open. ]

The boundary of V is denoted ∂V , and provided the embedding of the submanifold is smooth, then the boundary is of dimension (r − 1). Note that if V is oriented, then ∂V is also oriented. Hence ∂V is an (r − 1)-submanifold or disconnected sum of these.

An r-submanifold is

• an r-cycle if it is closed and has no boundary ie, ∂V = 0

• an r-boundary if it is closed and is the boundary of some other closed (r + 1)-submanifold i.e. V = ∂W with W a closed (r + 1)-submanifold.

A very important fact from topology is that a boundary has no boundary! This is obviously true for elementary submanifolds with boundary, such as an r-simplex in Rm, and more complicated submanifolds can be build out of many of these building blocks.

Hence if V is an r-boundary then ∂V = 0. We can think of ∂ as being an operator that is nilpotent. This is the basis for homology theory in topology.

106 5.2 Stokes’ theorem For ω ∈ Ωr−1(M) and r-submanifold V , then Stokes’ theorem states that, Z Z dω = ω (244) V ∂V where we emphasize that V is an r-submanifold (i.e. a closed oriented submanifold).

This is quite a remarkably elegant result.

The righthand side vanishes if ∂V = 0.

If ∂V is a disconnected sum of (r − 1)-submanifolds then the righthand side must be summed over each component (r − 1)-submanifold.

The proof is elementary but long winded - see Nakahara for details. One essentially must prove this is true for the ‘standard simplex’ and then extend the result to general simplices, and hence general submanifolds. A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle with its interior, a 3-simplex is a solid tetrahedron (a tetrahedron with its interior) and an r-simplex is an r-dimensional generalisation of the tetrahedron. The boundary of an r-simplex is made of r − 1 simplices, each of which has a boundary made of r − 2 simplices, and so on. In simplicial homology, a too- logical space is decomposed into simplices glued together along their ‘faces’.

An important property of Stokes’ theorem is ‘integration by parts’. Let m = dim(M). Then take ω ∈ Ωr−1, η ∈ Ωm−r. Then Z Z Z Z ω ∧ η = d (ω ∧ η) = dω ∧ η + (−1)r−1 ω ∧ dη (245) ∂M M M M

107 5.3 Example. Gauss’s and Stokes’ law Recall that grad, div and curl of 3-d vector calculus arose naturally considering forms in R3. In 3-d vector calculus we also have the Gauss law, Z Z ∇ · v d3x = v · dS (246) V S where S is the boundary of V and dS is the directed surface area element, and the Stoke’s law, Z Z (∇ × v) · dS = v · dl (247) S C for a surface S with bounding curve C, where again dS is the directed surface area element and dS is the directed line element.

Take M = R3. It is a simple exercise to show that Gauss’s law arises when we consider Stoke’s theorem for a 2-form ω, and a 3-simplex given by the geometric volume V , so that ∂V = S. Likewise Stoke’s law arises when we consider Stoke’s theorem for a 1-form ω, and a 2-simplex which is the geometric surface S, and so ∂S = C. These are the two instances of Stoke’s law for R3.

108 5.4 de Rham cohomology Recall for ω ∈ Ωr, ω is closed if dω = 0 and exact if ω = da. Then we deﬁne

• the space of closed r-forms forms, the vector space, Zr(M)

• the space of exact r-forms forms, the vector space, Br(M)

As we saw earlier the exterior derivative is nilpotent - hence d2ω = 0 for ω ∈ Ωr. Thus, Br ⊂ Zr. i.e. if ω ∈ Br then ω ∈ Zr.

r Two closed r-forms z1, z2 ∈ Z (M) are equivalent - ‘cohomologous’ - if

r z1 − z2 ∈ B (M) (248) i.e. the closed forms diﬀer by an exact form. We write the equivalence z1 ∼ z2.

We denote the set of equivalent closed r-forms - ‘equivalence classes’ - by [z] where z is a representative closed r-form in the class. Hence

[z] = {z0 ∈ Zr(M) where z0 ∼ z} (249) is the equivalence class.

The linear structure of Zr,Br is simply inherited from Ωr. However the set of equivalence classes of closed r-forms, {[z]}, also forms a vector space Hr(M) - the cohomology vector space. It inherits its linear structure from Zr(M) and the d operator.

r Take z1,2, w1,2 ∈ Z (M) so that z1 ∼ w1 and z2 ∼ w2. Hence we can write w1,2 = z1,2 + dc1,2 with c1,2 ∈ Ωr−1(M). Then for α, β ∈ R we have,

αw1 + βw2 = αz1 + βz2 + αdc1 + βdc2

= αz1 + βz2 + dc (250) where c = αc1 + βc2 ∈ Ωr−1(M) and so we see αw1 + βw2 ∼ αz1 + βz2.

Thus we ﬁnd the linear structure,

α[z1] + β[z2] = [αz1 + βz2] α, β ∈ R (251)

109 where we have shown above that this relation is independent of the choice of representative. i.e. for w1,2 ∼ z1,2 then

α[w1] + β[w2] = α[z1] + β[z2] = [αz1 + βz2] = [αw1 + βw2] (252)

While the space of closed and exact forms Zr and Br are infinite dimensional vector spaces, the cohomology vector space may be finite dimensional. i.e. there are only a finite number of inequivalent closed r-forms on a manifold.

We may provide a basis for the vector space Hr(M) by giving a set of inequivalent closed r-forms that give representatives for classes spanning Hr. Let us call these inequivalent closed r-forms {ωi}, i = 1,..., dim(Hr). Then the basis for Hr(M) is {[ωi]}.

For an m-dimensional manifold, Zr(M), Br(M) and Hr(M) are only non- trivial for 0 ≤ r ≤ m.

Remark: For any (connected) M, H0(M) = R.

Proof: For r = 0 we have Ω0(M) = F(M). For f ∈ F(M) we have, in some ∂f(x) µ ∂f(x) chart, df = ∂xµ dx and so if df = 0 then ∂xµ = 0 in all charts over M and hence f is constant.

Thus the space of closed 0-forms, Z0 = R, i.e. the coeﬃcient of the constant function. Since for 0-forms B0(M) = 0, then H0 = Z0 = R, with the representative for a basis being any constant function.

110 5.5 Example: Cohomology of R Consider H1(R). Any 1-form is closed as for any ω ∈ Ω1(R) then dω = 0.

However every 1-form is also exact. Take coordinate x. Then we can write,

Z x ω = f(x)dx = d f(x0)dx0 = dF (253) 0

R x 0 0 0 where F (x) = 0 f(x )dx ∈ Ω (R).

Hence any closed 1-form is equivalent to zero, i.e. for any ω ∈ Z1 then [ω] = 0. Therefore H1(R) = 0.

5.6 Example: Cohomology of S1

Let us now reconsider the above situation for an S1. We can embed S1 in C as S1 = {eiθ ∀ θ ∈ [0, 2π)}.

As for R, every 1-form is closed. However, now not every 1-form is exact. As above we can use our coordinate θ in the chart θ ∈ (0, 2π) to write a 1-form ω ∈ Ω1(S1) as

Z θ ω = f(θ)dθ = d f(θ0)dθ0 = dF (θ) (254) 0 Now the question is whether F (θ) deﬁned locally over the chart θ ∈ (0, 2π) can be extended to a smooth function globally on the S1. If it can, then ω is exact. Otherwise it is only closed. Clearly F (θ) can be extended to a smooth function only if F (0) = F (2π). Note that F (0) = 0, hence our condition is that Z Z 2π if ω = f(θ)dθ = 0 then ω ∈ Br(S1) (255) S1 0 Take two forms ω, ω0 ∈ Z1(S1) such that R ω 6= 0 i.e. ω∈ / B1(S1). Then R 0 R 0 1 0 0 take a = ( ω )/( ω) ∈ R. Now ω − a ω ∈ B . Hence [ω ] = a[ω] for any ω . Hence H1(S1) = R and ω gives a representative for the basis.

111 For example ω = dθ in the chart θ = (0, 2π), which is a volume form on S1, gives a representative. Choosing an atlas for S1, one can see that ω extends to a well-deﬁned 1-form on S1, but there is no globally deﬁned function f on the circle with ω = df.

112 5.7 Cohomology and Topology A very important result is that the cohomology groups of a manifold depend only on its topology. In particular, the dimension of the vector space Hr(M) is a topological invariant, known as the r’th Betti number br(M). For an m-dimensional manifold, br = 0 for r > m and the Euler number or Euler characteristic is deﬁned by

m χ = b0 − b1 + b2 − b3 + ... (−1) bm (256) de Rham’s theorem: If M is compact without boundary then Hr(M) is ﬁnite dimensional. Furthermore Hr(M) is dual to the r’th homology group Hr(M).

This means that, as the name suggests, homology is dual to cohomology. Homology is outside the scope of this course; see Nakahara for an exposition. Roughly speaking, homology is based on the idea of considering closed cycles modulo cycles that are boundaries.

De Rham’s theorem is a profound statement as it links topology and properties of diﬀerential forms. The cohomology groups are often straightforward to calculate and so can be a useful way of ﬁnding information about topology.

5.8 Stoke’s Theorem and Cohomology Consider an r-submanifold V and an r-form ω ∈ Ωr. We may integrate ω over V .

Firstly suppose that V is a cycle, so ∂V = 0, and that ω is exact so that ω = dα. Then Z Z Z ω = dα = α = 0 (257) V V ∂V Now suppose that B is a boundary so that B = ∂V . We may integrate a form β ∈ Ωr−1 over B. Suppose β is closed, so that dβ = 0. Then Z Z Z β = β = dβ = 0 (258) B ∂V V Hence Stoke’s theorem implies 2 important things for cohomology;

113 • Integrating an exact form over a cycle gives zero.

• Integrating a closed form over a boundary gives zero.

In practice these are useful. For example, if we integrate a closed form over a cycle and we get a non-zero answer, then the form is not exact.

Furthermore, due to the relation of the dimension of the cohomology to the r topological Betti numbers, dim H = br, there is a useful consequence of de Rham’s theorem. Recall that integrating an exact form over a cycle gives zero. Then

Theorem: Take M compact without boundary. Consider a closed r-form R ω. Then if V ω = 0 for every possible r-cycle V in M then ω is exact.

Map on cohomology spaces induced by cycles: Given an r-cycle V we may deﬁne the map

r ΛV : H → R Z [ω] → ΛV ([ω]) ≡ ω (259) V We note that while the deﬁnition depends on an explicit representative, the map is independent of the choice of this representative.

