Elements of differential geometry -...

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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.

19

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.

20

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.

21

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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