Directory UMM :Journals:Journal_of_mathematics:BJGA:

Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian

Abstract.On a Weyl manifold (M,_bg, w), we consider ◦

∇ the Levi-Civita connection associated to a metric g ∈ bg , ∇ the symmetric connection, compatible with the Weyl structurewand the family of linear connections

C={∇λ := ◦

∇+λ(∇ − ◦

∇) | λ ∈ R}. For ∇ ∈ Cλ , we investigate some

properties of the deformation algebraU(M, λ

∇ − ◦

∇). Next, we study the

case when ◦

∇ and∇λ determine the same Ricci tensor and the case when the curvature tensors of the connections

◦ ∇and

λ

∇ are proportional.

M.S.C. 2010: 53B20, 53B21.

Key words: deformation algebra, Weyl structure, Weyl manifold, conformal Weyl connection, families of linear connections.

1

Introduction

The general problem of ”Comparison geometry” is: to what extent topological, dif-ferential, differential affine, metric, etc invariants determine some structure on a manifold, up to a homeomorphism, diffeomorphism, isometry, etc. Usually, one com-pares a given manifold with some ”standard” manifolds, such as space forms, Einstein spaces, or special product manifolds.

The topic originated in the Erlangen Program of F. Klein and was founded ex-plicitely in the work of E. Cartan, under the so-called ”equivalence problem”([3]). In the first half of the 20-th century, local methods were developed by S. Chern, G. Vranceanu ([12]) and others (see [4] for a modern review). Global methods arrose in the second half of the 20-th century, especially in global Riemannian geometry.

Our paper finds sufficient conditions for the curvature tensor (or the Ricci tensor) determines the Levi-Civita connection. On a Weyl manifold, we consider some special connections and, with them, we construct several deformation algebras. Properties of these deformation algebras will determine ”how far apart” will be the involved curvature tensors, or the Ricci tensors. In particular, we characterize the situation when the curvature tensors, or the Ricci tensors, coincide.

∗

Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 98-110. c

2

Definitions and notations

LetMbe ann−dimensionalC∞

differentiable manifold. We denote byF(M) the ring of real-valued functions of classC∞

defined onM and byTr

s (M) theF(M)−module

of tensor fields of type (r, s). In particular, forT1

0 (M) (respectively T10(M)) we use the notationX(M) (respectively Λ1_(M)).

Let A ∈ T1

2 (M).The F(M)−module X(M) becomes an F(M)−algebra, de-noted byU(M, A), with the product of two vector fieldsX andY defined by

(2.1) X◦Y =A(X, Y).

In particular, ifA=∇ − ∇, where∇ and∇are two arbitrary linear connections onM, then U¡M,∇ − ∇¢is called the deformation algebra associated to¡∇,∇¢.

Let A ∈ T1

2 (M) and m > 0, m ∈ Z. An object X ∈ U(M, A) is called almost m-principal field if there exists an 1−form ω ∈ Λ1_{(M) and a functionf ∈ F(M)} such that ([7])

(2.2) A³Z, X(m)´=f Z+ω(Z)X, ∀Z∈ X(M),

whereX(m)_=X(m−1)_◦X,X(1)_=X.

If m = 1, then (2.2) shows that X is an almost principal field in the algebra U(M, A). Ifm= 1 andf = 0, thenX is a principal field in the algebraU(M, A). If

m= 1 andω= 0, then X is almost special field. Ifm= 1, f = 0 andω= 0, then X

is special field. IfA(X, X) = 0, thenX is 2−nilpotent field.

Let (M, g) be now an n−dimensional semi-Riemannian manifold and let bg the conformal structure generated byg, i.e. bg={eu_{g|u∈ F(M)}.}

Letw be a Weyl structure on the conformal manifold (M,_bg), i.e. an application

w:bg−→Λ1_{(M), which verifies ([1], [5], [6])w(e}u_{g) =w(g)−du,∀u∈ F(M). The}

triple (M,_bg, w) is called aWeyl manifold.

