Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian

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 ∇ ∈ C, we investigate some^λ 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 Balkan Society of Geometers, Geometry Balkan Press 2011.

2 Definitions and notations

LetMbe ann−dimensionalC^∞differentiable manifold. We denote byF(M) the ring of real-valued functions of classC^∞defined onM and byT_s^r(M) theF(M)−module of tensor fields of type (r, s). In particular, forT₀¹(M) (respectively T₁⁰(M)) we use the notationX(M) (respectively Λ¹(M)).

Let A ∈ T₂¹(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 ∈ T₂¹(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) A³

Z, 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={e^ug|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^ug) =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=∇ −∇ ∈ T^◦ ₂¹(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), for all X, Y, Z ∈ X(M). A natural problem is to deduce properties of two semi- Riemannian manifolds from properties of the deformation algebra of their Levi-Civita

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 ¹₂w(g), respectively. Then (2.3) can be written

(2.4) Aⁱ_jk=δⁱ_juk+δⁱ_kuj−gjkuⁱ. We define the family of linear connections ([8])

C=

½λ

∇:=∇^◦ +λA|λ∈R

¾ .

(In the affine space of all linear connections on M, C is the ”affine line” passing through the ”point”∇^◦ and of ”direction”A).

LetR^◦ andR^λ be the curvature tensor field of the connection∇^◦ and∇, respectively.^λ

3 The main results

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

(i) ∇^λ =∇;^◦

(ii) R^λ =R, if^◦ R^◦p:TpM×TpM×TpM −→TpM is surjective, ∀p∈M; (iii) ∇^λ and∇^◦ admit the same geodesics;

(iv) the algebraU µ

M,∇ −^λ ∇^◦

¶

is associative;

(v) if R^◦p : TpM ×TpM ×TpM −→ TpM is surjective, ∀p ∈ M and the 1−form w(g)is exact, then ∇^λ and∇^◦ have the same Ricci tensor.

Proof. (i)=⇒(ii), (i)=⇒(iii), (i)=⇒(iv), (i)=⇒(v) are trivial. (ii)=⇒(i). From (ii),

∇λX

Rλ =∇^λX

R,◦ ∀X ∈ X(M). Then,∀X, Y, Z, V ∈ X(M), (∇^λX

R)(Y, Z, Vλ ) = (∇^◦X

R)(Y, Z, V◦ ) +A(X,^λ R(Y, Z^◦ )V)−R(^◦ A(X, Y^λ ), Z)V

−R(Y,^◦ A(X, Z))V^λ −R(Y, Z^◦ )A(X, V^λ ), (3.1)

where we denoted byA^λ =∇ − ∇^λ =λA. Similarly, we obtain

(∇^λ_YR)(Z, X, V^λ ) = (∇^◦_YR)(Z, X, V^◦ ) +A(Y,^λ R(Z, X^◦ )V)−R(^◦ A(Y, Z), X)V^λ

−R(Z,^◦ A(Y, X))V^λ −R(Z, X^◦ )A(Y, V^λ ), (3.2)

(∇^λZ

R)(X, Y, Vλ ) = (∇^◦Z

R)(X, Y, V◦ ) +A(Z,^λ R(X, Y^◦ )V)−R(^◦ A(Z, X^λ ), Y)V

−R(X,^◦ A(Z, Y^λ ))V −R(X, Y^◦ )A(Z, V^λ ).

(3.3)

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

½ A

µ

X,R^◦(Y, Z)V

¶ +A

µ

Y,R^◦(Z, X)V

¶ +A

µ

Z,R^◦ (X, Y)V

¶

−R^◦(Y, Z)A(X, V)−R^◦(Z, X)A(Y, V)−R^◦ (X, Y)A(Z, V)

¾

= 0.

From this, we deduceλ= 0, so we proved (i); hence A

µ

X,R^◦(Y, Z)V

¶ +A

µ

Y,R^◦ (Z, X)V

¶ +A

µ

Z,R^◦(X, Y)V

¶

(3.4) −R^◦ (Y, Z)A(X, V)−R^◦(Z, X)A(Y, V)−R^◦(X, Y)A(Z, V) = 0.

In local coordinates, (3.4) is written (3.5)

µ

δ_i^sR^◦r_ljk+δ^s_kR^◦r_lij+δ^s_jR^◦r_lki

¶ ur+

µ gil

R◦^s_rjk+gjl

R◦^s_rki+gkl

R◦^s_rij

¶ u^r= 0.

