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On some families of linear connections
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
◦
Abstract. On a Weyl manifold (M, gb, w), we consider ∇ the Levi-Civita
connection associated to a metric g ∈ gb , ∇ the symmetric connection,
compatible with the Weyl structure w and the family of linear connections
λ
◦
λ
◦
C ={∇ := ∇ + λ(∇ − ∇) | λ ∈ R}. For ∇ ∈ C, we investigate some
λ
◦
properties of the deformation algebra U(M, ∇ − ∇). Next, we study the
◦
λ
case when ∇ and ∇ determine the same Ricci tensor and the case when
◦
λ
the curvature tensors of the connections ∇ and ∇ 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, differential, differential affine, metric, etc invariants determine some structure on a
manifold, up to a homeomorphism, diffeomorphism, isometry, etc. Usually, one compares 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 explicitely 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.
°
On some families of linear connections
2
99
Definitions and notations
Let M be an n−dimensional C ∞ differentiable manifold. We denote by F (M ) the ring
of real-valued functions of class C ∞ defined on M and by Tsr (M ) the F (M ) −module
of tensor fields of type (r, s). In particular, for T01 (M ) (respectively T10 (M )) we use
the notation X (M ) (respectively Λ1 (M )).
Let A ∈ T21 (M ).The F (M ) −module X (M ) becomes an F (M ) −algebra, denoted by U (M, A), with the product of two vector fields X and Y defined by
(2.1)
X ◦ Y = A (X, Y ) .
In particular,
¡
¢
¡ if A = ∇¢ − ∇, where ∇ and ∇ are two arbitrary linear connections
on M , 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 function f ∈ F (M )
such that ([7])
³
´
(2.2)
A Z, X (m) = f Z + ω (Z) X, ∀Z ∈ X (M ) ,
where X (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). If m = 1 and f = 0, then X is a principal field in the algebra U (M, A). If
m = 1 and ω = 0, then X is almost special field. If m = 1, f = 0 and ω = 0, then X
is special field. If A (X, X) = 0, then X is 2−nilpotent field.
Let (M, g) be now an n−dimensional semi-Riemannian manifold and let gb the
conformal structure generated by g, i.e. gb = {eu g | u ∈ F (M )}.
Let w be a Weyl structure on the conformal manifold (M, gb), i.e. an application
w : gb −→ Λ1 (M ), which verifies ([1], [5], [6]) w (eu g) = w (g) − du, ∀u ∈ F (M ). The
triple (M, gb, w) is called a Weyl manifold.
Let (M, gb, w) be a Weyl manifold. A linear connection ∇ on M is called compatible
with the Weyl structure w if ∇X g + w (g) (X) g = 0, for all X ∈ X (M ).
It is well known that there exists an unique symmetric linear connection ∇, defined
on M , compatible with the Weyl structure ([10]). This linear connection ∇ is called
the Weyl conformal connection and is defined by ([9])
2g(∇X Y, 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 all X, Y, Z ∈ X (M ). Let ∇ be the Levi-Civita connection associated to g and let
◦
A = ∇ − ∇ ∈ T21 (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 semiRiemannian manifolds from properties of the deformation algebra of their Levi-Civita
100
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
connections; a remarkable particular case is when the latter ones coincide (see also
[2]).
Let gij , Aijk and ui the components, in a system of local coordinates, of g, A and
of the 1−form 12 w (g), respectively. Then (2.3) can be written
Aijk = δji uk + δki uj − gjk ui .
(2.4)
We define the family of linear connections ([8])
½
¾
λ
◦
C = ∇ := ∇ + λA | λ ∈ R .
(In the affine space of all linear connections on M , C is the ”affine line” passing
◦
through the ”point” ∇ and of ”direction” A).
λ
◦
λ
◦
Let R and R be the curvature tensor field of the connection ∇ and ∇, respectively.
3
The main results
Theorem 3.1. Let M be an n−dimensional connected manifold, n > 3. The following
assertions are equivalent:
λ
◦
λ
◦
(i) ∇ = ∇;
◦
(ii) R = R, if Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, ∀p ∈ M ;
λ
◦
(iii) ∇ and ∇ admit the same geodesics;
µ
¶
λ
◦
(iv) the algebra U M, ∇ − ∇ is associative;
◦
(v) if Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, ∀p ∈ M and the 1−form
λ
◦
w (g) is exact, then ∇ and ∇ have the same Ricci tensor.