Let (M,_bg, w) be a Weyl manifold. A linear connection∇onM is calledcompatible with the Weyl structurewif∇Xg+w(g) (X)g= 0, for allX ∈ X(M).

It is well known that there exists an unique symmetric linear connection∇, defined onM, compatible with the Weyl structure ([10]). This linear connection∇ is called theWeyl conformal connectionand is defined by ([9])

2g(∇XY, Z) = X(g(Y, Z)) +Y(g(X, Z))−Z(g(X, Y)) +w(g)(X)g(Y, Z) +

+w(g) (Y)g(X, Z)−w(g) (Z)g(X, Y) +g([X, Y], Z) + +g([Z, X], Y)−g([Y, Z], X),

for allX, Y, Z∈ X(M). Let ◦

∇be the Levi-Civita connection associated tog and let

A=∇ − ◦ ∇ ∈ T1

2 (M). We have

(2.3) 2g(A(X, Y), Z) =w(g)(X)g(Y, Z) +w(g)(Y)g(X, Z)−w(g)(Z)g(X, Y),

connections; a remarkable particular case is when the latter ones coincide (see also [2]).

Letgij,Ai

jkandui the components, in a system of local coordinates, ofg,Aand

of the 1−form 1

2w(g), respectively. Then (2.3) can be written

(2.4) Ai

jk=δijuk+δikuj−gjkui.

We define the family of linear connections ([8])

C=

(In the affine space of all linear connections on M, C is the ”affine line” passing

through the ”point” ◦

∇and of ”direction”A).

Let ◦

RandRλ be the curvature tensor field of the connection ◦

∇and∇λ, respectively.

3

The main results

Theorem 3.1.LetM be ann−dimensional connected manifold,n >3. The following assertions are equivalent:

(i) ∇λ =

∇ admit the same geodesics;

(iv) the algebraU

µ

∇ have the same Ricci tensor.

(∇λZ

Using the Bianchi identities, from (3.1), (3.2) and (3.3), it follows

λ

In local coordinates, (3.4) is written

(3.5)

jil are the components of Ricci tensor. Multiplying (3.6) by gjl and

summing with respect toj andl, it follows

(3.7) (n−2)

Because we supposedn >3, from (3.8) we have

Becausegis non-degenerate, from (3.11) it followsA= 0, soAλ =λA= 0, i.e. ∇λ = ◦ ∇.

(iii)=⇒(i) The linear connections∇λ and ◦

∇ are symmetric, hence they admit the same geodesics if and only if there exists an 1−form onM such that

(3.12) ∇λXY =

◦

∇XY +ω(X)Y +ω(Y)X,∀X, Y ∈ X(M).

From (3.12) and fromAλ =λA, we obtain

(3.13) λA(X, X) = 2ω(X)X, ∀X ∈ X(M).

ForY =X, from (2.3) it follows

(3.14) 2g(A(X, X), Z) = 2w(g) (X)g(X, Z)−w(g) (Z)g(X, X) , ∀X, Z∈ X(M).

From (3.13) and (3.14), we obtain

(3.15) 4ω(X)g(X, Z)−2λw(g) (X)g(X, Z) =−λw(g) (Z)g(X, X) ,

for allX, Z∈ X(M).

Becausen >3, for everyp∈M and everyZp∈TpM − {0}, there exists a vector

Xp∈TpM− {0} such that we have

(3.16) gp(Xp, Zp) = 0,gp(Xp, Xp)6= 0.

From (3.15) and (3.16), we get

(w(g))_p(Zp) = 0,∀Zp∈TpM − {0},∀p∈M,

sow(g) = 0 and from (2.3) we obtain thatA= 0, soAλ = 0, i.e. ∇λ = ◦ ∇.

(iv)=⇒(i) Forλ= 0 is trivial. We supposeλ6= 0. Because the algebraU

µ M,

λ

∇ − ◦ ∇

¶

is commutative, it follows thatU

µ M,

λ

∇ − ◦ ∇

¶

is associative if and only if we have

(3.17) A(X, A(Y, Z)) =A(Y, A(X, Z)) , ∀X, Y, Z∈ X(M).