114 5.9 Poincar´e’sLemma [We will not prove the Lemma here, although the proof is elementary. Naka- hara provides a proof.]

An open set U in M is contractible if it can be smoothly contracted to a point, p0 ∈ M. This means there is a map f

f : R × M → M λ, p → p0 = f(λ, p) (260)

0 such that U = f(0,U) and p0 = f(1,U) and f is continuous (C ).

E.g. A disc is contractible but an annulus is not.

Then consider forms over the open set U, which itself is a manifold (diﬀeo- morphic to an open subset of Rdim(M)). Then Zr(U),Br(U) are the space of closed, exact forms deﬁned over U (not over all M).

Poincar´e’sLemma: For contractible U ⊂ M, any closed form over U is also exact, i.e. ω ∈ Zr(U) ⇒ ω ∈ Br(U).

We say that any closed form is locally exact - where locally is meant in the sense above (i.e. in a contractible open set).

We may then think of De Rham cohomology as being the topological ‘ob- struction’ to closed forms being globally exact on M.

115 5.10 Example: Electromagnetism (again!) Recall we claimed the EM ﬁeld strength is a 2 form F = Ω2(M). Then locally we may write in some contractible open set F = dA for A ∈ Ω1(U). Hence the usual statement in components Fµν = ∂µAν − ∂νAµ.

Globally F is closed, not exact. Hence we may characterize a solution of EM by the cohomology class of F . Given a set of closed 2-forms {ωi} which give 2 i a basis of H , {[ω ]} for i = 1, . . . , b2(M) then the general ﬁeld strength can µ be globally written in terms of a dynamical ‘vector potential’ A = Aµdx as

i 1 F = αiω + dA, αi ∈ R,A ∈ Ω (M) (261)

Why were we not told of this? Simply because H2(Rm) = 0, by Poincar´e’s lemma - i.e. any open set in Rm is contractible to a point, so any closed form is also exact.

Suppose we are not on Rm. We may take a 2d submanifold M with no bound- R ary - say a 2-sphere - and integrate, M F to measure the ‘flux’ through this surface. If this ‘flux’ does not vanish then F is not exact. Note then that m R the space cannot be R ! The flux M F represents the magnetic charge contained within M, so this would be non-zero if there is a magnetic monopole contained within M.

116 5.11 Poincar´eduality Take an m-dimensional compact manifold M. Then let us deﬁne an inner product

r m−r < ·, · >: H (M) × H (M) → R Z [ω] , [α] → < [ω], [α] >≡ ω ∧ α (262) M This inner product is bilinear in the two arguments. It is simple to show that this deﬁnition is independent of the representatives of the cohomology classes Z Z Z < [ω], [α + dβ] > = ω ∧ α + ω ∧ dβ =< [ω], [α] > −(−1)r dω ∧ β M M M = < [ω], [α] > (263) and likewise in the ﬁrst argument. We have used the ‘integration by parts’ with ∂M = 0 as M is compact.

Theorem (Poincar´e): The inner product < ·, · > is non-degenerate.

Hence we see that Hm−r(M) is the dual vector space to Hr(M). Note that this implies the relation of the Betti numbers br = bm−r.

117 6 Riemannian Geometry I: The metric

6.1 The metric

0 Take a manifold M.A Riemannian metric is a (0, 2)-tensor ﬁeld g ∈ J2 that is symmetric and positive, i.e. at any point p ∈ M and for any vectors U, V ∈ TpM,

g(U, V ) = g(V,U) g(U, U) ≥ 0 (264) with equality in the latter only if U = 0.

The manifold M together with a metric g form the pair, (M, g), a Rieman- nian manifold.

A pseudo-Riemannian manifold is a manifold together with a pseudo-Riemannian metric which is a symmetric (0, 2)-tensor ﬁeld g, now not positive, but instead with the property that at a point p ∈ M

if g(U, V ) = 0 ∀ U ∈ TpM ⇒ V = 0 (265)

This pseudo-Riemannian metric divides the elements of TpM into 3 classes:

< 0 timelike g(U, U) = 0 null > 0 spacelike (266)

118 µ µ Taking a chart with coordinates {x } we can express the metric g = gµνdx ⊗ ν dx . In the Riemannian case positivity implies that the matrix gµν is invertible - i.e. the metric is non-degenerate. Likewise in the pseudo-Riemannian case the property in equation (265) ensures invertibility of gµν. Then Rie- mannian and pseudo-Riemannian manifolds both have an inverse metric, i.e. −1 2 µν a (2, 0)-tensor ﬁeld g ∈ J0 that is symmetric and whose components g in any chart form a symmetric matrix that is the inverse of the matrix gµν given by the components of the metric.

The metric has components gµν = gνµ which form a real symmetric matrix, so the eigenvalues of the matrix gµν are real, and are non-zero since the metric is invertible. Then at any point we may compute the signature of the metric;

signature = (# of negative eigenvalues, # of positive eigenvalues) (267)

This is independent of the point at which we evaluate it. For a d-dimensional M, we ﬁnd for a Riemannian metric the signature (0, d) and for a pseudo- Riemannian metric the signature (1, d − 1).

−1 µν We denote the inverse metric components (gµν) = g so that,

αµ α g gµβ = δ β (268)

119 6.2 Metric inner product The metric tensor gives the non-degenerate inner product at the point p

g : TpM × TpM → R U, V → g(U, V ) (269)

It thus provides the map (‘lowering an index’)

∗ TpM → Tp M U → g(·,U) ∂ uµ → (g uν) dxµ (270) ∂xµ µν and also the map (‘raising an index’)

∗ Tp M → TpM ω → U s.t. g(·,U) = ω(·) ∂ ω dxµ → (gµνω ) (271) µ ν ∂xµ This extends to general (q, r) tensors, as we are familiar with from physics.

120 6.3 Volume element The metric g on an orientable m-dim’l manifold gives us a canonical volume form, Ωg. This is deﬁned as

q 1 m Ωg ≡ | det gµν| dx ∧ ... ∧ dx (272)

It is not a priori obvious that this statement is independent of the coordinates - we note the explicit dependence on the components gµν. However the statement is invariant under change to other coordinates, although we require all the coordinates to have the same orientation (hence the requirement the manifold is orientable). Let us now show this.

Recall from our previous discussion of volume elements that under a change of coordinates xµ → yµ,

∂xα dx1 ∧ ... ∧ dxm = det dy1 ∧ ... ∧ dym (273) ∂yβ where det ∂xα/∂yβ is the Jacobian. Since our manifold is orientable we may choose charts with coordinates so that all such Jacobians on any chart overlaps are positive i.e. they all have the same orientation. We recall that the metric components transform as

∂xµ ∂xν g = g dxµ ⊗ dxν =g ˜ dyα ⊗ dyβ = g dyα ⊗ dyβ (274) µν αβ ∂yα ∂yβ µν so that,

Hence we see that,

q ∂xρ Ω = | det g | det dy1 ∧ ... ∧ dym g µν ∂yα ∂xρ q = sign det | detg ˜ | dy1 ∧ ... ∧ dym ∂yα αβ

q 1 m = | detg ˜αβ| dy ∧ ... ∧ dy (276)

121 since we have chosen coordinates so that the sign of the Jacobian is +1.

Thus we have seen that provided we have coordinates which preserve orientation the deﬁnition of the volume element above is indeed independent of the coordinate chart.

The volume element is called a pseudo-tensor as the sign of the tensor flips if you define it using coordinate charts covering the manifold with the opposite orientation. Once defined, it behaves as an m−form tensor in all respects.

The component function is always positive by the deﬁnition, and since the determinant of the metric never vanishes, we see Ωg indeed deﬁnes a volume form.

122 6.4 Epsilon symbol Let us deﬁne the epsilon symbol,

+1 for (µ1, µ2, . . . µm) an even permutation of (1, 2, . . . , m)

µ1µ2...µm = −1 for (µ1, µ2, . . . µm) an odd permutation of (1, 2, . . . , m) 0 otherwise (277)

Then we may raise indices using the metric to give,

ν1...νr ν1µ1 νrµr µr+1...µm = g . . . g µ1µ2...µm (278) We note that the epsilon symbol does not transform as the components of a p µ1 tensor. As we have just seen we require the combination |g|µ1...µm dx ∧ µm ... ∧ dx which then is a (pseudo-) tensor, where g = det gµν. In fact the epsilon symbol is said to transform as a tensor density, namely as the components of a tensor up to multiplication by g.

We note the following useful identities:

ν1 νm β aµ1 . . . aµm ν1...νm = det aα µ1...µm µ1...µm −1 = (det g) µ1...µm µ1...µm µ1...µm = m! det g α1...αrαr+1...αm = r!(m − r)!(det g)−1 δ[αr+1 δαr+2 . . . δαm] α1...αrβr+1...βm β1+r βr+2 βm where T = 1 P sign(p)T . [α1...αr] r! p∈Perm(1,2,...,r) αp1 ...αpr

123 6.5 Hodge Star Recall that dim Ωr = dim Ωm−r on an m-dimensional manifold. With a metric we have a natural map;

? :Ωr(M) → Ωm−r(M) 1 p dxµ1 ∧ ... ∧ dxµr → ? (dxµ1 ∧ ... ∧ dxµr ) ≡ |g|µ1...µr dxµr+1 ∧ ... ∧ dxµm (m − r)! µr+1...νm

Remarks

1 µ1 µr • For a general r-form ω = r! ωµ1...µr dx ∧ ... ∧ dx , we have

p|g| ? ω = ω µ1...µr dxµr+1 ∧ ... ∧ dxµm (279) r!(m − r)! µ1...µr µr+1...µm

• The volume element can be written as ?1 = Ωg • The components of ?ω are

p|g| (? ω) = ω µ1...µr (280) µr+1...µm r! µ1...µr µr+1...µm

• For ω ∈ Ωr we ﬁnd

Riem (0, m): ? ? ω = +(−1)r(m−r)ω Lorentzian (1, m − 1) : ? ? ω = −(−1)r(m−r)ω (281)

Ex: show these are true.

124 6.6 Inner product on r-forms Take α, β ∈ Ωr(M), so that, 1 1 α = α dxµ1 ∧ ... ∧ dxµr β = β dxµ1 ∧ ... ∧ dxµr (282) r! µ1...µr r! µ1...µr and then 1 α ∧ (?β) = β ∧ (?α) = (α βµ1...µr )Ω (283) r! µ1...µr g which is natural to integrate over M.