We makes:=i and sum with respect toi; it follows (3.6) (n−2)R^◦r_ljkur+

µ◦

Rlrjk−gjl

R◦rk+gkl

R◦rj

¶ u^r= 0,

where R^◦jl =R^◦i_jil are the components of Ricci tensor. Multiplying (3.6) by g^jl and summing with respect toj andl, it follows

(3.7) (n−2)R^◦rku^r= 0.

From (3.6) and (3.7) we get

(3.8) (n−3)R^◦r_ljkur= 0.

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

(3.9) R^◦r_ljkur= 0.

The relations (3.9) show that, for everyp∈M, (3.10) (w(g))_p

µ_◦

Rp(Xp, Yp)Zp

¶

= 0,∀Xp, Yp, Zp∈TpM.

Because R^◦p : TpM ×TpM ×TpM −→ TpM is surjective, from (3.10) we obtain w(g))_p= 0, ∀p∈M, sow(g) = 0 and using (2.3), we find

(3.11) g(A(X, Y), Z) = 0, ∀X, Y, Z∈ X(M).

Becausegis non-degenerate, from (3.11) it followsA= 0, soA^λ =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 fromA^λ =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, soA^λ = 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) Aⁱ_skA^s_jl−Aⁱ_slA^s_jk= 0.

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

(3.19) δ_kⁱujul−δⁱ_lujuk+gjluⁱuk−gjkuⁱul+δⁱ_lgjkusu^s−δ_kⁱgjlu^sus= 0.

In (3.19), we makei:=kand sum with respect toi; it follows (n−2) (ujul−gjlusu^s) = 0.

Becausen >3, we obtain

(3.20) ujul−gjlu^sus= 0.

Multiplying (3.20) byg^jl and summing with respect toj andl, we get

(3.21) u^sus= 0.

From (3.20) and (3.21) we obtainujul = 0, ∀j, l ∈ {1,2, ..., n}. Thereforew(g) = 0 and, from (2.3), it followsA= 0, so∇^λ =∇.^◦

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

(3.22) Γⁱ_jk=¯

¯ⁱ_jk¯

¯+δ_jⁱuk+δ_kⁱuj−gjkuⁱ, where

¯¯

¯ⁱ_jk

¯¯

¯are the Christoffel symbols of the second kind, constructed using the me- tric components gij. From (3.22) and from ∇^λ = ∇^◦ +λ

µ

∇ −∇^◦

¶

, we obtain the components of the linear connection∇:^λ

(3.23) Γ^λi_jk=¯

¯ⁱ_jk¯

¯+δ_jⁱψk+δ_kⁱψj−gjkψⁱ,

whereψi =λui. The curvature tensor of the connection∇^λ has the components (3.24) R^λi_jkl=R^◦i_jkl+δⁱ_j(ψ_kl−ψ_lk) +δ_kⁱψ_jl−δ_lⁱψ_jk−g_jkψ_lⁱ+g_jlψ_kⁱ, where we denoted

ψjl =∂ψj

∂x^l +¯

¯^r_jl¯

¯ψr+ψjψl−1 2φ²gjl,

ψ_lⁱ=g^ijψjl,φ²=grsψ^rψ^s=ψiψⁱ. If we assigni=kand sum, from (3.24) we obtain

(3.25) R^λjl=R^◦jl+ (n−1)ψjl−ψlj+gjlϕ,

whereR^λjl=R^λi_jil,ϕ=ψⁱ_i=g^jlψjl,R^◦jl=R^◦i_jil. Because the 1−formw(g) is exact, it follows thatψjl=ψlj. From (3.25), we have

(3.26) R^λjl=R^◦jl+ (n−2)ψjl+gjlϕ.

Using the hypothesis,R^λjl=R^◦jl. Therefore, from (3.26) we get

(3.27) (n−2)ψjl+gjlϕ= 0.

Multiplying (3.27) byg^jl and summing with respect toj andl, it follows

(3.28) ϕ= 0.

From (3.27) and (3.28), we have

(3.29) ψjl= 0, ψ_lⁱ= 0.

The relations (3.28) and (3.29) show that Rλⁱ_jkl=R^◦i_jkl.