Proof. (i)=⇒(ii), (i)=⇒(iii), (i)=⇒(iv), (i)=⇒(v) are trivial. (ii)=⇒(i). From (ii),
λ
λ
λ
◦
∇X R = ∇X R, ∀X ∈ X (M ). Then, ∀X, Y, Z, V ∈ X (M ),
λ
λ
λ
◦
◦
◦
◦
λ
(∇X R)(Y, Z, V ) = (∇X R)(Y, Z, V ) + A(X, R(Y, Z)V ) − R(A(X, Y ), Z)V
λ
◦
(3.1)
λ
◦
−R(Y, A(X, Z))V − R(Y, Z)A(X, V ),
λ
λ
where we denoted by A = ∇ − ∇ = λA. Similarly, we obtain
λ
λ
◦
λ
◦
◦
◦
λ
(∇Y R)(Z, X, V ) = (∇Y R)(Z, X, V ) + A(Y, R(Z, X)V ) − R(A(Y, Z), X)V
◦
(3.2)
λ
◦
λ
−R(Z, A(Y, X))V − R(Z, X)A(Y, V ),
101
On some families of linear connections
λ
λ
◦
(∇Z R)(X, Y, V )
λ
◦
λ
◦
(3.3)
◦
◦
λ
= (∇Z R)(X, Y, V ) + A(Z, R(X, Y )V ) − R(A(Z, X), Y )V
λ
◦
−R(X, A(Z, Y ))V − R(X, Y )A(Z, V ).
Using the Bianchi identities, from (3.1), (3.2) and (3.3), it follows
½ µ
¶
µ
¶
µ
¶
◦
◦
◦
λ A X, R (Y, Z) V + A Y, R (Z, X) V + A 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 deduce λ = 0, so we proved (i); hence
µ
¶
µ
¶
µ
¶
◦
◦
◦
A X, R (Y, Z) V + A Y, R (Z, X) V + A 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
¶
µ
¶
µ
◦
◦
◦
◦
◦
◦
(3.5)
δis Rrljk + δks Rrlij + δjs Rrlki ur + gil Rsrjk + gjl Rsrki + gkl Rsrij ur = 0.
We make s := i and sum with respect to i; it follows
¶
µ
◦
◦
◦
◦
r
(3.6)
(n − 2) Rljk ur + Rlrjk − gjl Rrk + gkl Rrj ur = 0,
◦
◦
where Rjl = Rijil are the components of Ricci tensor. Multiplying (3.6) by g jl and
summing with respect to j and l, it follows
◦
(n − 2) Rrk ur = 0.
(3.7)
From (3.6) and (3.7) we get
◦
(n − 3) Rrljk ur = 0.
(3.8)
Because we supposed n > 3, from (3.8) we have
◦
Rrljk ur = 0.
(3.9)
The relations (3.9) show that, for every p ∈ M ,
µ
¶
◦
(3.10)
(w (g))p Rp (Xp , Yp ) Zp = 0, ∀Xp , Yp , Zp ∈ Tp M.
◦
Because Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, from (3.10) we obtain
w (g))p = 0, ∀p ∈ M , so w (g) = 0 and using (2.3), we find
(3.11)
g (A (X, Y ) , Z) = 0, ∀X, Y, Z ∈ X (M ) .
102
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
λ
λ
◦
Because g is non-degenerate, from (3.11) it follows A = 0, so A = λA = 0, i.e. ∇ = ∇.
λ
◦
(iii)=⇒(i) The linear connections ∇ and ∇ are symmetric, hence they admit the
same geodesics if and only if there exists an 1−form on M such that
(3.12)
λ
◦
∇X Y = ∇X Y + ω (X) Y + ω (Y ) X, ∀X, Y ∈ X (M ) .
λ
From (3.12) and from A = λA, we obtain
(3.13)
λA (X, X) = 2ω (X) X, ∀X ∈ X (M ) .
For Y = 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 all X, Z ∈ X (M ).
Because n > 3, for every p ∈ M and every Zp ∈ Tp M − {0}, there exists a vector
Xp ∈ Tp M − {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 ∈ Tp M − {0} , ∀p ∈ M ,
λ
λ
◦
so w (g) = 0 and from (2.3) we obtain that A = 0, so A = 0, i.e. ∇ = ∇. µ
¶
λ
◦
(iv)=⇒(i) For λ = 0 is trivial. We suppose λ 6= 0. Because the algebra U M, ∇ − ∇
µ
¶
λ
◦
is commutative, it follows that U M, ∇ − ∇ 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
Aisk Asjl − Aisl Asjk = 0.