In local coordinates, (3.17) becomes

(3.18) AiskA

s jl−A

i slA

s jk= 0.

From (3.18), using (2.4), we can write

(3.19) δkiujul−δ i

lujuk+gjlu i

uk−gjkuiul+δilgjkusu s₋

δkigjlu s

us= 0.

In (3.19), we makei:=kand sum with respect toi; it follows

(n−2) (ujul−gjlusus) = 0.

Becausen >3, we obtain

Multiplying (3.20) bygjl _{and summing with respect toj andl, we get}

(v)=⇒(i) In local coordinates, the conformal Weyl connection has the components

(3.22) Γi

¯are the Christoffel symbols of the second kind, constructed using the

me-tric components gij. From (3.22) and from

λ

components of the linear connection∇λ:

(3.23) Γλi

whereψi =λui. The curvature tensor of the connection∇λ has the components

(3.24) Rλi

where we denoted

ψjl =∂ψj

Using the hypothesis,Rjlλ = ◦

Rjl. Therefore, from (3.26) we get

(3.27) (n−2)ψjl+gjlϕ= 0.

Multiplying (3.27) bygjl _{and summing with respect toj andl, it follows}

(3.28) ϕ= 0.

From (3.27) and (3.28), we have

The relations (3.28) and (3.29) show that

λ Rijkl=

◦

Rijkl.

From the last relation, we find (i). ¤

Remark 3.1. For n = 3, the theorem remains true if we replace the condition

” ◦

Rp :TpM×TpM×TpM −→TpM is surjective,∀p∈M” by ”the Ricci tensor of the semi-Riemannian manifold (M, g) is non-degenerate”. This is easy to see from (3.7).

Theorem 3.2. LetU

µ M,∇ −λ

◦ ∇

¶

be the deformation algebra as above. We suppose

that the manifoldM is connected andn >2. The following assertions are equivalent:

(i) ∇λ = ◦ ∇;

(ii) all elements of the deformation algebra U

µ M,

λ

∇ − ◦ ∇

¶

are almost principal

fields;

(iii) all elements of the deformation algebra U

µ M,∇ −λ

◦ ∇

¶

are principal fields;

(iv) all elements of the deformation algebraU

µ M,∇ −λ

◦ ∇

¶

are almost special fields;

(v) all elements of the deformation algebra U

µ M,∇ −λ

◦ ∇

¶

are special fields;

(vi) all elements of the deformation algebraU

µ M,∇ −λ

◦ ∇

¶

are 2−nilpotent fields.

Proof. (i)=⇒(ii), (i)=⇒(iii), (i)=⇒(iv), (i)⇐⇒(v), (i)⇐⇒(vi) are obvious.

(iv)=⇒(i). BecauseAλ(Z, X) =fXZ,∀X, Z ∈ X(M), it follows that there exists

an 1−form θ onM such that θ(X) = fX. We have, therefore

λ

A(Z, X) =θ(X)Z,

∀X, Z∈ X(M). BecauseAλ is symmetric, we getθ(X)Z=θ(Z)X,∀X, Z∈ X(M). In local coordinates, the last equality is writtenθiδj_k =θkδ_ij. If we make j =i and

sum, we obtain (n−1)θk= 0, soθ= 0 and, finally,

λ

∇=

◦ ∇.

(iii)=⇒(i). We have Aλ(Z, X) = ω(Z)X, ∀X, Z ∈ X(M). Because Aλ(Z, X) =

λ

A(X, Z), ∀X, Z ∈ X(M), we obtain ω(Z)X = ω(X)Z, ∀X, Z ∈ X(M), which

impliesω= 0, i.e. ∇λ = ◦ ∇.

(ii)=⇒(i). Using the hypothesis, we have

λ

From here, it follows that there exists an 1−formη, defined onM, such thatη(X) =

fX. Therefore, we haveAλ(Z, X) =η(X)Z+ω(Z)X,∀X, Z∈ X(M). BecauseAλ is symmetric, we find that

(η(X)−ω(X))Z= (η(Z)−ω(Z))X, ∀X, Z∈ X(M).

