Elements of differential geometryR.Beig (Univ. Vienna)

ESI-EMS-IAMP School on Mathematical GR, 28.7. - 1.8. 2014

1. tensor algebra

2. manifolds, vector and covector fields

3. actions under diffeos and flows

4. connections

5. pseudo-Riemannian manifolds

6. geodesics

7. curvature

1

Tensor algebra

Let T be an n-dimensional vector space over R and T∗ its dual.

Elements u, v of T are called vectors, elements ω, µ of T∗ are called

covectors. In a basis {ei} the vector u has the form u =∑n

1 uiei =

uiei (note summation convention!) and the covector ω reads

ω = ωiei in the dual basis given by ei(ej) = δij . The action of ω on

the v reads ω(v) = ωivi. Although vi and ωi depend on the choice

of basis, ωivi does not. Reading ωiv

i ’from right to left’ gives the

identification T ∼= T∗∗. The spaces Trs consist of multilinear forms on

T∗ × ...T∗ × ..T (r copies of T∗, s copies of T). They have r upper

and s lower indices: U = U i1...irj1..js ei1 ⊗ ...eir ⊗ ej1 ⊗ ..ejs . In

particular T ∼= T1 and T∗ ∼= T1 and T11∼= L(T,T).

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A scalar product γ on T is given by γ(u, v) = γijuivj with

γij = γji non-degenerate. γ of signature (++, ..+) is positive

definite, γ with (−,+,+, ...+) a Lorentzian metric. γ gives rise to a

unique quadratic form on V∗ given by γij where γijγjk = δik. The

quantities γij and γij yield an isomorphism between elements v ∈ Tand ω ∈ T∗ by means of ’raising and lowering of indices’, e.g.

vi(ω) = γijωj =: ωi. γ Lorentzian: a non-zero vector v ∈ T is

• timelike: γ(v, v) = γijvivj = −(v0)2 +

∑n−11 (vi)2 < 0

• null: γ(v, v) = γijvivj = −(v0)2 +

∑(vi)2 = 0

• spacelike γ(v, v) = γijvivj = −(v0)2 +

∑(vi)2 > 0

Null vectors form a double cone (’past and future light cone’) C,

timelike inside.

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Subspaces W ⊂ T are

• spacelike: γ(., .)|W pos.def.

• null: γ(., .)|W degenerate, same as T tangent to C

• timelike: γ(., .)|W Lorentzian

Fundamental inequalities:

• ’reverse C-S inequality’: (γ(u, v))2 ≥ γ(u, u)γ(v, v), provided

that u or v (or both) are causal

• ’reverse ∆ inequ.: |u+ v| ≥ |u|+ |v], where

|u| =√

−γ(u, u) and u, v are causal, both future or both past

directed. Is essence behind the ’twin paradox’

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5

Manifolds

Stated somewhat informally a (C∞, n-dimensional manifold) M is a

topological space (Hausdorff, 2nd countable), equipped with a set of

coordinate charts (U, xi), i.e. U open and xi map U bijectively into

an open set in Rn. These charts should cover M , s.th on overlapping

charts (U, xi), (U , xi), U ∩ U = {0} they are smoothly related:

xi = F i(xj), F i ∈ C∞(Rn,R), i = 1, ...n. Smoothness of

functions f :M → R is defined ’chartwise’, likewise smoothness of

maps between more general manifolds.

(T, γ), viewed as an affine space, is a manifold, namely Minkowski

spacetime, the realm of Special Relativity. Lorentzian manifolds, see

later, are the realm of General Relativity.

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Let p ∈M . The tangent space Tp(M) at p can be defined as the

vector space of derivations (see below) acting on smooth functions

defined near p. Elements v ∈ Tp(M) can be shown to be the same,

in local coordinates (U, xi) with p ∈M , as directional derivatives,

i.e. v(f) = vi ∂f∂xi |p. Thus the ’coordinate vectors’ ∂

∂xi |p = ∂i|p form

a basis of Tp(M). It follows that, under a change of chart:

vi = (∂jxi)vj .

Example for tangent vector: a smooth curve γ : I →M with

γ(0) = p gives an element in Tp(M) via its tangent vector defined

by γ′(0)(f) = ddtf ◦ γ(t)|0. By the chain rule γ′(0) = dxi

dt(0)∂i|p.

All tangent vectors can be gotten in this way.

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A vector field v on M is a smooth assignment to each p ∈M of a

vector vp ∈ Tp(M). Or, v maps C∞(M) into itself subject to

• v(af + bg) = av(f) + bv(g) (a, b ∈ R, f, g ∈ C∞(M))

• v(fg) = fv(g) + gv(f) ’Leibniz rule’

Here v(f)(p) = vp(f). The set of smooth vector fields is denoted by

X(M). It is a module over C∞(M), addition and scalar

multiplication being defined in the obvious way. In local coordinates

v ∈ X(M) can be written as v = vi(x)∂i or v(f) = vi(x)∂if .