(3.18)
From (3.18), using (2.4), we can write
(3.19)
δki uj ul − δli uj uk + gjl ui uk − gjk ui ul + δli gjk us us − δki gjl us us = 0.
In (3.19), we make i := k and sum with respect to i; it follows
(n − 2) (uj ul − gjl us us ) = 0.
Because n > 3, we obtain
(3.20)
uj ul − gjl us us = 0.
103
On some families of linear connections
Multiplying (3.20) by g jl and summing with respect to j and l, we get
us us = 0.
(3.21)
From (3.20) and (3.21) we obtain uj ul = 0, ∀j, l ∈ {1, 2, ..., n}. Therefore w (g) = 0
λ
◦
and, from (2.3), it follows A = 0, so ∇ = ∇.
(v)=⇒(i) In local coordinates, the conformal Weyl connection has the components
¯ ¯
(3.22)
Γijk = ¯ijk ¯ + δji uk + δki uj − gjk ui ,
¯ ¯
¯ ¯
where ¯ijk ¯ are the Christoffel symbols of the second kind, constructed using the meµ
¶
λ
◦
◦
tric components gij . From (3.22) and from ∇ = ∇ + λ ∇ − ∇ , we obtain the
λ
components of the linear connection ∇:
λ
¯ ¯
Γijk = ¯ijk ¯ + δji ψk + δki ψj − gjk ψ i ,
(3.23)
λ
where ψi = λui . The curvature tensor of the connection ∇ has the components
λ
◦
Rijkl = Rijkl + δji (ψkl − ψlk ) + δki ψjl − δli ψjk − gjk ψli + gjl ψki ,
(3.24)
where we denoted
ψjl =
1
∂ψj ¯¯r ¯¯
+ jl ψr + ψj ψl − φ2 gjl ,
∂xl
2
ψli = g ij ψjl , φ2 = grs ψ r ψ s = ψi ψ i .
If we assign i = k and sum, from (3.24) we obtain
λ
(3.25)
◦
Rjl = Rjl + (n − 1) ψjl − ψlj + gjl ϕ,
λ
λ
◦
◦
where Rjl = Rijil , ϕ = ψii = g jl ψjl , Rjl = Rijil . Because the 1−form w (g) is exact, it
follows that ψjl = ψlj . From (3.25), we have
λ
(3.26)
◦
Rjl = Rjl + (n − 2) ψjl + gjl ϕ.
λ
◦
Using the hypothesis, Rjl = Rjl . Therefore, from (3.26) we get
(3.27)
(n − 2) ψjl + gjl ϕ = 0.
Multiplying (3.27) by g jl and summing with respect to j and l, it follows
(3.28)
ϕ = 0.
From (3.27) and (3.28), we have
(3.29)
ψjl = 0, ψli = 0.
104
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
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 : Tp M × Tp M × Tp M −→ Tp M 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).
µ
¶
λ
◦
Theorem 3.2. Let U M, ∇ − ∇ be the deformation algebra as above. We suppose
that the manifold M is connected and n > 2. The following assertions are equivalent:
λ
◦
(i) ∇ = ∇;
µ
¶
λ
◦
(ii) all elements of the deformation algebra U M, ∇ − ∇ are almost principal
fields;
(iii) all elements of the deformation algebra U
(iv) all elements of the deformation algebra U
µ
¶
λ
◦
M, ∇ − ∇ are principal fields;
µ
λ
◦
M, ∇ − ∇
¶
are almost special fields;
µ
¶
λ
◦
(v) all elements of the deformation algebra U M, ∇ − ∇ are special fields;
µ
¶
λ
◦
(vi) all elements of the deformation algebra U M, ∇ − ∇ are 2− nilpotent fields.
Proof. (i)=⇒(ii), (i)=⇒(iii), (i)=⇒(iv), (i)⇐⇒(v), (i)⇐⇒(vi) are obvious.
λ
(iv)=⇒(i). Because A (Z, X) = fX Z, ∀X, Z ∈ X (M ), it follows that there exists
λ
an 1−form θ on M such that θ (X) = fX . We have, therefore A (Z, X) = θ (X) Z,
λ
∀X, Z ∈ X (M ). Because A is symmetric, we get θ (X) Z = θ (Z) X, ∀X, Z ∈ X (M ).
In local coordinates, the last equality is written θi δkj = θk δij . If we make j = i and
λ
◦
sum, we obtain (n − 1) θk = 0, so θ = 0 and, finally, ∇ = ∇.