We use this to deﬁne the following inner product.

r r (·, ·):Ω (M) × Ω (M) → R Z α , β → (α, β) ≡ α ∧ (?β) (284) M We see that this product is symmetric:(α, β) = (β, α)

For the case of a Riemannian metric, signature (0, m), the product is positive deﬁnite so that (ω, ω) ≥ 0 with equality only if ω = 0. This is easily seen using Riemann normal coordinates.

125 6.7 Adjoint of d: d† We deﬁne the adjoint exterior derivative as d† :Ωr(M) → Ωr−1(M) ω → d†ω ≡ −(−1)m(r+1) ? d ? ω (0, m) Riem d†ω ≡ +(−1)m(r+1) ? d ? ω (1, m − 1) Lorentzian We ﬁnd that d† is nilpotent so (d†)2 = 0. This is since d†d† = ?d ? ?d? = ± ? d2? = 0 since d2 = 0.

Why all the annoying signs in the deﬁnitions? All the signs have been to ensure the following is nice.

Take a compact orientable manifold with metric (M, g). Take α ∈ Ωr(M), β ∈ Ωr−1(M) relation; (dβ, α) = (β, d†α) (285) Hence d† is the adjoint to d with respect to the inner product (·, ·).

Let us now show this for the Riemannian (0, m) case - I leave the Lorentzian (1, m − 1) case as an exercise. Using Stokes we have, Z Z Z Z 0 = β ∧ (?α) = d (β ∧ (?α)) = dβ ∧ (?α) + (−1)r+1 β ∧ d ? α ∂M M M M and so Z (dβ, α) = (−1)r(−1)r˜(m−r˜) β ∧ (? ? d ? α) (286) M where (d ? α) is anr ˜-form, sor ˜ = m − r + 1 and we have inserted the identity (−1)r˜(m−r˜) ?? acting on this. Then we have, Z (dβ, α) = (−1)(m−r+1)(r−1)+r β ∧ (? ? d ? α) M Z = (−1)mr−r2+r−m−1 β ∧ (? ? d ? α) M Z = −(−1)mr+m(−1)r(r−1) β ∧ (? ? d ? α) M Z = (−1)r(r−1) β ∧ (?d†α) = (β, d†α) M

126 in gory detail.

Exercise Show that for a 1-form v with components in a chart given by µ † v = vµdx , d v is the 0-form given in that chart by † 1 q µ d v = − p ∂µ | det gµν| v | det gµν|

µ µν where v = g vν.

127 6.8 3d vector calculus again. Forms generalize 3d vector calculus. Recall for a function f(x) we may construct a vector field using grad, ∇f. For a vector field v(x), we may take the curl to give another vector field ∇ × v, or take the divergence to get a function ∇ · v.

3 i Take M = R with the usual cartesian coordinates x and metric gij = δij.

The exterior derivative d of a function f ∈ Ω0, df gives a one-form with components that are those of the gradient of f.

i df = ∇if dx

i A one-form v = vidx has 3 components vi, and there is a corresponding i ij 3-vector ﬁeld v = δ vj. The Hodge star gives the 2-form 1 ∗v = v i dxj ∧ dxk 2 i jk The divergence of v is given by

† i d v = − ∗ d ∗ v = −∂iv = −∇.v

The Hodge star of a 2-form 1 ω = ω dxj ∧ dxk 2 ij is the one-form 1 ∗ω = (ω ij ) dxk 2 ij k The d of a one-form gives a 2 form, again with 3 components, and taking the Hodge star gives a 1-form whose components are the curl of the vector ﬁeld:

ij k i ∗dv = (∂ivj) kdx = (∇ × v)i dx

128 6.9 The Laplacian We may now write down the Laplacian, a map operating on r-forms.

4 :Ωr(M) → Ωr(M) ω → 4ω ≡ (dd† + d†d)ω (287)

Note that 4 = dd† + d†d = (d + d†)2 since both d2 = (d†)2 = 0.

For example, for a function, a 0-form, f ∈ Ω0(M), one ﬁnds using a coordinate chart that,

† 1 p µν 4f = d d f = − ∂µ |g|g ∂νf(x) (288) p|g|

The Laplacian is one of the most important diﬀerential operators in physics and mathematics.

129 6.10 Ex. Electromagnetism again....

αβµν Recall that the equation ∂βFµν = 0 can be recast in the statement that Fµν deﬁnes a closed 2-form F , so

dF = 0 (289)

µ µ Recall the remaining Maxwell equations ∂ Fµν = jν for a current j = (ρ, j). One can check that we may write this as

d†F = j (290)

µ 1 where j = jµdx ∈ Ω (M).

Acting with d† on this we ﬁnd d†j = 0. One can check that when written in components this is nothing but current conservation. The equations dF = 0, d†F = j can be written on any 4-dimensional manifold and give Maxwell’s equations on any space-time. Note that generalising to m dimensions is also possible, but requires j to be an m − 3 form.

130 6.11 Hodge theory We now deﬁne ‘harmonic’ , ‘co-closed’ and ‘co-exact’ forms; An r-form ω ∈ Ωr(M) is

closed if d ω = 0 exact ” ω = dα α ∈ Ωr−1(M) co − closed ” d†ω = 0 co − exact ” ω = d†β β ∈ Ωr+1(M) harmonic ” 4ω = 0

As an exact form is closed, a co-exact form is co-closed. Recall we denoted the space of closed and exact r-forms as Zr(M),Br(M). Likewise we denote the space of co-closed and co-exact r-forms as Z†r(M),B†r(M). The space of harmonic r-forms we denote Harmr(M)

Consider a Riemannian manifold (M, g). Then since the inner product (·, ·) is positive, so that (α, α) ≥ 0 with equality on if α = 0, then we have,

(ω, 4ω) = (ω, d†dω) + (ω, dd†ω) = (dω, dω) + (d†ω, d†ω) ≥ 0 (291)

Hence we say the Laplacian is a positive operator.

We see then that ω is harmonic, so 4ω = 0, if and only if dω = 0 and d†ω = 0 i.e. a form is harmonic iﬀ it is closed and co-closed.

Remarks

• The spaces Br(M), B†r(M) and Harmr(M) are othogonal wrt to (·, ·). i.e. for any α ∈ Br(M), β ∈ B†r(M) and γ ∈ Harmr(M) then (α, β) = (α, γ) = (β, γ) = 0.

• The vector space of r-forms can be decomposed into the vector subspace of harmonic forms plus its orthogonal complement: Ωr(M) = Harmr(M) ⊕ Harmr(M). Let us call the projector onto harmonic r- forms, P ,

P :Ωr(M) → Harmr(M) ω → P ω (292)

131 Hodge decomposition theorem

On a compact Riemannian manifold the space of r-forms admits a unique decomposition into exact, co-exact and harmonic forms; i.e.

Ωr = Br(M) ⊕ B†r(M) ⊕ Harmr(M) (293)

This implies any r-form ω ∈ Ωr can be written uniquely as

ω = dα + d†β + γ (294) for some α ∈ Ωr−1, β ∈ Ωr+1 and γ ∈ Harmr.

‘Proof’

We will not prove the result but recast the result in terms of another: Con- sider φ, j ∈ Ωr on a compact Riemannian manifold, such that (φ, γ) = (j, γ) = 0 for all γ ∈ Harmr i.e. both solution φ and source j are orthogonal to the harmonic forms. Then one may prove that we can uniquely solve Poisson’s equation 4φ = j to give,

φ = 4−1j (295)

This result requires hard analysis to prove.

Note that it is simple to make the solution φ orthogonal to the harmonic forms by adding the correct complementary function i.e. φ + γ solves the same equation for any γ ∈ Harmr.

A side remark is that one cannot solve the Poisson equation 4φ = j if the source j is not orthogonal to the harmonic forms. This follows from (4φ, γ) = 0 for any γ ∈ Harmr, which follows from 4 = dd† + d†d.

Example: The harmonic 0-forms are constant functions. Hence for functions φ, ρ one cannot solve 4φ = ρ unless (ρ, c) = 0 for any constant function c R and hence, recalling that ?c = c Ωg, we require M ρ = 0.

Given the above result, we may proceed to prove the Hodge decomposition theorem. Now given a form ω ∈ Ωr we may use the projector P above to

132 compute the harmonic component. Subtracting this harmonic component as ω0 = ω − (P ω) gives an r-form in the complement to the harmonic forms. Then

(ω0, γ) = 0 ∀ γ ∈ Harmr(M) (296)

From the above result about Poisson’s equation we see that we may then write ω0 = 4φ for a unique φ. Hence,

ω = 4φ + P ω = d(d†φ) + d†(dφ) + P ω = dα + d†β + γ (297) for α = d†φ, β = dφ and γ = P ω ∈ Harmr. Hence we have explicitly constructed the decomposition claimed above.

133 6.12 Harmonic representatives for cohomology Consider a closed r-form ω ∈ Zr on a compact Riemannian manifold. Hodge’s theorem tells us we may write any r-form as ω = dα+d†β +γ for harmonic γ. However we wish this to be closed. Then dω = d2α + dd†β + dγ = dd†β since γ being harmonic implies it is closed and co-closed, so dγ = 0. Consider

(dω, β) = (dd†β, β) = (d†β, d†β) (298)

Since we want dω = 0 this must vanish. However the RHS can only vanish if d†β = 0. Hence we see Zr = Br ⊕ Harmr, so we may write a closed form as

ω = dα + γ (299) for γ ∈ Harmr.

Therefore an important consequence of Hodge’s theorem is that every cohomology class [ω] ∈ Hr(M) has a unique harmonic representative on a compact Riemannian manifold. Take a given representative, ω, which then has a unique decomposition, ω = dα + γ for γ ∈ Harmr. Then γ is the harmonic representative, so [ω] = [γ].

Note that any other representative of the same class, say ω0, so [ω] = [ω0] will be written ω0 = dα0 + γ for the same γ, since all representatives diﬀer by an exact form.

We then see that Harmr ' Hr, and the space of harmonic r-forms is a finite dimensional vector space whose dimension is given by the topological invariant br. This gives the remarkable fact that the number of linearly independent solutions to the equation ∆ω = 0 for r-forms ω depends only on the topology of the manifold. There are a number of similar results (e.g. in index theorems) where the number of solutions to a differential equation on a manifold is a topological invariant, showing a profound link between apparently different areas of mathematics.