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

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

”R^◦p :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). BecauseA^λ(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 A^λ(Z, X) =θ(X)Z,

∀X, Z∈ X(M). BecauseA^λ is symmetric, we getθ(X)Z=θ(Z)X,∀X, Z∈ X(M).

In local coordinates, the last equality is writtenθiδ^j_k =θkδ_i^j. If we make j =i and sum, we obtain (n−1)θk= 0, soθ= 0 and, finally,∇^λ =∇.^◦

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

Aλ(Z, X) =fXZ+ω(Z)X,∀X, Z∈ X(M).

From here, it follows that there exists an 1−formη, defined onM, such thatη(X) = fX. Therefore, we haveA^λ(Z, X) =η(X)Z+ω(Z)X,∀X, Z∈ X(M). BecauseA^λ 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)δ^j_k−(ηk−ωk)δ^j_i = 0.

If we assignj=kand sum, we get

(n−1) (ηi−ωi) = 0,∀i∈ {1,2, ..., n}. Thereforeη=ω and we have

∇λZX=∇^◦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∈ T₃¹(M) the curvature conformal Weyl tensor. LetC_jklⁱ the components ofC in a system of local coordinates. Then, we have ([11], [12])

C_jklⁱ = R^◦i_jkl− 1 n−2

µ

δ_kⁱR^◦jl−δ_lⁱR^◦jkgjkg^isR^◦sl+gjlg^isR^◦sk

¶

+ g^rsR^◦rs

(n−1) (n−2)

¡δⁱ_kgjl−δ_lⁱgjk

¢.

Theorem 3.3. With the above notations, we suppose that (i) the1−formw(g)is exact;

(ii) R^◦_p:T_pM ×T_pM ×T_pM −→T_pM 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 R^λ = fR, then^◦

∇λ =∇.^◦

Proof. In local coordinates, the linear connection∇^λ has the components Γλⁱ_jk=¯

¯ⁱ

jk

¯¯+δ_jⁱψk+δ_kⁱψj−gjkψⁱ,

where

¯¯

¯ⁱ_jk

¯¯

¯are the Christoffel symbols of the second kind, constructed using the metric componentsgij andψi are the components of 1−form ^λ₂w(g). Because the 1−form w(g) is exact, the last equalities imply

(3.30) R^λi_jkl=R^◦i_jkl+δ_kⁱψjl−δ_lⁱψjk−gjkψⁱ_l+gjlψ_kⁱ, where

(3.31) ψjl=∂ψj

∂x^l +¯

¯^r

jl

¯¯ψr+ψjψl−1

2gjlψ^sψs=ψlj,ψⁱ_k=g^ijψjk. From (3.30), we have

(3.32) R^λjl=R^◦jl+ (n−2)ψjl+gjlg^rsψrs,

where R^λjl =R^λk_jkl = R^λlj. Multiplying (3.32) by g^jl and summing with respect toj andl, we obtain

(3.33) g^jlR^λjl =g^jlR^◦jl+ 2 (n−1)g^rsψrs. Becausen >3, from (3.32) and (3.33) we deduce that

(3.34) g^rsψrs = 1

2 (n−1) µλ

Rrs−R^◦rs

¶ g^rs

(3.35) ψjl= 1 n−2

µλ

Rjl−R^◦jl

¶

− gjl

2 (n−1) (n−2)g^rs µλ

Rrs−R^◦rs

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

Rλⁱ_jkl− 1 n−2

µ

δⁱ_kR^λjl−δⁱ_lR^λjk+gjkg^isR^λsl+gjlg^isR^λsk

¶

(3.36) + g^rsR^λrs

(n−1) (n−2)

¡δ_kⁱgjl−δⁱ_lgjk

¢=C_jklⁱ ,

where

C_jklⁱ = R^◦i_jkl− 1 n−2

µ

δⁱ_kR^◦jl−δⁱ_lR^◦jk−gjkg^isR^◦sl+gjlg^isR^◦sk

¶

+ g^rsR^◦rs

(n−1) (n−2)

¡δ_kⁱgjl−δⁱ_lgjk

¢

are the components of the curvature conformal Weyl tensorC. On the other hand, Rλ =fR^◦ implies that

Rλⁱ_jkl=fR^◦i_jkl,R^λjl=fR^◦jl, g^rsR^λrs=f g^rsR^◦rs,

Rλⁱ_jkl− 1 n−2

µ

δⁱ_kR^λjl−δⁱ_lR^λjk−gjkg^isR^λsl+gjlg^isR^λsk

¶

(3.37) + g^rsR^λrs

(n−1) (n−2)

¡δⁱ_kgjl−δ_lⁱgjk

¢=f C_jklⁱ .