λ
λ
(iii)=⇒(i). We have A (Z, X) = ω (Z) X, ∀X, Z ∈ X (M ). Because A (Z, X) =
λ
A (X, Z), ∀X, Z ∈ X (M ), we obtain ω (Z) X = ω (X) Z, ∀X, Z ∈ X (M ), which
λ
◦
implies ω = 0, i.e. ∇ = ∇.
(ii)=⇒(i). Using the hypothesis, we have
λ
A (Z, X) = fX Z + ω (Z) X, ∀X, Z ∈ X (M ) .
105
On some families of linear connections
From here, it follows that there exists an 1−form η, defined on M , such that η (X) =
λ
λ
fX . Therefore, we have A (Z, X) = η (X) Z + ω (Z) X, ∀X, Z ∈ X (M ). Because A 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 ) δkj − (ηk − ωk ) δij = 0.
If we assign j = k and sum, we get
(n − 1) (ηi − ωi ) = 0, ∀i ∈ {1, 2, ..., n} .
Therefore η = ω and we have
λ
◦
∇Z X = ∇Z X + η (X) Z + η (Z) X, ∀X, Z ∈ X (M ) .
λ
◦
The last equality shows that the symmetric linear connections ∇ and ∇ admit the
λ
◦
same geodesics. Using Theorem 3.1, we obtain ∇ = ∇.
¤
Remark 3.2. In the following, we will denote by C ∈ T31 (M ) the curvature conformal
i
Weyl tensor. Let Cjkl
the components of C in a system of local coordinates. Then,
we have ([11], [12])
i
Cjkl
◦
=
Rijkl
1
−
n−2
¶
µ
◦
◦
◦
◦
is
is
i
i
δk Rjl − δl Rjk gjk g Rsl + gjl g Rsk
◦
¡ i
¢
g rs Rrs
δk gjl − δli gjk .
+
(n − 1) (n − 2)
Theorem 3.3. With the above notations, we suppose that
(i) the 1−form w (g) is exact;
◦
(ii) Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, ∀p ∈ M ;
(iii) the curvature conformal Weyl tensor is nowhere vanishing, i.e. Cp 6= 0, ∀p ∈ M .
λ
◦
If there exists a function f ∈ F (M ), f (p) 6= 0, ∀p ∈ M , such that R = f R, then
λ
◦
∇ = ∇.
λ
Proof. In local coordinates, the linear connection ∇ has the components
λ
¯ ¯
Γijk = ¯ijk ¯ + δji ψk + δki ψj − gjk ψ i ,
106
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
¯ ¯
¯ ¯
where ¯ijk ¯ are the Christoffel symbols of the second kind, constructed using the metric
components gij and ψi are the components of 1−form
w (g) is exact, the last equalities imply
λ
λ
2 w (g).
Because the 1−form
◦
Rijkl = Rijkl + δki ψjl − δli ψjk − gjk ψli + gjl ψki ,
(3.30)
where
(3.31)
ψjl =
∂ψj ¯¯r ¯¯
1
+ jl ψr + ψj ψl − gjl ψ s ψs = ψlj , ψki = g ij ψjk .
l
∂x
2
From (3.30), we have
λ
◦
Rjl = Rjl + (n − 2) ψjl + gjl g rs ψrs ,
(3.32)
λ
λ
λ
where Rjl = Rkjkl = Rlj . Multiplying (3.32) by g jl and summing with respect to j
and l, we obtain
λ
◦
g jl Rjl = g jl Rjl + 2 (n − 1) g rs ψrs .
(3.33)
Because n > 3, from (3.32) and (3.33) we deduce that
¶
µ
λ
◦
1
rs
(3.34)
g ψrs =
Rrs − Rrs g rs
2 (n − 1)
(3.35)
ψjl =
1
n−2
¶
µ
λ
◦
Rjl − Rjl −
µ
¶
λ
◦
gjl
g rs Rrs − Rrs .