134 6.13 Maxwell’s Equations on a Compact Riemannian Manifold Consider Maxwell’s equations

dF = 0 d†F = j on a compact Riemannian Manifold, so that Hodge’s decomposition theorem applies. As we saw before we may write

F = ω + dA (300) where A ∈ Ω1(M) and ω is ﬁxed and gives the cohomology class of the ﬁeld strength; [F ] = [ω]. The ‘gauge’ transformation is then written as A0 = A + dλ, for a function λ ∈ Ω0(M). We see that F remains invariant when A transforms, since d2 = 0.

Recall the equation d†F = j. Earlier we had commented that this implies local current conservation d†j = 0. However, we see now a stronger statement is implied, namely that in fact j is globally a co-exact form.

We now use our discussion of Hodge theory to choose a canonical gauge and a canonical representative ω.

Lorentz gauge

Let us consider the gauge d†A = 0 - the ‘Lorentz’ gauge. Under a gauge transform we see,

d†A0 = d†A + d†dλ = d†A + (d†d + dd†)λ = d†A + 4λ (301) where we have used the fact that d†λ = 0 since λ is a zero form. Hence we see that by choosing λ = −4−1(d†A), then d†A0 = 0.

Note that (on a compact Riemannian manifold) we may always invert the Laplacian, since the source, d†A being co-exact is always orthogonal to the harmonic forms.

Harmonic representative

135 Recall d†F = j. Hence if we choose ω ∈ Harm2(M) as the representative of the cohomology class of F , so [F ] = [ω], then since a harmonic form is co-closed we simply have,

d†dA = j (302)

Furthermore using Lorentz gauge, d†A = 0, we can write this as

4A = j (303) and hence the gauge potential can be solved as A = 4−1j. Recall (on a compact Riemannian manifold) we require the source j to be orthogonal to the harmonic forms for this to be well deﬁned. However, as we said above, j must be co-exact, and therefore satisﬁes this restriction.

Much of physics is governed by the Laplacian.

136 7 Riemannian Geometry II: Geometry

A metric allows us to give a geometry to a manifold. In GR we often see the formula

2 µ ν ds = gµνdx dx (304)

What does this mean!? Where is the ⊗?! Where is the geometry?!

m At a point p the metric is an inner product on the vector space TpM = R , and (in Riemannian signature) it deﬁnes the dot product of tangent vectors, and hence determines their lengths, and the angle between them, just as usual for vectors in Rm.

However, the metric tensor ﬁeld allows us a notion of geometry in M itself, not just in its tangent spaces. The key is through the theory of integration that we have developed, which can be applied to give the length of a curve.

137 7.1 Induced metric and volume for a submanifold Given an m-dimensional manifold M and metric g, let us consider an embedded submanifold, diﬀeomorphic to an n-dimensional manifold N (n ≤ m), deﬁned by a smooth (one-to-one) map,

f : N → M (305)

∗ The we may induce a metric gN on N by the pull-back gN = f g. This is called the induced metric.

Take coordinates {xµ} on M and {yα} on N . Then we can write g = µ ν gµνdx ⊗ dx , and f is deﬁned by x = x(y). Then the induced metric is ∗ ∗ α β explicitly given as f g = (f g)αβ dy ⊗ dy , with ∂xµ ∂xν (f ∗g) = g (306) αβ ∂yα ∂yβ µν We may think of ‘classical geometry’ as the study of induced metrics on sub- m i j manifolds of M = R equipped with the Euclidean metric g = δijdx ⊗ dx . All notions of distance and curvature we have and will discuss give the correct classical result in this case.

The induced metric can be used to deﬁne a volume form on N

p 1 2 n ΩN = | det gN | dy ∧ dy ... ∧ dy (307) which can be integrated to give the volume of N with respect to the metric ∗ 0 gN = f g. However, N is diffeomorphic to N = f(N ) and the metric gN on N can be pushed forward to a metric gN 0 = f∗gN on f(N ) which is the metric on the embedded submanifold inherited from the metric g on M. The 0 corresponding volume element ΩN 0 on N = f(N ) is the push-forward of ΩN , 0 ΩN 0 = f∗(ΩN ). As N and N are diffeomorphic, Z Z ΩN 0 = ΩN (308) N 0 N and defines the volume of the submanifold N 0 = f(N ), and this can be calculated as an integral over N .

138 7.2 Length of a Curve and the Line Element Consider a coordinate chart and a curve C given explicitly by xµ(λ). This an example of the set-up of the last subsection, with C = N 0 an embedded 1-dimensional submanifold, with n = 1 and λ = y1 the coordinate on (an open subset of) the parameter space N = R. Then gN is the 1 × 1 matrix dxµ dxν g = g (309) N µν dλ dλ and p ΩN = |gN | dλ (310) Then the length of the curve is s Z Z µ ν p dx dx s = |gN | dλ = gµν dλ (311) dλ dλ

Then at any point p ∈ C, we have

ds 2 dxµ dxν = g (312) dλ µν dλ dλ

The metric allows us to associate a geometric distance to a portion of the 0 R R ds m curve C ⊂ C by the integral C0 ds = C0 dλ dλ. For M = R with a Euclidean metric this agrees with the usual notion of distance, and for a general manifold and metric provides the generalization of it.

For an inﬁnitesimal change in parameter along the curve, δλ, we have

δs 2 δxµ δxν = g (313) δλ µν δλ δλ

We call δs the line element, and identify it with a physical distance measure along the curve. The GR notation

2 µ ν ds = gµνdx dx (314) is an abuse of notation for the line element where we have cancelled the δλ2’s !

139 7.3 Integration of n-forms over oriented n-dimensional submanifolds We have an m-dimensional manifold M, and a one-to-one map f deﬁning an embedding of an n-dimensional orientable manifold N into M, where n ≤ m. Thus,

f : N → M (315) and the image N 0 = f(N ) is a submanifold of M (which is diﬀeomorphic to N ).

Given such an n-dimensional oriented submanifold of M, we deﬁned the integral of an n-form α ∈ Ωn(M) over it using the pull back Z Z α ≡ (f ∗α) (316) f(N ) N

We now give the coordinate expression for this. Consider a chart (U, ψ) for (M with coordinates xµ(p) = ψ(p)(µ = 1, . . . , m) and a chart (V, φ) for N such that f(V ) ⊂ U with coordinates yα(q) = φ(q)(α = 1, . . . , n). Then in U 1 α = α dxµ1 ∧ ... ∧ dxµn n! µ1...µn and in V 1 ∂xµ1 ∂xµn ∗ α1 αn (f α) = αµ ...µn ... dy ∧ ... ∧ dy n! 1 ∂yα1 ∂yαn ∂xµ1 ∂xµn = α ... dy1 ∧ ... ∧ dyn µ1...µn ∂y1 ∂yn ! 1 ∂xµ1 ∂xµn = α ... Ω (317) p µ1...µn 1 n N | det gN | ∂y ∂y where the induced volume form is

p 1 2 n ΩN = | det gN | dy ∧ dy ... ∧ dy (318)

140 A point q in N with coordinates yα is mapped to a point f(q) in N 0 = f(N ) µ with coordinates x (y). Consider the n vector ﬁelds tα,(α = 1, . . . n)

∂ ∂xµ ∂ t ≡ f = α ∗ ∂yα ∂yα ∂xµ

µ 1 For example, t1 is the tangent to the curve C1, which is the curve x (y ) given by the functions xµ(yα) with y1 varying and y2, y3, . . . yn ﬁxed, so that

d t = 1 dy1

µ Similarly, tα0 for some ﬁxed α0 is the tangent to the curve Cα0 given by x (y) and varying the coordinate yα0 while keeping the other n − 1 coordinates yα with α 6= α0 constant. The tangent vector to this curve at f(q) is d ∂xµ ∂ tα ≡ = 0 dyα0 ∂yα0 ∂xµ

The n vectors tα deﬁned in this way are tangent vectors to the submanifold N 0 = f(N ).

141 7.4 Hypersurfaces A hypersurface in an m-dimensional manifold is an (m − 1)-dimensional submanifold. Consider then the case of the previous subsections with n = m − 1, so that N 0 = f(N ) is an m − 1 dimensional submanifold of M (which is diffeomorphic to N ). The induced volume element ΩN on N defines an 0 m − 1 form ΩN 0 defined for all p ∈ N given by the push-forward of ΩN , 1 ΩN 0 = f∗(ΩN ). The Hodge dual of this defines a 1-form n(p) ∈ Ω (M) defined for all p ∈ N 0 m−1 n = λ(−1) ? ΩN 0 where λ = 1 for Riemannian signature, and λ = −1 for Lorentzian signature. This then implies ΩN 0 = ∗n ∗ 0 The 1-form n ∈ Tf(p)M annihilates all vectors tangent to N , i.e. to all vectors in the m − 1 dimensional subspace f∗(TpN ) of Tf(p)M,

hn, f∗V i = 0 for all V ∈ TpN . and is a smooth nowhere vanishing 1-form ﬁeld on f(N ). The pull-back of n is zero, f ∗(n) = 0

Using the metric to rise the index on n gives a vector N ∈ Tf(p)M with µ µν components N = g nµ. For a pseudo-Riemannian manifold, the vector N is timelike, spacelike or null, and the corresponding hypersurface is then said to be a timelike, spacelike or null hypersurface, respectively. This is orthogonal to all vectors tangent to the hypersurface at f(p),

g(N, f∗V ) = 0 for all V ∈ TpN and is called a normal vector. The normal vector N(p) then give a vector ﬁeld over the hypersurface. For a null hypersurface, the induced metric gN has vanishing determinant and as a result the volume form ΩN = 0. Then the volume of a null hypersurface is zero.

142 The normal vector is normalised so that g(N,N) is 1 for a spacelike hypersurface, 0 for a null one and −1 for a timelike one. (For the Riemannian case, it is 1.) We now show this. Using (283)

2 2 µν n ∧ (?n) = n Ωg, n = g nµnν (319) and

2 ΩN 0 ∧ (?ΩN 0 ) = Ω Ωg (320)

0 where Ωµ1...µm−1 are the components of ΩN , i.e. 1 µ1 µm−1 Ω 0 = Ω dx ∧ ... ∧ dx N (m − 1)! µ1...µm−1 and 1 Ω2 = Ω Ωµ1...µm−1 (m − 1)! µ1...µm−1

From ΩN 0 = ∗n we obtain Ω2 = λn2 using ∗ ∗ n = λ(−1)m−1n. For a null hyper surface, n2 = 0. If n2 6= 0, then as ΩN 0 is the push-forward of

We will now assume that the hypersurface is not null. Then

Ωg = n ∧ ΩN 0 The normal vector is normalised so that g(N,N) = 1 if it is spacelike or g(N,N) = −1 if it is timelike. If N(f(p)) is not null, then it does not lie in

143 the tangent space f∗(TpN ) to the hypersurface.

