Estadísticas de síntesis y no sectoriales

Recall that a bilinear form on a real-valued,k-dimensional vector space V is a binary mapα:V ×V →R, such that:

α(u+v, w) =α(u, w) +α(v, w)

α(u, v+w) =α(u, v) +α(u, w) α(r·u, v) =r·α(u, v) =α(u, r·v)

for everyu, w, v∈V andr∈R. We will denote the space of all bilinear forms onV byT2(V∗_).14_{The spaceT}2_(V∗_{) can be turned into a vector space using}

the operations of pointwise addition and scalar multiplication of functions.15

The dimension of the spaceT2_(V∗_{) is equal tok}2_{, with a natural basis given}

by the collection{ǫi⊗ǫj |i, j≤n},where{ǫi}ni=1 forms a basis for the dual

spaceV∗_.16

In order to eventually define what it means for a smooth manifoldM to be a spacetime, we will need to discuss the bilinear forms built from elements T M. In this case, we can use the fibresT2_(T∗

pM) and apply the Bundle chart

lemma to equip the set

T2(T∗_{M) :=} G

p∈M

T2(T∗

pM)

with a vector bundle structure. The exact details of this construction are not too important for our purposes, though we will remark that the local trivialisations ofT2_(T∗_{M) are given by:}

Φ(p, α) =Φ(p, rijdϕi⊗dϕj) = (p, r11, ..., r1n, ..., rn1, ..., rnn),

where we here we denote by dϕi _{the dual-basis element associated to the}

directional derivative ∂/∂ϕi_{. Local charts of the bundle T}2_(T∗_{M) are then}

given by:

π−1_(U)7→U×

Rn2 7→ϕ(U)×Rn2

in the obvious way, and the resulting spaceT2_(T∗_{M) becomes a vector bundle}

of rankn2_overM.17

Given a smooth mapf between smooth manifolds M and N, there is a natural sense in which (some of) the bilinear forms onN can be transferred toM. Suppose we have some element of the form (f(p), α) inT2_(T∗

f(p)N). We

can then define the objectf∗_αinT2_(T∗

pM) by:

f∗α(v, w) =α(dfp(v), dfp(w))

for allv, w∈TpM. Observe that sincef is smooth, by Proposition A.10.4 the

images of v and w under the differential dfp are indeed elements ofTf(p)N,

so this definition makes sense. Moreover, the formf∗_{αinherits its bilinearity}

fromα, so (p, f∗_{α) lies in the fibreT}2_(T∗

pM). We will refer to the objectf∗α

as the (pointwise) pullback ofαunder f.18

14_{This space is also commonly denoted asV}∗

⊗V∗

, orT(0,2)V, orT20(V). 15_{That is, we define (α+β)(v, w) :=α(v, w) +β(v, w) and (r·α)(v, w) =r·α(v, w).} 16_{The dual space for a vector space V is the set of all linear mapsα:V →R. If}

{ei} is a basis forV, then the collection {ǫi} of maps defined byǫi(ej) = 1 iff

i=j andǫi(ej) = 0 iffi6=jforms a basis forV ∗

.

17_{When viewed as a smooth manifold,T}2_(T∗

M) is of dimensionn+n2_.

18_{There is a sense in which fields of bilinear forms (and in general, covariant tensor}

There is also a sense in which the pointwise pullback has a restricted dual. In the case where M and N have the same dimension and f is a smooth embedding (which by Lemma A.12 means that M and f(M) are diffeomor- phic), we can also push bilinear forms of M forward into N. Given some (p, β)∈T2_(T∗

pM), we can define the mapf∗β to act as:

f∗β(v, w) =β(v′, w′), wherev=dfp(v′) andw=dfp(w′).

Observe that wheneverf is a diffeomorphism, the pointwise differentialdfpis

a bijection (see Prop. A.10.4), and since f(M) is a submanifold of the same dimension as N, we can use the maps as in the proof of Lemma A.12 to conclude that every element of the tangent spaceTf(p)N is of the formdfp(v)

for some vectorv in TpM, and thus f∗β is a well-defined bilinear form that

lives in the fibre T2_(T∗

f(p)N). We have the following facts about pointwise

pullbacks and pushforwards.

Lemma A.19.Let f : M →N andg :N →P be smooth embeddings, with dim(M) = dim(N) = dim(P), and let α, β and δ be elements of the fibres T2_(T∗ pM), T2(Tf(p)∗ N)andT 2_(T∗ g◦f(p)P)respectively. 1.(idM)∗α= (idM)∗α=α 2.f∗α= (f−1)∗α 3.f∗_{β = (f}−1₎ ∗β 4.(f∗_◦g∗_{)δ= (g◦f)}∗_δ 5.f∗_(f ∗α) =α 6.g∗◦f∗(α) = (g◦f)∗α

We saw in Proposition A.18 that wheneverf :M →N is a smooth map, the mapdf (defined fibrewise as the pointwise differential mapsdfp) becomes

a smooth bundle morphism from T M toT N. Similarly, we can also define a map that acts on fibres as the pointwise pullback (pushforward) does. Just as the differential map associated to a diffeomorphism is a bundle isomorphism between tangent bundles, there is an associated result for the bundle of bilinear forms.

Lemma A.20.Iff :M →N is a diffeomorphism, then the bundlesT2(T∗_M)

andT2_(T∗_{N)are isomorphic.}

We remark that in the proof of the above result, the isomorphism is given by the map ξ: T2_(T∗_{M)→ T}2_(T∗_{N) defined by ξ}−1_{(p, α) = (f(p), f}

∗α). This

will be important in Section 7.1, when we adjoin bundles of bilinear forms.