2 (n − 1) (n − 2)
Introducing (3.34), (3.35) in (3.30), we find
¶
µ
λ
λ
λ
λ
λ
1
δki Rjl − δli Rjk + gjk g is Rsl + gjl g is Rsk
Rijkl −
n−2
λ
(3.36)
+
¡ i
¢
g rs Rrs
i
δk gjl − δli gjk = Cjkl
,
(n − 1) (n − 2)
where
i
Cjkl
◦
=
Rijkl
1
−
n−2
¶
µ
◦
◦
◦
◦
is
is
i
i
δk Rjl − δl Rjk − gjk g Rsl + gjl g Rsk
◦
¢
¡ i
g rs Rrs
δk gjl − δli gjk
+
(n − 1) (n − 2)
are the components of the curvature conformal Weyl tensor C. On the other hand,
λ
◦
R = f R implies that
λ
◦
λ
◦
λ
◦
Rijkl = f Rijkl , Rjl = f Rjl , g rs Rrs = f g rs Rrs ,
107
On some families of linear connections
λ
Rijkl −
1
n−2
µ
¶
λ
λ
λ
λ
δki Rjl − δli Rjk − gjk g is Rsl + gjl g is Rsk
λ
(3.37)
+
¡ i
¢
g rs Rrs
i
δk gjl − δli gjk = f Cjkl
.
(n − 1) (n − 2)
i
Because Cp 6= 0, ∀p ∈ M , it follows that the functions Cjkl
are nowhere vanishing.
Taking this into the account, the relations (3.36) and (3.37) show that we have f = 1,
λ
λ
◦
◦
so R = R. Using Theorem (3.1), we obtain ∇ = ∇.
¤
Theorem 3.4. Let (M, g) be a connected n−dimensional semi-Riemannian manifold,
with n ≥ 3. We consider two Weyl structures w and w′ on the conformal manifold
(M, gb). Let ∇ (resp. ∇′ ) be the symmetric conformal Weyl connection, compatible
with the Weyl structure w (resp. w′ ). Define the family of linear connections
e {∇ + λ (∇′ − ∇) | λ ∈ R} .
C=
λ
e The following
For λ ∈ R, we consider the linear connection ∇ = ∇ + λ (∇′ − ∇) ∈ C.
assertions are equivalent:
λ
(i) ∇ = ∇;
(ii) the deformation algebra U
µ
¶
λ
M, ∇ − ∇ is associative;
λ
(iii) ∇ and ∇ admit the same geodesics;
(iv) all the elements of the deformation algebra U
µ
¶
λ
M, ∇ − ∇ are almost principal
fields.
Proof. (i)=⇒(ii), (i)=⇒(iii),µ(i)=⇒(iv) ¶
are obvious.
µ
¶
λ
λ
(ii)=⇒(i) The algebra U M, ∇ − ∇ is commutative. It follows that U M, ∇ − ∇
is associative if and only if we have
µ
¶
µ
¶
λ
λ
λ
λ
(3.38)
A X, A (Y, Z) − A Y, A (X, Z) = 0, ∀X, Y, Z ∈ X (M ) ,
λ
λ
λ
where A = ∇ − ∇ = λA′ , A′ = ∇ − ∇.
For λ = 0 the assertion is trivial. We suppose λ 6= 0. The linear connection ∇′ is
defined by
2g(∇′X Y, 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), ∀X, Y, Z ∈ X (M ) .
108
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
From here, it follows that A′ 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).
′
Let A′i
jk (resp. ϕi ) the components of A (resp. ϕ) in a system of local coordinates.
Then we can rewrite (3.39) as
i
i
i
A′i
jk = δj ϕk + δk ϕj − gjk ϕ ,
(3.40)
where ϕi = g ik ϕk . In local coordinates, (3.38) can be written
¡ i
¢
¡
¢
δk ϕl − δli ϕk ϕj + δli gjk − δki gjl ϕs ϕs + (gjl ϕk − gjk ϕl ) ϕi = 0.
For i = k, we sum and obtain
(n − 2) (ϕj ϕl − gjl ϕs ϕs ) = 0.
(3.41)
Because n ≥ 3, from (3.41) we get
ϕj ϕl − gjl ϕs ϕs = 0.
(3.42)
Multiplying (3.42) by g jl and summing with respect to j and l, we have ϕs ϕs = 0.
Taking into account this, from (3.42) follows ϕj ϕl = 0, ∀j, l ∈ {1, 2, ..., n}. We obtain
λ
λ
ϕ = 0, i.e. w = w′ . From (3.39) we find A′ = 0, so A = 0, i.e. ∇ = ∇.
λ
(iii)=⇒(i). Because the linear connections ∇ and ∇ are symmetric, it follows that
they admit the same geodesics if and only if there exists an 1−form σ, defined on M ,
such that
(3.43)
λ
∇X Y = ∇X Y + 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 ) .