µ For a 1-form ﬁeld w = wµ dx on M with corresponding vector ﬁeld W = wµ∂/∂xµ, there is an (m − 1)-form

iW ΩV = ?w which can be written as

iW (n ∧ ΩN ) = n(W )ΩN 0 − n ∧ iW ΩN 0 (322) The pull-back of this is ∗ f (?w) = n(W )ΩN (323) ∗ ∗ µ using f (ΩN 0 ) = ΩN and f (n) = 0. Here n(W ) = nµW (x(y) is a function on N . Then the integral of ?w over the embedded hypersurface is Z Z ∗w = n(W )ΩN (324) f(N ) N Physics notation

In section 4, the integral of a top form over a chart was deﬁned. For a function φ on a manifold M, φ Ωg is a top form and for a chart (U, ψ) with coordinates xµ in U 0 = ψ(U) ⊆ Rm the integral of the top form over U is given by an ordinary integral over U 0 Z Z m p φ Ωg ≡ d x | det g| φ (325) U U 0 The integral over U 0 is sometimes written as Z Z φ dV = dmx p| det g| φ (326) U 0 U 0 where here dV is not a form but represents an inﬁnitesimal volume element.

R R For a hyper surface, the integral f(N ) ∗w gives N w(N)ΩN . For a chart (V, ψ) for N with coordinates yα in V 0 = ψ(V ) ⊆ Rm−1, Z Z m−1 p µ w(N)ΩN = d y | det gN | nµW (327) V V 0 and the right hand side is sometimes written as Z Z Z m−1 p µ µ µ d y | det gN | nµW = W nµ dS = W dSµ (328) V 0 V 0 V 0

144 7.5 Electric and Magnetic Flux

1 µ ν 0 The Maxwell 2-form is 2 Fµνdx ∧ dx . At ﬁxed time x =constant, we have the R3 with coordinates xi, and

k k Fij = ijkB , (?F )ij = ijkE (329) where Ei,Bi are the electric and magnetic ﬁelds. Then 1 1 F dxi ∧ dxj = ?B, (?F ) dxi ∧ dxj = ?E, (330) 2 ij 2 ij where i i B = Bidx ,E = Eidx Then for a 2-surface N 0 ⊂ R3 Z Z Z F = ?B = n(B)ΩN (331) N 0 N 0 N In a coordinate chart, the right hand side can be written in the conventional physics form for the magnetic ﬂux, Z i BidS

Similarly, Z Z Z ∗F = ?E = n(E)ΩN (332) N 0 N 0 N gives the conventional form for electric ﬂux Z i EidS

7.6 Stokes’s Theorem in 3D We can now find the original Stokes’s theorem in 3D as a special case of the general theorem referred to as Stokes’s theorem. Consider a 2-dimensional surface S bounded by a curve C = ∂S which is embedded in flat 3-dimensional 3 i space, R . For a 1-form v = vidx , the 2-form dv can be integrated over S and Stokes’s theorem used to find Z Z dv = v S C

145 The left-hand-side can be rewritten as Z Z Z i i dv = (∗dv)i dS = (∇ × v)i dS S S S and for the right hand side the notation Z i vidl C is sometimes used, where dli represents an inﬁnitesimal element of length tangent to the curve. Then we have the familiar formula Z Z (∇ × v).dS = v.dl S C

7.7 Gauss’s Theorem We can now combine the above with Stokes’s theorem to obtain the generalisation of Gauss’s Theorem to any number of dimensions. For a 1-form ﬁeld µ w = wµ dx on M, d∗w is a top form. Suppose the hypersurface f(N ) is the boundary of some m-dimensional submanifold P ⊆ M, so that f(N ) = ∂P. Then by Stokes’s theorem Z Z Z d ∗ w = ∗w = n(W )ΩN P ∂P ∂P Using d ∗ w = − ∗ d†w this can be rewritten as Z Z † (− ∗ d w) = n(W )ΩN P ∂P The notation µ † ∇µw ≡ −d w then allows us to rewrite Gauss’s theorem in the more familiar form Z Z µ (∇.w)dV = wµ dS P ∂P

146 7.8 Connections

Recall that given a vector at a point p, say V ∈ TpM, we can take the directional derivative of a function f ∈ F(M) at that point as V [f].

However we currently have no way to take a directional derivative of a vector field (or indeed any tensor field). (Recall using the Lie derivative we may 1 differentiate a tensor field using a vector field, V ∈ J0 , but at a point, p, this derivative doesn’t just depend on the direction vector V |p. Hence it is not a directional derivative.)

In order to take the directional derivative of a tensor we ﬁrst introduce an aﬃne connection, ∇. This is a map

1 1 1 ∇ : J0 (M) × J0 (M) → J0 (M)

X,Y → ∇X Y (333)

1 with the following properties: for any X,Y,Z ∈ J0 (M) and any f ∈ F(M) the map is bilinear so that,

∇X (Y + Z) = ∇X Y + ∇X Z

∇(X+Y )Z = ∇X Z + ∇Y Z and also

∇fX Y = f∇X Y

∇X (f Y ) = X[f] Y + f ∇X Y

Of these latter two properties, the ﬁrst shows the derivative is a directional one - i.e. at a point p,(∇fX Y )|p only depends on X|p, and not the value of X at any other point. The second shows the connection obeys a Liebnitz rule.

Since it is linear, we may specify this map by its action on basis vector ﬁelds. ∂ Taking a coordinate basis {eµ} = { ∂xµ } in a chart (U, ψ), we can specify ∇ by its components

α ∇µeν ≡ Γµν eα (334)

α where {Γµν } are the connection components, a set of smooth functions de- ﬁned over the open set U. To specify the connection we must supply these

147 connection components in all the charts in our atlas.

Using these components together with the properties of a connection above, we may then calculate ∇X Y in a coordinate chart (U, ψ) as

µ ν ∇X Y = X ∇µ(Y eν) (using directional deriv property) µ ν ν = X (eµ[Y ]eν + Y ∇µeν) (using Leibnitz property) ∂ = Xµ Y ν + Γ νY α e (using connection components) ∂xµ µα ν which is the familiar expression we are used to.

Consider an overlapping chart with coordinates {yµ} and corresponding co- ∂ ordinate basis {e˜µ} = { ∂yµ }. Then we can write the connection components ˜ α in the new coordinates, {Γµν }, as

˜ α ∇e˜µ e˜ν = Γµν e˜α (335)

∂xα Using the relation betweens these bases,e ˜µ = ∂yµ eα, we can compute,

β α β ρ ∂x ∂x ∂x ˜ α Γµν e˜ρ = ∇ ∂x e eβ = ∇eα eβ ∂yµ α ∂yν ∂yµ ∂yν ∂xα ∂xβ ∂xβ = ∇ e + e e ∂yµ ∂yν eα β α ∂yν β ∂xα ∂xβ ∂xσ = Γ σe +e ˜ e ∂yµ ∂yν αβ σ µ ∂yν σ ∂xα ∂xβ ∂ ∂xβ ∂yρ = Γ σ + e˜ (336) ∂yµ ∂yν αβ ∂yµ ∂yν ∂xσ ρ and so see that the two sets of connection components are related as

∂xα ∂xβ ∂yρ ∂2xα ∂yρ Γ˜ ρ = Γ σ + (337) µν ∂yµ ∂yν ∂xσ αβ ∂yµ∂yν ∂xα We see that the connection components almost transform under coordinates changes as the components of a (1, 2) tensor, but not quite due to the last term above.

148 As a comment, suppose we have two connections ∇, ∇0 which in some co- α 0 α ordinate chart have connection components {Γµν } and {Γµν } respectively. α α 0 α Then consider the diﬀerence of their components {δΓµν = Γµν − Γµν }. From above we see these do transform as components of a (1, 2) tensor,

∂xα ∂xβ ∂yρ δΓ˜ ρ = δΓ σ (338) µν ∂yµ ∂yν ∂xσ αβ since the last term in equation (337) cancels. Hence the diﬀerence between two connections does deﬁne a (1, 2) tensor.

1 Alternatively, given a connection ∇ and a tensor field H ∈ J2 (M) we may 0 α define a new connection ∇ . In a chart the connection has components {Γµν } α 0 and the tensor has components Hµν , and we define a new connection ∇ to act on the coordinate basis as

0 α α ∇µeν = Γµν + Hµν eα (339)

149 7.9 Torsion We decompose the connection ∇ into symmetric and antisymmetric parts 1 1 Γ ρ = S ρ + T ρ S ρ ≡ Γ ρ + Γ ρ µν µν 2 µν µν 2 µν νµ ρ ρ ρ Tµν ≡ Γµν − Γνµ (340)

From the transformation of the connection components (337), we see that the symmetric part S transforms like the components of a connection

∂xα ∂xβ ∂yρ ∂2xα ∂yρ S˜ ρ = S σ + (341) µν ∂yµ ∂yν ∂xσ αβ ∂yµ∂yν ∂xα while the antisymmetric part T transforms like the components of a tensor

∂xα ∂xβ ∂yρ T˜ ρ = T σ (342) µν ∂yµ ∂yν ∂xσ αβ

ρ Then the antisymmetric components Tµν deﬁne a (1, 2)-tensor, ∂ T = T ρ ⊗ dxµ ⊗ dxν (343) µν ∂xρ

ρ called the Torsion tensor. The symmetric components {Sµν } are given by ρ ρ 1 ρ 1 Sµν = Γµν − 2 Tµν where Γ is a connection and − 2 T is a tensor, and so ρ Sµν transform as the components of a connection, ∇S.

Hence we may think of a general connection ∇ as being deﬁned from a symmetric connection, ∇S, together with a torsion tensor, T .