Thus

vi(x) =∂xi

∂xj(x(x))vj(x(x)) (∗)

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Given v, w ∈ X(M), the map f ∈ C∞(M) 7→ v(w(f)) does not

define a vector field, but the Lie bracket

[v, w] = vw − wv = (vj∂jwi − wj∂jv

i)∂i

does. Note [∂i, ∂j] = 0.

Jacobi identity: [v, [w, z]] + [z, [v, w]] + [w, [z, v]] = 0

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Covectors: a covector at p is an element ωp of T ∗p (M). A covector

field or 1-form ω is defined in the obvious way. It is smooth if

ω(v)(p) = ωp(vp) is smooth for all v ∈ X(M). Let f ∈ C∞(M).

The 1-form df is defined by df(v) = v(f). In particular

dxi(∂j) = δij ...dual basis. ω = ωi(x)dxi, where ωi = ω(∂i).

E.g. df = ∂if dxi.

Under change of chart: ωi(x) =∂xj

∂xi (x)ωj(x(x)).

Higher order tensors (tensor fields) are defined in the obvious way,

e.g. the (1, 1)-tensor t = tij ∂i ⊗ dxj . Contraction, in a basis, is by

summation over a pair of up-and downstairs indices.

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The operation d sending functions to 1-forms is a special case of an

operation d, sending p-forms, i.e. covariant, totally antisymmetric

tensors ωi1...ip , p < n into p+ 1- forms. E.g. when p = 1 we define

(check this is a 2-tensor!)

dω(u, v) = u(ω(v))− v(ω(u))− ω([u, v])

i.e. dωij = ∂iωj − ∂jωi. We have that ddf = 0, and dω = 0

implies ω = df when M is simply connected.

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Action under flows

Let Φ :M → N be a diffeomorphism, i.e. a (smooth) mapping with

smooth inverse. The push-forward Φ∗v ∈ X(N) of v ∈ X(M) is

defined by (f ∈ C∞(N))

(Φ∗v)(f)(p) = v(f ◦ Φ)(Φ−1(p))

Locally yA = ϕA(xi) and

(Φ∗v)A(y) =

∂ϕA

∂xj(ϕ−1(y))vj(ϕ−1(y))

Next let Φ :M → N be smooth and ω a 1-form on N . Then the

pull-back ϕ∗ω on M is defined as (Φ∗ω)(v)(p) = ω(Φ∗v(Φ(p)).

(Note: does not require Φ invertible.)

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In coordinates

(Φ∗ω)i(x) =∂ϕA

∂xi(x)ωA(ϕ(x))

Pull-back on higher covariant tensor fields analogous. Pull-back

Φ∗f ∈ C∞(N) of functions f ∈ C∞(N) is simply Φ∗f = f ◦ Φ.

For mixed tensors, say on N , their pull-back to M is defined by

’pull-back under Φ w.r. to the downstairs indices’ and ’pull-back under

Φ−1 w.r. to the upstairs indices’.

Vector fields define a local(-in-t) 1-parameter family Ψt of maps

M →M via their flow, i.e.

dψit

dt= vi(ψt), Ψ0(p) = id

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The Ψt’s are local diffeomorphisms of M into itself in that they map

small neighbourhoods of each p ∈M diffeomorphically onto their

image. This is enough in order for the Lie derivative of w w.r. to v, i.e.

Lvw = ddt|t=0(Ψ−t)∗w to be defined. It turns out that

Lvw = [v, w] or

(Lvw)i = vj∂jw

i − wj∂jvi

Next Lvω is defined by Lv ω = ddt|t=0(Ψt)

∗ω. It turns out that,

locally,

(Lvω)i = vj∂jωi + ωj∂ivj, Lvf = v(f) = vi∂if

Similarly, for a 2-tensor gij ,

(Lvg)ij = vk∂kgij + gik ∂jvk + gkj ∂iv

k

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Geometrically, the equation Lvt = 0 means that the structure defined

by the object t is invariant under the flow generated by v. E.g. for gij

a symmetric tensor of Riemannian or Lorentz signature the Killing

vector field v satisfying Lvg = 0 generates a flow leaving the

Riemannian (Lorentzian) structure invariant.

The operations d and Lv are ’natural’ in that they, appropriately,

commute with general diffeomorphisms. This means they require no

structure an M . In contrast, ∇v, defined presently, is not natural.