λ
For λ = 0, we obtain σ = 0 and from (3.34) we have ∇ = ∇. We shall consider the
case for λ 6= 0. For Y = 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 ) .
Because n ≥ 3, for any p ∈ M and any Zp ∈ Tp M − {0}, there exists Xp ∈ Tp M − {0}
such that we have
(3.47)
gp (Xp , Zp ) = 0, gp (Xp , Xp ) 6= 0.
109
On some families of linear connections
From (3.46) and (3.47) we obtain ϕp (Zp ) = 0, for any Zp ∈ Tp M − {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
λ
we find A′ (X, Y ) = 0, ∀X, Y ∈ X (M ), so A′ = 0. In conclusion, A = λA′ = 0, i.e.
λ
∇ = ∇.
(iv)=⇒(i). Because all elements of the algebra U
µ
¶
λ
M, ∇ − ∇ are almost principal
fields, it follows that for any X ∈ X (M ) there exists a function fX ∈ F (M ) and an
1−form ω ∈ Λ1 (M ) such that
λ
(3.48)
A (Z, X) = fX Z + ω (Z) X, ∀Z, X ∈ X (M ) .
From (3.48), there exists an 1−form η ∈ Λ1 (M ) such that η (X) = fX . From (3.48),
we obtain
λ
(3.49)
A (Z, X) = η (X) Z + ω (Z) X, ∀Z, X ∈ X (M ) .
µ
¶
λ
Because the algebra U M, ∇ − ∇ is abelian, from the last equality follows η = ω
and from (3.49) we obtain
λ
∇Z X = ∇Z X + η (X) Z + η (Z) X, ∀Z, X ∈ X (M ) .
λ
The last equality shows that the symmetric and linear connections ∇ and ∇ admit
λ
the same geodesics. From here, ∇ = ∇.
¤
References
[1] B. Alexandrov and S. Ivanov, Weyl structures with positive Ricci tensor, Diff.
Geom. Appl. 18, 3 (2003), 343-350.
[2] G. Bercu, Pseudo-Riemannian structures with the same connection, Balkan
J.Geom. Appl., 11, no.1 (2006), 23-38.
[3] E. Cartan, Les problèmes d’équivalence, Oeuvres complètes, vol.III, C.N.R.S.
Paris, 1984, 1311-1334.
[4] R. Gardner, The method of equivalence and its applications, SIAM, Philadelphia,
1989.
[5] I. E. Hirică and L. Nicolescu, Some results on Weyl manifolds, Ann. Univ.
Bucharest, Mathematics, LVI (2007), 247-252.
[6] I. E. Hirică and L. Nicolescu, Conformal connections on Lyra manifolds, Balkan
J.Geom. Appl., 13, no.2 (2008), 43-49.
[7] L. Nicolescu, Sur les courbes m−principales associées à un champ tensoriel de
type (1, 2), Ann. Univ. Bucharest, Mathematics, LIII (2004), 63-72.
[8] L. Nicolescu, G.T. Pripoae and R. Gogu, On some families of semi-symmetric
metric connections, Ann. Univ. Bucharest, Mathematics, LVI (2007), 303-312.
110
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
[9] T. Oprea, On the Geometry of Weyl Manifolds, Ann. Univ. Bucharest, Mathematics, LIV (2005), 123-126.
[10] D.K. Sen and J.R. Vanstone, On Weyl and Lyra Manifolds, J. Math Phys., 13
(1972), 990-993.
[11] C. Udrişte, Geometric Dynamics, Mathematics and Its Applications, 297, Kluwer
Academic Publishers, Dordrecht, Boston, London, 1994.
[12] G. Vrănceanu, Lectures on Differential Geometry (in Romanian), vol. II, E.D.P.,
Bucharest, 1964.
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
Politehnica University of Bucharest, Faculty of Applied Sciences,
Splaiul Independentei 313, RO-060042, Bucharest, Romania.
E-mail: virgil.damian@gmail.com
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
◦
Abstract. On a Weyl manifold (M, gb, w), we consider ∇ the Levi-Civita
connection associated to a metric g ∈ gb , ∇ the symmetric connection,
compatible with the Weyl structure w and the family of linear connections
λ
◦
λ
◦
C ={∇ := ∇ + λ(∇ − ∇) | λ ∈ R}. For ∇ ∈ C, we investigate some
λ
◦
properties of the deformation algebra U(M, ∇ − ∇). Next, we study the
◦
λ
case when ∇ and ∇ determine the same Ricci tensor and the case when
◦
λ
the curvature tensors of the connections ∇ and ∇ 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, differential, differential affine, metric, etc invariants determine some structure on a
manifold, up to a homeomorphism, diffeomorphism, isometry, etc. Usually, one compares 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 explicitely 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.