There is an elegant basis invariant deﬁnition given by the following map,

1 1 1 T : J0 × J0 → J0

X,Y → T (X,Y ) ≡ ∇X Y − ∇Y X − [X,Y ] (344)

In addition to being bilinear, the ‘torsion map’ has the following important property (Exercise to check);

T (a X, b Y ) = a b T (X,Y ) ∀ a, b ∈ F(M) (345)

150 which implies that it deﬁnes a (1, 2)-tensor with components in a basis {eµ},

ρ ρ Tµν = < e , T (eµ, eν) > (346)

α This map gives the torsion tensor since T (eµ, eν) = ∇µeν − ∇νeµ = 2Γ[µν] eα. We note the torsion map is antisymmetric,

T (X,Y ) = −T (Y,X) (347)

151 7.10 Parallel transport and Geodesics

1 Consider a vector ﬁeld X ∈ J0 (M) and a curve C ⊂ M given by p(λ). We may ask how the vector ﬁeld at a point on the curve varies, i.e. how X|p(λ) varies. We say the vector X|p(λ) doesn’t change along the curve, or more formally is parallelly transported along the curve if

∇ d X = 0 (348) dλ p(λ)

µ µ Explicitly in coordinates {x } so that X = X (x)eµ and the curve C is given d dxµ(λ) by x(λ) with tangent dλ = ( dλ )eµ then

α µ d µ dx β µ ∇ d X = ∇ d (X eµ) = [X ] + X Γαβ eµ (349) dλ dλ dλ dλ

An important point is that whilst we have defined X as a vector field over the manifold, we see that actually since ∇ is a directional derivative, we actually only require the vector field to be defined over the curve C - since we only d µ require dλ [X (x(λ))].

Note that solving the above ODEs set to zero explicitly allows us to parallel transport a vector along a curve, by specifying the components Xµ at one end and integrating.

152 Geodesics

A curve C is said to be geodesic if the tangent vector at p, V (p) = d/dλ|p, is parallelly transported along the curve

∇V V |p = 0 ∀ p ∈ C (350) Intuitively we may think of the curve as being ‘straight’ with respect to the connection, since its tangent is parallelly transported along the curve.

In coordinates we have,

d2xµ dxα dxβ + Γ µ = 0 (351) dλ2 dλ dλ αβ

A geodesic is a curve where xµ(λ) satisfies this second order differential equa- µ tion. Note that in general the connection components Γαβ (x) will be functions of the coordinate x, so that the differential equation will be non-linear in general. Conversely, given a vector V ∈ TpM at a point p with coordinates µ x0 , we can solve the above second order differential equation to generate a curve x(λ) with

µ µ µ d dx (λ) x (0) = x0 ; = eµ = V dλ λ=0 dλ λ=0 to ﬁnd a geodesic through p with tangent vector V at p. If the connec- µ tion components vanish, Γαβ = 0, then the diﬀerential equation becomes d2xµ µ µ µ dλ2 = 0 with solution given by the straight line x = x0 + λV through p in the direction V .

7.11 Interpretation of Torsion A remark is that the geodesic curves of a connection ∇ only depend on the µ symmetric part of Γαβ . Hence two connections that diﬀer by a torsion will have the same geodesics. Knowing all the geodesics of a connection allows µ us to precisely reconstruct the symmetric components Sαβ .

153 Given a geodesic, we may think of torsion as eﬀecting how a vector othogonal to the tangent parallel transports.

Start at p with two vectors X,Y ∈ Tp. Flow a small parameter δλ along the geodesic of X to the point px. Parallel transport Y along this curve to 0 0 obtain Y ∈ Tpx . Now ﬂow δλ along the geodesic generated by Y starting at px to reach the point pyx. Repeat the same operations but interchanging X and Y to end at the point pxy.

Torsion measures how close the points pxy and pyx are to each other - i.e. how near the parallelogram is to closing. In coordinates one ﬁnds to linear order in λ,

xµ(p ) − xµ(p ) = δλ T µY αXβ (352) yx yx αβ p We remark that in ‘classical geometry’ torsion vanishes.

154 7.12 Covariant derivative We now define a directional derivative which acts on a general tensor field, 1 called the covariant derivative. Using a vector field X ∈ J0 we denote the derivative ∇X . It is the bilinear map

1 q q ∇ : J0 × Jr → Jr

X,T → ∇X T (353) and we encapsulate it being a directional derivative by requiring, as we did for the connection, that ∇fX T = f∇X T We require our covariant derivative to obey a Leibnitz rule

∇X (T1 ⊗ T2) = (∇X T1) ⊗ T2 + T1 ⊗ (∇X T2) (354)

q1 q2 where T1 ∈ Jr1 and T2 ∈ Jr2 are two tensor ﬁelds. Furthermore we require that under contracting indices we have

µ µ (∇X T )(· ... ·, e , · ... ·, eµ, · ... ·) = ∇X (T (· ... ·, e , · ... ·, eµ, · ... ·)) (355)

0 Thinking of functions F(M) as (0, 0)-tensor fields, J0 (M), then we may define the covariant derivative on a (0, 0)-tensor field simply as

∇X f ≡ X[f] (356)

This implies that

∇X ∇Y f − ∇Y ∇X f = X[Y [f]] − Y [X[f]] = ([X,Y ])[f] (357) where [X,Y ] is the Lie bracket.

The covariant derivative acting on a (1, 0)-tensor ﬁeld we take to be the connection. Then recall we required our connection to have the property: 0 1 ∇X (f Y ) = X[f] Y + f ∇X Y for f ∈ J0 and X,Y ∈ J0 . We can now think of this as a giving the Leibnitz rule required by the covariant derivative,

∇X (f ⊗ Y ) = (∇X f) ⊗ Y + f ⊗ (∇X Y ) (358)

155 0 where f ⊗ Y = fY as f ∈ J0 . Using these two deﬁnitions and the Liebnitz property, we can induce the action of the covariant derivative on any tensor ﬁeld.

Consider a 1-form ω and a vector ﬁeld V , so that

∇X (ω ⊗ V ) = (∇X ω) ⊗ V + ω ⊗ (∇X V ) (359)

ν µ This has components ωµV . Contracting the indices gives ωµV , the components of the function < ω, V >= ω(V ). The condition that the covariant derivative of a contraction is the contraction of a covariant derivative then implies that

∇X hω, V i = h(∇X ω),V i + hω, (∇X V )i (360)

Then for all ω, V, X we have

h(∇X ω),V i = X [hω, V i] − hω, (∇X V )i (361) and this deﬁnes ∇X ω.

Consider the above with

ρ X = eµ,V = eν, ω = e with the usual coordinate basis vectors ∂ e = , eρ = dxρ µ ∂xµ

∂ ν ν Then we use the fact that < ∂xµ , dx >= δµ in a coordinate chart to ﬁnd

ρ ρ ρ h(∇µe ), eνi = eµ [he , eνi] − he , (∇µeν)i (362) ρ ρ σ = ∂µ [he , eνi] − e , Γµνeσ ρ σ ρ = ∂µδν − Γµν he , eσi ρ = −Γµν

ρ µ Now ∇µe is a one-form and so is a linear combination of basis covectors e

µ µ ν ∇αe = Hαν e (363)

156 µ for some components Hαν . We then ﬁnd

ν ν Hαµ = −Γαµ (364)

Thus we have ∂ ∂ ∇ = +Γ ν , ∇ dxµ = −Γ µdxν (365) α ∂xµ αµ ∂xν α αν and using these and Liebnitz we can compute the covariant derivative of a α1...αq ∂ ∂ β1 βr general tensor T = T ⊗ ... ⊗ α ⊗ dx ⊗ ... ⊗ dx where ∇ β1,...βr ∂xα1 ∂x q X acts on the component T α1...αq function and each vector and co-vector in the β1,...βr basis of this (q, r)-form ﬁeld.

For example

ν ∇µω = ∇µ(ωνe ) ν ν = ∂µ(ων)e + ων∇µe ν ν ρ = ∂µ(ων)e − ωνΓµρe ρ ν = ∂µων − Γµνωρ e

157 7.13 Metric connection For a given metric g, a connection ∇ is a metric connection if at each point p ∈ M it obeys

∇X g = 0 ∀ X ∈ Tp(M) (366)

Another way to say this is that the metric is parallel transported along any curve.

Given a coordinate basis we may compute

ρ ρ α β ∇µg = ∂µgαβ − Γµα gρβ − Γµβ gαρ e ⊗ e (367) and hence the metric connection must obey

ρ ρ ∂µgαβ − Γµα gρβ − Γµβ gαρ = 0 ∀ µ, α, β (368) Cyclic permutations of this give,

ρ ρ ∂αgβµ − Γαβ gρµ − Γαµ gβρ = 0 ρ ρ ∂βgµα − Γβµ gρα − Γβα gµρ = 0 (369) Adding both equations in (369) and subtracting the one in (368) we obtain

ρ ρ ρ ρ Γ(αβ) gρµ − Γ[µα] gρβ − Γ[µβ] gρα = Cαβ gρµ (370)

ρ where Cαβ is called the Christoffel connection, defined as 1 C ρ ≡ gρσ (∂ g + ∂ g − ∂ g ) (371) αβ 2 α βσ β ασ σ αβ Hence we see that the symmetric part of the connection can simply be written in terms of the Christoffel connection and the antisymmetric part, given by the torsion tensor,

ρ ρ ρ Sαβ = Cαβ + T (αβ) (372)

ρ ρµ ν where we note that T αβ = g gβνTµα .

In summary the condition that ∇ is a metric connection completely determines the symmetric part of the connection in terms of the antisymmetric

158 part, which is unconstrained by the condition. Thus a metric connection is determined entirely by the metric g (which determines the Christoﬀel connection) and torsion tensor T .

Levi-Civita connection

In the case of vanishing torsion, T = 0, we see there is a unique metric con- ρ ρ ρ ρ nection Γµν = Cµν , which is symmetric Γµν = Γνµ . This is called the Levi-Civita connection.

Hence for vanishing torsion the metric g completely determines the metric connection. In the absence of physics to dynamically determine the torsion, this is the natural connection arising in GR. In particular this arises from variation of the Einstein-Hilbert action.

We note that for the Levi-Civita connection one ﬁnds that geodesics have extremal length. One can compute a geodesic curve C by functional variation of, Z Z q µ ν I[C] = ds = dλ x˙ x˙ gµν[x(λ)] (373) C C

µ µ dxµ(λ) µ µ α β with respect to x (λ) wherex ˙ = dλ . One obtains:x ¨ + Cαβ x˙ x˙ = 0.

159 7.14 Curvature We deﬁne the curvature map as

1 1 1 1 R : J0 × J0 × J0 → J0

X,Y,Z → R(X,Y,Z) ≡ ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z

Firstly this map is antisymmetric R(X,Y,Z) = −R(Y,X,Z).

Secondly we note it is trilinear and has the following important property (Exercise to check),

R(a X, b Y, c Z) = a b c R(X,Y,Z) ∀ a, b, c ∈ F(M) (374)

α ρ µ ν which implies the map deﬁnes a (1, 3)-tensor R ρµνeα ⊗ e ⊗ e ⊗ e , by

α α R ρµν = < e ,R(eµ, eν, eρ) > (375)

This (1, 3) tensor is called the Riemann curvature tensor.