BecauseCp 6= 0,∀p∈M, it follows that the functions C_jklⁱ are nowhere vanishing.

Taking this into the account, the relations (3.36) and (3.37) show that we havef = 1, soR^λ =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 w⁰ on the conformal manifold (M,bg). Let ∇ (resp. ∇⁰) be the symmetric conformal Weyl connection, compatible with the Weyl structurew (resp. w⁰). Define the family of linear connections

C=e {∇+λ(∇⁰− ∇)|λ∈R}.

Forλ∈R, we consider the linear connection∇^λ =∇+λ(∇⁰− ∇)∈C. The followinge assertions are equivalent:

(i) ∇^λ =∇;

(ii) the deformation algebraU µ

M,∇ − ∇^λ

¶

is associative;

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

(iv) all the elements of the deformation algebraU µ

M,∇ − ∇^λ

¶

are almost principal fields.

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

(ii)=⇒(i) The algebraU µ

M,∇ − ∇^λ

¶

is commutative. It follows thatU µ

M,∇ − ∇^λ

¶

is associative if and only if we have (3.38) A^λ

µ

X,A^λ(Y, Z)

¶

−A^λ µ

Y,A^λ(X, Z)

¶

= 0, ∀X, Y, Z∈ X(M),

whereA^λ =∇ − ∇^λ =λA⁰,A⁰ =∇ − ∇.^λ

Forλ= 0 the assertion is trivial. We supposeλ6= 0. The linear connection ∇⁰ is defined by

2g(∇⁰_XY, Z) = X(g(Y, Z)) +Y(g(X, Z))−Z(g(X, Y)) +w⁰(g)(X)g(Y, Z) + +w⁰(g)(Y)g(X, Z)−w⁰(g)(Z)g(X, Y) +g([X, Y], Z) + +g([Z, X], Y)−g([Y, Z], X),∀X, Y, Z∈ X(M) .

From here, it follows thatA⁰ is given by

(3.39) g(A⁰(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=w⁰(g)−w(g).

LetA⁰ⁱ_jk(resp. ϕi) the components ofA⁰ (resp. ϕ) in a system of local coordinates.

Then we can rewrite (3.39) as

(3.40) A⁰ⁱ_jk=δⁱ_jϕk+δ_kⁱϕj−gjkϕⁱ, whereϕⁱ=g^ikϕk. In local coordinates, (3.38) can be written

¡δ_kⁱϕl−δⁱ_lϕk

¢ϕj+¡

δⁱ_lgjk−δ_kⁱgjl

¢ϕsϕ^s+ (gjlϕk−gjkϕl)ϕⁱ= 0.

Fori=k, we sum and obtain

(3.41) (n−2) (ϕjϕl−gjlϕsϕ^s) = 0.

Becausen≥3, from (3.41) we get

(3.42) ϕjϕl−gjlϕsϕ^s= 0.

Multiplying (3.42) byg^jl 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=w⁰. From (3.39) we findA⁰ = 0, soA^λ = 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) ∇^λ_XY =∇_XY +σ(X)Y +σ(Y)X,∀X, Y ∈ X(M). From (3.43) we get

(3.44) λA⁰(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(A⁰(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

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

From (3.46) and (3.47) we obtain ϕp(Zp) = 0, for any Zp ∈ TpM − {0} and any p ∈ M. Therefore, ϕ = 0, so w⁰ = w. From (3.39) follows g(A⁰(X, Y), Z) = 0,

∀X, Y, Z ∈ X(M). Because the metric g is non-degenerate, from the last equality we find A⁰(X, Y) = 0,∀X, Y ∈ X(M), so A⁰ = 0. In conclusion, A^λ =λA⁰ = 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) A^λ(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) A^λ(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

∇λZX=∇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 =∇. ¤

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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: [email protected] Virgil Damian

Politehnica University of Bucharest, Faculty of Applied Sciences, Splaiul Independentei 313, RO-060042, Bucharest, Romania.

E-mail: [email protected]