°
On some families of linear connections
2
99
Definitions and notations
Let M be an n−dimensional C ∞ differentiable manifold. We denote by F (M ) the ring
of real-valued functions of class C ∞ defined on M and by Tsr (M ) the F (M ) −module
of tensor fields of type (r, s). In particular, for T01 (M ) (respectively T10 (M )) we use
the notation X (M ) (respectively Λ1 (M )).
Let A ∈ T21 (M ).The F (M ) −module X (M ) becomes an F (M ) −algebra, denoted by U (M, A), with the product of two vector fields X and Y defined by
(2.1)
X ◦ Y = A (X, Y ) .
In particular,
¡
¢
¡ if A = ∇¢ − ∇, where ∇ and ∇ are two arbitrary linear connections
on M , 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 function f ∈ F (M )
such that ([7])
³
´
(2.2)
A Z, X (m) = f Z + ω (Z) X, ∀Z ∈ X (M ) ,
where X (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). If m = 1 and f = 0, then X is a principal field in the algebra U (M, A). If
m = 1 and ω = 0, then X is almost special field. If m = 1, f = 0 and ω = 0, then X
is special field. If A (X, X) = 0, then X is 2−nilpotent field.
Let (M, g) be now an n−dimensional semi-Riemannian manifold and let gb the
conformal structure generated by g, i.e. gb = {eu g | u ∈ F (M )}.
Let w be a Weyl structure on the conformal manifold (M, gb), i.e. an application
w : gb −→ Λ1 (M ), which verifies ([1], [5], [6]) w (eu g) = w (g) − du, ∀u ∈ F (M ). The
triple (M, gb, w) is called a Weyl manifold.
Let (M, gb, w) be a Weyl manifold. A linear connection ∇ on M is called compatible
with the Weyl structure w if ∇X g + w (g) (X) g = 0, for all X ∈ X (M ).
It is well known that there exists an unique symmetric linear connection ∇, defined
on M , compatible with the Weyl structure ([10]). This linear connection ∇ is called
the Weyl conformal connection and is defined by ([9])
2g(∇X Y, 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 all X, Y, Z ∈ X (M ). Let ∇ be the Levi-Civita connection associated to g and let
◦
A = ∇ − ∇ ∈ T21 (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 semiRiemannian manifolds from properties of the deformation algebra of their Levi-Civita
100
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
connections; a remarkable particular case is when the latter ones coincide (see also
[2]).
Let gij , Aijk and ui the components, in a system of local coordinates, of g, A and
of the 1−form 12 w (g), respectively. Then (2.3) can be written
Aijk = δji uk + δki uj − gjk ui .
(2.4)
We define the family of linear connections ([8])
½
¾
λ
◦
C = ∇ := ∇ + λA | λ ∈ R .
(In the affine space of all linear connections on M , C is the ”affine line” passing
◦
through the ”point” ∇ and of ”direction” A).
λ
◦
λ
◦
Let R and R be the curvature tensor field of the connection ∇ and ∇, respectively.
3
The main results
Theorem 3.1. Let M be an n−dimensional connected manifold, n > 3. The following
assertions are equivalent:
λ
◦
λ
◦
(i) ∇ = ∇;
◦
(ii) R = R, if Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, ∀p ∈ M ;
λ
◦
(iii) ∇ and ∇ admit the same geodesics;
µ
¶
λ
◦
(iv) the algebra U M, ∇ − ∇ is associative;
◦
(v) if Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, ∀p ∈ M and the 1−form
λ
◦
w (g) is exact, then ∇ and ∇ have the same Ricci tensor.