In local coordinates, the last equality may be written

(ηi−ωi)δj_k−(ηk−ωk)δj_i = 0.

If we assignj=kand sum, we get

(n−1) (ηi−ωi) = 0,∀i∈ {1,2, ..., n}.

Thereforeη=ω and we have

λ

∇ZX=

◦

∇ZX+η(X)Z+η(Z)X,∀X, Z∈ X(M).

The last equality shows that the symmetric linear connections∇λ and ◦

∇ admit the

same geodesics. Using Theorem 3.1, we obtain∇λ = ◦

∇. ¤

Remark 3.2. In the following, we will denote byC∈ T1

3 (M) the curvature conformal Weyl tensor. LetCi

jkl the components ofC in a system of local coordinates. Then,

we have ([11], [12])

Cjkli =

◦

Rijkl−

1

n−2

µ δki

◦

Rjl−δli

◦

Rjkgjkgis

◦

Rsl+gjlgis

◦

Rsk ¶

+ g

rs

◦

Rrs

(n−1) (n−2)

¡

δikgjl−δ i lgjk

¢ .

Theorem 3.3. With the above notations, we suppose that

(i) the1−formw(g)is exact;

(ii)

◦

Rp:TpM ×TpM ×TpM −→TpM is surjective,∀p∈M;

(iii) the curvature conformal Weyl tensor is nowhere vanishing, i.e. Cp6= 0,∀p∈M.

If there exists a function f ∈ F(M), f(p) 6= 0, ∀p∈ M, such that Rλ = f

◦

R, then

λ

∇= ◦ ∇.

Proof. In local coordinates, the linear connection∇λ has the components

λ

Γijk= ¯ ¯i

jk ¯

¯+δjiψk+δ i

kψj−gjkψ i

where¯¯_¯i jk

¯ ¯

¯are the Christoffel symbols of the second kind, constructed using the metric componentsgij andψi are the components of 1−form λ

2w(g). Because the 1−form

w(g) is exact, the last equalities imply

(3.30)

From (3.30), we have

(3.32) Rjlλ =

Rlj. Multiplying (3.32) by gjl _{and summing with respect toj}

andl, we obtain

Becausen >3, from (3.32) and (3.33) we deduce that

(3.34) grsψrs = 1

Introducing (3.34), (3.35) in (3.30), we find

λ

are the components of the curvature conformal Weyl tensorC. On the other hand,

λ R=f

◦

Rimplies that

λ

jkl are nowhere vanishing.

Taking this into the account, the relations (3.36) and (3.37) show that we havef = 1,

soRλ = ◦

R. Using Theorem (3.1), we obtain∇λ = ◦

∇. ¤

Theorem 3.4. Let(M, g)be a connectedn−dimensional semi-Riemannian manifold, with n≥3. We consider two Weyl structures w and w′

on the conformal manifold

(M,bg). Let ∇ (resp. ∇′

) be the symmetric conformal Weyl connection, compatible with the Weyl structurew(resp. w′

). Define the family of linear connections

e

C={∇+λ(∇′

− ∇)|λ∈R}.

Forλ∈R, we consider the linear connection

λ

∇=∇+λ(∇′

− ∇)∈Ce. The following assertions are equivalent:

(i) ∇λ =∇;

(ii) the deformation algebraU

µ

(iii) ∇λ and∇ admit the same geodesics;

(iv) all the elements of the deformation algebraU

µ M,

λ

∇ − ∇

¶

are almost principal

fields.

is commutative. It follows thatU

µ

M,∇ − ∇λ

¶

is associative if and only if we have

From here, it follows thatA′

is given by

(3.39) g(A′

(X, Y), Z) =ϕ(X)g(Y, Z) +ϕ(Y)g(X, Z)−ϕ(Z)g(X, Y) ,

∀X, Y, Z∈ X(M),

where we used the notation 2ϕ=w′

(g)−w(g). LetA′i

jk(resp. ϕi) the components ofA

′

(resp. ϕ) in a system of local coordinates. Then we can rewrite (3.39) as

(3.40) A′i

jk=δ i jϕk+δ

i

kϕj−gjkϕ i

,

whereϕi_=gik_{ϕk. In local coordinates, (3.38) can be written} ¡

δkiϕl−δ i lϕk

¢

ϕj+¡δilgjk−δ i kgjl

¢

ϕsϕs+ (gjlϕk−gjkϕl)ϕi= 0.