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Pseudo-Riemannian manifolds

A manifold is called pseudo-Riemannian if it is provided with a

symmetric (0, 2)- tensor field g = gijdxi ⊗ dxj with

gij = g(∂i, ∂j) = gji non-degenerate. It is called Riemannian if g is

furthermore positive definite and Lorentzian if it has Lorentzian

signature at each p ∈M . Note that, e.g. in the Lorentzian case, there

is in general no chart near p ∈M , for which gij(x) = ηij = const.

This phenomenon is related to the presence of curvature.

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Linear connections

A linear connection ∇ on M is an R-bilinear map

∇ : X(M)× X(M) → X(M)

(u, v) 7→ ∇uv with (f ∈ C∞(M))

• ∇fuv = f∇uv

• ∇u fv = u(f)v + f∇uv

So ∇uv is tensorial w.r. to u, i.e. defines a (1, 1) tensor. In local

coordinates (xi), ∇uv = uj(∇jvi)∂i, where

∇ivj = vj ;i = ∂iv

j + Γjikv

k , ∇∂j∂k = Γijk∂i

Note ∇Φ∗uΦ∗v = Φ∗∇uv except if Φ leaves connection invariant.

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∇ can be naturally extended to act on tensor fields as follows:

∇uf := u(f) for f ∈ C∞(M). Then, for 1-forms ω, require that

∇u(v(ω)) = (∇uv)(ω) + u(∇uω), finally that ∇ satisfy the

Leibniz rule w.r. to ⊗ and be linear under addition. E.g.

(∇ut)ji∂j ⊗ dxi = uk(∂kt

ji + Γj

kltli − Γl

kitjl︸ ︷︷ ︸

∇ktji

)∂j ⊗ dxi

∇ is symmetric (torsion-free) if [u, v] = ∇uv −∇vu for all

u, v ∈ C∞(M). This in a local chart means that Γijk = Γi

kj .

Let Cijk = Ci

kj be a globally defined (1,2)-tensor field. Then, given ∇,

∇′uv = ∇uv + Ci

jkujvk∂i is also a symmetric connection.

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∇ on pseudo-Riemannian manifold (M, g) is called metric when

∇u g(v, w) = g(∇uv, w) + g(v,∇uw)

Given g, there ∃ unique symmetric linear (’Levy-Civita’)connection

which is metric. It is given by

Γijk =

1

2gil(∂jgkl + ∂kgjl − ∂lgjk)

Henceforth ∇ will be Levy-Civita.

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Geodesics

Let γ : I →M be a smooth curve on M and v a vector field along

γ. The covariant derivative of v along γ is defined as

Dv

Dt= ∇γ′v =

(dvi

dt+ Γi

jk

dxj

dtvk)∂i

The curve t 7→ γ(t) is geodesic when Dγ′

Dtis zero, i.e.

d2xi

dt2+ Γi

jk

dxj

dt

dxk

dt= 0

Prop.: Given p ∈M and v ∈ Tp(M), there ∃ interval I about t = 0

and a unique geodesic γ : I →M , s.th. γ(0) = p and γ′(0) = v.

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Because ofd g(u, v)

dt= g(

Du

Dt, v) + g(u,

Dv

Dt)

the causal character of the geodesic, in the Lorentzian case, is

preserved. Timelike geodesics model freely falling pointlike bodies,

null geodesics play the role of light rays.

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Curvature

Curvature, i.e. deviation from flatness, can be measured by the

degree of non-commutativity of ∇ acting on tensor fields, which in

turn is measured by the Riemann tensor. The basic observation is the

Prop.: The vector field given by (u, v, w ∈ X(M))

R(u, v)w := ∇u∇vw −∇v∇uw −∇[u,v]w

is tensorial in (u, v, w), i.e. defines a (1, 3)- tensor field Rijkl:

Rijkl = 2 ∂[iΓ

kj]l + 2Γk

m[iΓmj]l

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Rijkl = gkmRijm

l can be shown to have the following symmetries

• Rijkl = −Rjikl = −Rijlk

• Rijkl +Rkijl +Rjkil = 0 ’1st Bianchi identity’

• Rijkl = Rklij

Here the last property follows from the other ones. For n = 4 the

number of algebraically independent components of Rijkl is 20.

Furthermore there is the following (’2nd Bianchi’) differential identity

∇iRjklm +∇kRijlm +∇jRkilm = 0

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One can infer the Bianchi identities from the equivariance (see

Kazdan, 1981)

Rijkl[Φ∗g] = (Φ∗R)ijkl[g]

where Φ is a diffeomorphism M →M .

The identities fulfilled by Rijkl imply that the Ricci tensor

Rij = Rkikj satisfies Rij = Rji. Furthermore the Einstein tensor

Gij = Rij − 12gij g

klRkl is divergence-free, i.e.

gij∇iGjk = 0

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