Taking the basis {eα} to be a coordinate basis, so that [eµ, eν] = 0, then we can compute,

α α R ρµν = < e , ∇µ∇νeρ − ∇ν∇µeρ) > α β β = < e , ∇µ Γνρ eβ − ∇ν Γµρ eβ ) > α β β β β = < e , eµ[Γνρ ]eβ + Γνρ ∇µeβ − eν[Γµρ ]eβ − Γµρ ∇νeβ) > ∂ ∂ = Γ α − Γ α + Γ βΓ α − Γ βΓ α ∂xµ νρ ∂xν µρ νρ µβ µρ νβ

α and from R(X,Y,Z) = −R(Y,X,Z) we have the antisymmetry, R ρµν = α −R ρνµ.

Geometric interpretation

µ Consider a point p and coordinates {x } and the two curves C1, C2 deﬁned by µ µ µ µ µ µ xp + λ , xp + λδ where λ ∈ [0, 1] and we take , δ << 1. We close these curves into a small parallelogram using two more curves, C10 , C20 , deﬁned by µ µ µ µ µ µ 0 µ µ µ xp + +λδ and xp +λ +δ . Call point p that with coordinates xp + +δ .

160 µ Then take a vector V at x . Parallel transport along C1 then C10 to obtain a vector V1 ∈ Tp0 . Parallel transport along C2 then C20 to obtain a vector V2 ∈ Tp0 . Then the diﬀerence in these is given by the curvature,

µ µ α β µ ρ V1 − V2 = δ R ραβV (376) See the book by Nakahara for a derivation of this result.

7.15 Holonomy A loop is a closed curve

C : S1 → M λ ∈ [0, 1] → p = C(λ); p(1) = p(0) (377)

Consider the loops passing through a point p. Given a connection, any vector X ∈ TpM can be parallel transported round a curve C through p to give a new vector XC ∈ TpM. This deﬁnes a linear transformation

MC : TpM → TpM

MC : X → XC (378)

This is the holonomy at p of the connection for the loop C. To see this, choose a contractible patch U and note that each basis vector eµ will be transported to a new vector (eµ)C which can be written as a linear combination of basis vectors: ν (eµ)C = (MC)µ eν (379) µ so that X = X eµ transforms to

µ ν XC = X (MC)µ eν (380)

(as parallely transporting a function Xµ round a loop leaves it unchanged). Going back round the loop in the opposite direction should give the inverse transformation, so the matrix MC is invertible, MC ∈ GL(m, R).

Two curves C1, C2 through p can be combined to give a curve C3 through p, where you go round C1 ﬁrst, then round C2. The matrices clearly combine as

MC3 = MC2 MC1 (381)

161 The set of all linear transformations MC for all curves C through p then form a group Hp, the holonomy group at p. Clearly this is a subgroup of GL(m, R).

If two points p, q are connected by a curve D, then parallel transporting any vector X ∈ TpM along D to a vector XD ∈ TqM deﬁnes a linear transformation LD : TpM → TqM (382)

Given a curve Cq through q, there is then a curve Cp through p given by going from p to q along D, then round Cq, then back along D to p. This can −1 be written as Cp = D CqD. Then the linear transformations are related by conjugation −1 MCp = LD MCq LD (383) −1 so that Hp = LD HqLD and Hp and Hq are isomorphic. Then if M is (path- wise) connected, i.e. any two points can be joined by a curve in M, then the holonomy group H is independent of the point p.

For a connected manifold, the holonomy H of a connection is then H ⊆ GL(m, R). If the manifold is orientible and the connection preserves the orientation, then H ⊆ SL(m, R). For a metric connection, the lengths of vectors are preserved by the holonomies, so that H ⊆ O(m) for Riemannian manifolds or H ⊆ O(m−1, 1) for Lorentzian ones. If the manifold is orientible and the connection is metric then H ⊆ SO(m) for Riemannian manifolds or H ⊆ SO(m − 1, 1) for Lorentzian ones, while for a metric of signature (s, t), H ⊆ SO(s, t). For a product space M1 × M2 with connection ∇1 + ∇2 where ∇i is a connection on Mi with holonomy Hi (i = 1, 2), the holonomy is H1 × H2.

162 Berger’s Classiﬁcation In 1955, Berger gave a complete classiﬁcation of possible holonomy groups for the Lev-Civita connection for simply connected, Riemannian manifolds which are irreducible (not locally a product space) and nonsymmetric (not locally a coset space G/H with Lie groups G, H). Berger’s list is as follows:

Holonomy dim Type of manifold Comments SO(n) n Orientable manifold — U(n) 2n Kähler manifold Kähler Ricci-flat, SU(n) 2n Calabi–Yau manifold Kähler Sp(n) · Quaternion-Kähler 4n Einstein Sp(1) manifold Ricci-flat, Sp(n) 4n Hyperkähler manifold Kähler

G2 7 G2 manifold Ricci-flat Spin(7) 8 Spin(7) manifold Ricci-flat

Kähler,hyperkählerand Calabi-Yau manifolds are all complex manifolds. Quaternion-Kählermanifolds are sometimes called Quaternionic manifolds.

163 7.16 Non-coordinate basis Given a chart U with coordinates xµ there is a natural choice of basis for ∗ TpM and Tp M at every point p in that chart, namely the coordinate basis ∂ µ µ {eµ = ∂xµ } and its dual {e = dx }. However, it is sometimes useful to consider a general basis. At each point p ∈ U we choose a set of basis vectors {eâ(p)} for TpM with a = 1, . . . m and require that and require that thee â(p) are smooth vector fields on U.

Each basis vectore ˆa can be written as a linear combination of the coordinate basis vectors eµ:

µ µ eâ =e â eµ , eâ ∈ GL(m, R) (384)

µ with componentse ˆa (x).

Terminology

µ The matricese ˆa are called the zweibeins/dreibeins/vierbeins/vielbeins for a 2/3/4/> 4 dimensional manifold.

Given a metric g, the components of g in the non-coordinate basis will be

gab = g(ˆea, eˆb) (385)

At a point p, this is diagonalisable: one can change to a new orthonormal 0 basise ˆa

0 b b eˆa = Λa eˆb , Λa ∈ GL(m, R) (386) such that

0 0 0 gab = g(ˆea, eˆb) (387) is diagonal with eigenvalues ±1. This can be done for all points p ∈ U, with b a local GL(m, R) transformation Λa which is a smooth function of x. We will always restrict to such orthonormal bases.

164 For a Riemannian metric, we have (in an orthonormal basis)

g(ˆea, eˆb) = δab (388) and for pseudo-Riemannian signature (1, m−1) we require (in an orthonormal basis)

g(ˆea, eˆb) = ηab (389) with ηab = diag(−1, +1,...+1) the Minkowski metric. For pseudo-Riemannian signature (t, s) we require (in an orthonormal basis)

g(ˆea, eˆb) = ηab (390) with ηab = diag(−1,... − 1, +1,... + 1) the metric with t −1’s and s +1’s. Hence we see the metric is the constant ﬂat metric in the non-coordinate basis.

We will deal with the case of signature (t, s) in what follows and use ηab for the metric ηab = diag(−1,... − 1, +1,... + 1) with t −1’s and s +1’s, so that for the Riemannian case with t = 0, we have ηab = δab.

We will also require that the non-coordinate basis has the same orientation as the coordinate basis, ie.

µ dete ˆa > 0 (391)

We shall see later that this is useful.

165 Index notation

We will use roman labels for non-coordinate basis indices and greek for coordinate basis indices. Our relation above g(ˆea, eˆb) = ηab translates to the following component statement,

µ ν gµν eˆa eˆa = ηab (392)

Let us introduce the following notation to represent the inverse of the m × m µ matrixe ˆa as

a µ −1 eˆ µ ≡ (ˆea ) (393) so that we may conveniently write

a µ µ a a eˆ µ eˆb =e ˆb eˆ µ = δb (394) Using this notation we may invert the relation above to give the metric in terms of thee ˆ’s

a b gµν =e ˆ µ eˆ ν ηab (395)

As an example we can write a vector as

µ ˆ a V = V eµ = V eˆa (396)

ˆ a a µ where the non-coordinate components are given as V =e ˆ µV .

166 Dual non-coordinate basis

We may deﬁne a dual non-coordinate basis {θˆa} so that,

ˆa a < θ , eˆb >= δb (397)

(for any signature). Then we see,

ˆa a µ θ =e ˆ µdx (398)

For example we can now write a covector as,

µ ˆa ω = ωµdx =ω ˆaθ (399)

µ with the non-cordinate componentsω ˆa =e ˆa ωµ.

We see our index notation has been so that we can think of multiplication µ a bye ˆa ande ˆ µ as converting the coordinate basis tensor components to non- coordinate basis ones.

167 Form of the metric

The metric now takes a very simple form in the non-coordinate basis,

µ ν ˆa ˆb g = gµνdx ⊗ dx = ηabθ ⊗ θ (400)

(in the Riemannian case). However, we must recall that the non-coordinate basis is not a coordinate basis! Hence,

c [eµ, eν] = 0 , but [ˆea, eˆb] = cab (p)ˆec 6= 0 (401)

c where the coeﬃcients cab (p) (sometimes called the object of anholonomy) µ depend on the matricese ˆa (x) and their derivatives.

ˆa a µ In a non-coordinate basis, so that θ =e ˆ µdx then we ﬁnd,

ˆ1 ˆ2 ˆm Ωg = θ ∧ θ ∧ ... ∧ θ (402)

Let us check this.

ˆ1 ˆm 1 m µ1 µm θ ∧ ... ∧ θ =e ˆ µ1 ... eˆ µm dx ∧ ... ∧ dx a 1 m = det eˆ µ dx ∧ ... ∧ dx (403)

a b t a 2 Now as gµν =e ˆ µeˆ νηab then we have det gµν = (−1) dete ˆ µ . Recall that a p we restricted our vierbeins to have dete ˆ µ > 0, and hence | det gµν| = a dete ˆ µ, showing the result claimed above is indeed true.