Fori=k, we sum and obtain

(3.41) (n−2) (ϕjϕl−gjlϕsϕs_{) = 0.}

Becausen≥3, from (3.41) we get

(3.42) ϕjϕl−gjlϕsϕs= 0.

Multiplying (3.42) bygjl _{and summing with respect to j and l, we have ϕsϕ}s _{= 0.}

Taking into account this, from (3.42) followsϕjϕl= 0,∀j, l∈ {1,2, ..., n}. We obtain

ϕ= 0, i.e. w=w′

. From (3.39) we findA′

= 0, soAλ = 0, i.e. ∇λ =∇.

(iii)=⇒(i). Because the linear connections∇λ and∇are symmetric, it follows that they admit the same geodesics if and only if there exists an 1−formσ, defined onM, such that

(3.43) ∇λXY =∇XY +σ(X)Y +σ(Y)X,∀X, Y ∈ X(M).

From (3.43) we get

(3.44) λA′

(X, X) = 2σ(X)X,∀X∈ X(M).

Forλ= 0, we obtainσ= 0 and from (3.34) we have ∇λ =∇. We shall consider the case forλ6= 0. ForY =X, from (3.39), we find

(3.45) g(A′

(X, X), Z) = 2ϕ(X)g(X, Z)−ϕ(Z)g(X, X) , ∀X, Z∈ X(M).

From (3.44) and (3.45), we get

(3.46) {2σ(X)−2λϕ(X)}g(X, Z) +λϕ(Z)g(X, X) = 0,∀X, Z∈ X(M).

Becausen≥3, for anyp∈M and anyZp∈TpM− {0}, there existsXp∈TpM− {0} such that we have

From (3.46) and (3.47) we obtain ϕp(Zp) = 0, for any Zp ∈ TpM − {0} and any

p ∈ M. Therefore, ϕ = 0, so w′

= w. From (3.39) follows g(A′

(X, Y), Z) = 0, ∀X, Y, Z ∈ X(M). Because the metric g is non-degenerate, from the last equality

we find A′

(X, Y) = 0,∀X, Y ∈ X(M), so A′

= 0. In conclusion, Aλ =λA′

= 0, i.e.

λ

∇=∇.

(iv)=⇒(i). Because all elements of the algebraU

µ M,

λ

∇ − ∇

¶

are almost principal

fields, it follows that for anyX ∈ X(M) there exists a functionfX∈ F(M) and an 1−formω∈Λ1_{(M) such that}

(3.48)

λ

A(Z, X) =fXZ+ω(Z)X,∀Z, X ∈ X(M).

From (3.48), there exists an 1−formη∈Λ1_{(M) such thatη(X) =fX. From (3.48),} we obtain

(3.49)

λ

A(Z, X) =η(X)Z+ω(Z)X,∀Z, X∈ X(M).

Because the algebraU

µ M,

λ

∇ − ∇

¶

is abelian, from the last equality follows η =ω

and from (3.49) we obtain

λ

∇ZX=∇ZX+η(X)Z+η(Z)X,∀Z, X∈ X(M).

The last equality shows that the symmetric and linear connections∇λ and ∇ admit

the same geodesics. From here,∇λ =∇. ¤

References

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[4] R. Gardner,The method of equivalence and its applications, SIAM, Philadelphia, 1989.

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Author’s address:

Liviu Nicolescu, Gabriel-Teodor Pripoae

University of Bucharest, Faculty of Mathematics and Informatics, Str. Academiei nr.14, RO- 010014, Bucharest, Romania.

E-mail: gpripoae@fmi.unibuc.ro

Virgil Damian