168 7.17 Connections and curvature in non-coordinate bases â Using a non-coordinate basis {eâ} and dual {θ } we define the connection components,

c ∇aeˆb ≡ ∇eˆa eˆb = Γab eˆc (404) We note that ∂ ∇ eˆ =e ˆ µ∇ (ˆe νe ) =e ˆ µ eˆ ν +e ˆ αΓ ν e (405) a b a µ b ν a ∂xµ b b µα ν and hence the connection components are related as, ∂ Γ c =e ˆ µeˆc eˆ ν +e ˆ αΓ ν (406) ab a ν ∂xµ b b µα

For a metric connection, ∇ag = 0, which implies

ˆb ˆc 0 = ∇a ηbcθ ⊗ θ

b ˆd ˆc c ˆb ˆd ˆd ˆc = ηbc Γad θ ⊗ θ + Γad θ ⊗ θ θ ⊗ θ ˆb ˆc = 2Γa(bc)θ ⊗ θ (407)

d where Γabc = Γab ηdc, and hence Γa(bc) = 0, so that

Γabc = −Γacb (408)

This enables us to write the covariant derivative of a vector field µ ˆ a V = V eµ = V eâ (409) with respect to a vector field µ ˆ a X = X eµ = X eâ (410) as ˆ a ∇X V = ∇X (V eâ) ˆ a ˆ a = X[V ]êa + V ∇X eâ µ ˆ a ˆ a ˆ b c = (X ∂µV )êa + V X Γbaeˆc µ ˆ a a b = X (∂µV +ω ˆµ bV )êa (411)

169 where

a c b ωˆµ b =e ˆµΓca (412)

The components of the torsion tensor are given by

a â â Tbc = < θ ,T (êb, eˆc) >=< θ , ∇beˆc − ∇ceˆb − [êb, eˆc] > a a a = Γbc − Γcb − cbc (413)

a where [ˆeb, eˆc] = cbc (p)ˆea.

Likewise the components of the Riemann tensor are given as,

a â â R dbc = < θ ,R(êb, eˆc, eˆd) >=< θ , ∇b∇ceˆd − ∇c∇beˆd − ∇[êb,eˆc]eˆd >

170 7.18 Cartan’s structure equations Since both the torsion and curvature have an explicit antisymmetry, it is natural to deﬁne forms from them. We do so as follows;

Torsion 2-form We define the vector valued 2-form; 1 Tâ ≡ T a θˆb ∧ θˆc (414) 2 bc a a where we note that the antisymmetry Tbc = −Tcb makes this natural to define.

Curvature 2-form We define the matrix valued 2-form; 1 Râ ≡ Ra θˆb ∧ θˆc (415) d 2 dbc a a where again the antisymmetry R dbc = −R dcb makes this natural to define.

Together with these 2-forms we also deﬁne a connection 1-form.

Spin connection 1-form We deﬁne the matrix valued 1-form a a ˆb ωˆ c ≡ Γbc θ (416)

a c b Usingω ˆµ b =e ˆµΓca this can also be written as a a µ ωˆ c =ω ˆµ b dx (417) Cartan’s structure equations This enables us to write the torsion and curvature in a particularly elegant way using differential form ‘technology’, â â a b T = dθ +ω ˆ b ∧ θ â a a c R b = dωˆ b +ω ˆ c ∧ ωˆ b (418) and these are known as Cartan’s structure equations.

[Ex. conﬁrm these give the correct form for the torsion and curvature].

171 We note that for a metric connection, since Γabc = −Γacb we have

ωˆab = −ωˆba (419)

c ˆ withω ˆab ≡ ηacωˆ b. Then Cartan’s structure equations imply that Rab = ˆ −Rba. In a coordinate basis this translates into the result for the curvature for a metric connection

Rµναβ = −Rνµαβ (420) which is familiar from GR.

Bianchi identities

Taking the exterior derivative of both of these equations and using the fact that d is nilpotent, we obtain

â a ˆb â b dT +ω ˆ b ∧ T = R b ∧ θ â a ˆc â c dR b +ω ˆ c ∧ R b − R c ∧ ωˆ b = 0 (421)

These are the Bianchi identities that are familiar from GR, although we note we have included torsion.

No torsion

In the absence of torsion, we see from the structure equations and Bianchi identities

â a b dθ +ω ˆ b ∧ θ = 0 â b R b ∧ θ = 0 â a ˆc â c dR b +ω ˆ c ∧ R b − R c ∧ ωˆ b = 0 (422)

In a coordinate basis, the 2nd and 3rd equations here become the familiar expressions from GR,

α R [µνβ] = 0 α ∇[ρR µν]β = 0 (423)

172 Solving Einstein’s equations using vierbeins

Taking the Levi-Civita connection, which is a metric connection with no torsion, then,

ˆa a b ωab = −ωba and dθ +ω ˆ b ∧ θ = 0 (424)

Given a metric, one may then choose a convenient non-coordinate basis, and ˆ a hence the set {θa}, and the above conditions then determine ω b, and hence the connection – a method introduced by Misner. One then solves for the curvature from the second Cartan structure equation.

An alternative approach is to observe that for vanishing torsion we my use equation (413) together with the metric condition Γabc = −Γacb to solve for c the Γab components.

173 7.19 Change of Basis and the Local Frame

0 In a patch U, any 2 sets of basis vector fieldse â,e â will be related at a point p by

0 ˜ b ˜ b eˆa = Λa eˆb , Λa ∈ GL(m, R) (425) ˜ b for some Λa (p) ∈ GL(m, R) which will depend smoothly on the point p, i.e. these will be local GL(m, R) transformations. For orthonormal frames, the ˜ b local transformations Λa (p) will be restricted to be in SO(m) for an oriented Riemannian manifold or in SO(m − 1, 1) for an oriented Lorentzian mani- ˜ b fold. For an oriented manifold of signature (t, s), Λa (p) is in SO(t, s). These are then referred to as local frame rotations or local Lorentz transformations.

0 0 0 In an overlap U ∩ U , the framese â in U ande â in U must be related by a ˜ b local Λa eˆb transformation of this kind (in GL(m, R), SO(m) or SO(m−1, 1)) 0 ˜ b ˜ b eâ = Λa eˆb , Λa ∈ GL(m, R) (426) There will be corresponding transformation of the dual bases; in order that

ˆa ˆ0a 0 a < θ , eˆb >=< θ , eˆb >= δb (427) be preserved,

ˆ0a a ˆb a ˜ −1 b θ = Λ bθ , Λ b ≡ (Λ )a (428)

A vector ﬁeld

ˆ a ˆ 0a 0 V = V eˆa = V eˆa (429) will then have frame components transforming as

ˆ 0a a ˆ b a ˜ −1 b V = Λ bV , Λ b ≡ (Λ )a (430) ˆ a ˆ 0a 0 so that V eˆa = V eˆa.

The torsion 2-form

ˆa ˆa a ˆb T = dθ +ω ˆ b ∧ θ

174 transforms to

ˆ0a ˆ0a 0a ˆ0b T = dθ +ω ˆ b ∧ θ a ˆc c ˆb = Λ c dθ +ω ˆ b ∧ θ (431) so that, using matrix notation,

ωˆ0 = ΛˆωΛ−1 − dΛΛ−1 (432)

The vielbein defines a local frame at a point p, a frame in which the metric is the flat metric η. For a Lorentzian manifold in GR, this can be thought of as defining a local inertial frame at a point p in the spacetime. The local c frame rotations provide a local SO(t, s) gauge symmetry, withω ˆ b acting as the gauge field. The metric gµν is a symmetric matrix and so has m(m+1)/2 a 2 degrees of freedom. The vielbeine ˆµ has m degrees of freedom, but the gauge a parameters Λ b have m(m − 1)/2 independent components. The local gauge symmetry can be used to remove m(m − 1)/2 of the m2 degrees of freedom a of the vielbeine ˆµ, leaving m(m + 1)/2 independent degrees of freedom. In GR, the m(m+1)/2 (off-shell) degrees of freedom of the gravitational field in m dimensions can be represented by the components of a symmetric metric 2 a gµν or by the m degrees of freedom of the vielbeine ˆµ, identified under the action of the gauge group of dimension m(m − 1)/2.

175 7.20 Tangent Bundle

At every point p in M (an m-dim’l manifold) we have a tangent space TpM.

The tangent bundle of M, denoted T M is simply the collection of all these tangent spaces. T M ≡ ∪p∈MTpM (433)

An element Vp ∈ T M gives both a point p ∈ M and a tangent vector Vp at that point. It is also a manifold, with dimension 2m and we will now construct it explicitly.

Locally the tangent space is a product manifold. Consider a chart Ui on M. µ µ ∂ Take coordinates {x(i)}. Then we can represent some vector Vp = v µ |p ∂x(i) µ m using the coordinate basis, where the components v = (v1, . . . , vm) = R .

An open set Ui is also a manifold, and the tangent space over this open set is given by the product manifold,

m TUi = Ui × R (434) Hence we can use this fact to provide an altas for T M as follows.

m Construct charts on T M as {Vi, ξi}, where Vi = Ui × R . These are homeomorphic to Rm × Rm = R2m - hence T M is a 2m-dim’l manifold.

176 m µ µ A point in Ui × R is labelled by the pair (p, v ), where p ∈ Ui and v = m (v1, . . . , vm) labels a point in R . Our homeomorphism ξi is given by

m 2m ξi : Ui × R → R µ µ µ 1 m 1 m (p, v ) → (x(i), v ) = (x(i), . . . , x(i), v , . . . , v ) (435) We now have an altas.

Hence we see that a point in TUi can be associated with a point p ∈ M µ and vector Vp ∈ TpM whose components are v in the coordinate basis i.e. µ ∂ Vp = v µ |p ∂x(i) We must check the transition functions for our altas is smooth. Consider a point q ∈ Vi ∩Vj, which corresponds to a point p ∈ Ui ∩Uj. Then any tangent vector Vp in TpM can be written as

∂xµ µ ∂ µ ∂ ν (j) ∂ Vp = v(i) µ |p = v(j) µ |p = v(i) ν µ |p (436) ∂x(i) ∂x(j) ∂x(i) ∂x(j)

0 0 Hence we see the transition function taking coordinates from Vi to Vj is given by

µ µ µ µ ν φij :(x(i), v ) → (x(j)(x(i)),M νv ) (437) where, ∂xµ µ ν (j) M νv = ν |p ∈ GL(m, R) (438) ∂x(i) and so the matrix M is an element of the general linear group (i.e. an m×m matrix with non-zero determinant so it is invertible). Are these transition µ functions smooth? Yes - since M is invertible, and x(j)(x(i)) are smooth, M is also smooth.

∗ The co-tangent bundle has an analogous construction and is Up∈MTp M. It is denoted T ∗M.

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