Proof. (i)=⇒(ii), (i)=⇒(iii), (i)=⇒(iv), (i)=⇒(v) are trivial. (ii)=⇒(i). From (ii),
λ
λ
λ
◦
∇X R = ∇X R, ∀X ∈ X (M ). Then, ∀X, Y, Z, V ∈ X (M ),
λ
λ
λ
◦
◦
◦
◦
λ
(∇X R)(Y, Z, V ) = (∇X R)(Y, Z, V ) + A(X, R(Y, Z)V ) − R(A(X, Y ), Z)V
λ
◦
(3.1)
λ
◦
−R(Y, A(X, Z))V − R(Y, Z)A(X, V ),
λ
λ
where we denoted by A = ∇ − ∇ = λA. Similarly, we obtain
λ
λ
◦
λ
◦
◦
◦
λ
(∇Y R)(Z, X, V ) = (∇Y R)(Z, X, V ) + A(Y, R(Z, X)V ) − R(A(Y, Z), X)V
◦
(3.2)
λ
◦
λ
−R(Z, A(Y, X))V − R(Z, X)A(Y, V ),
101
On some families of linear connections
λ
λ
◦
(∇Z R)(X, Y, V )
λ
◦
λ
◦
(3.3)
◦
◦
λ
= (∇Z R)(X, Y, V ) + A(Z, R(X, Y )V ) − R(A(Z, X), Y )V
λ
◦
−R(X, A(Z, Y ))V − R(X, Y )A(Z, V ).
Using the Bianchi identities, from (3.1), (3.2) and (3.3), it follows
½ µ
¶
µ
¶
µ
¶
◦
◦
◦
λ A X, R (Y, Z) V + A Y, R (Z, X) V + A 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 deduce λ = 0, so we proved (i); hence
µ
¶
µ
¶
µ
¶
◦
◦
◦
A X, R (Y, Z) V + A Y, R (Z, X) V + A 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
¶
µ
¶
µ
◦
◦
◦
◦
◦
◦
(3.5)
δis Rrljk + δks Rrlij + δjs Rrlki ur + gil Rsrjk + gjl Rsrki + gkl Rsrij ur = 0.
We make s := i and sum with respect to i; it follows
¶
µ
◦
◦
◦
◦
r
(3.6)
(n − 2) Rljk ur + Rlrjk − gjl Rrk + gkl Rrj ur = 0,
◦
◦
where Rjl = Rijil are the components of Ricci tensor. Multiplying (3.6) by g jl and
summing with respect to j and l, it follows
◦
(n − 2) Rrk ur = 0.
(3.7)
From (3.6) and (3.7) we get
◦
(n − 3) Rrljk ur = 0.
(3.8)
Because we supposed n > 3, from (3.8) we have
◦
Rrljk ur = 0.
(3.9)
The relations (3.9) show that, for every p ∈ M ,
µ
¶
◦
(3.10)
(w (g))p Rp (Xp , Yp ) Zp = 0, ∀Xp , Yp , Zp ∈ Tp M.
◦
Because Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, from (3.10) we obtain
w (g))p = 0, ∀p ∈ M , so w (g) = 0 and using (2.3), we find
(3.11)
g (A (X, Y ) , Z) = 0, ∀X, Y, Z ∈ X (M ) .
102
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
λ
λ
◦
Because g is non-degenerate, from (3.11) it follows A = 0, so A = λA = 0, i.e. ∇ = ∇.
λ
◦
(iii)=⇒(i) The linear connections ∇ and ∇ are symmetric, hence they admit the
same geodesics if and only if there exists an 1−form on M such that
(3.12)
λ
◦
∇X Y = ∇X Y + ω (X) Y + ω (Y ) X, ∀X, Y ∈ X (M ) .
λ
From (3.12) and from A = λA, we obtain
(3.13)
λA (X, X) = 2ω (X) X, ∀X ∈ X (M ) .
For Y = 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 all X, Z ∈ X (M ).
Because n > 3, for every p ∈ M and every Zp ∈ Tp M − {0}, there exists a vector
Xp ∈ Tp M − {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 ∈ Tp M − {0} , ∀p ∈ M ,
λ
λ
◦
so w (g) = 0 and from (2.3) we obtain that A = 0, so A = 0, i.e. ∇ = ∇. µ
¶
λ
◦
(iv)=⇒(i) For λ = 0 is trivial. We suppose λ 6= 0. Because the algebra U M, ∇ − ∇
µ
¶
λ
◦
is commutative, it follows that U M, ∇ − ∇ 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
Aisk Asjl − Aisl Asjk = 0.
(3.18)
From (3.18), using (2.4), we can write
(3.19)
δki uj ul − δli uj uk + gjl ui uk − gjk ui ul + δli gjk us us − δki gjl us us = 0.
In (3.19), we make i := k and sum with respect to i; it follows
(n − 2) (uj ul − gjl us us ) = 0.
Because n > 3, we obtain
(3.20)
uj ul − gjl us us = 0.
103
On some families of linear connections
Multiplying (3.20) by g jl and summing with respect to j and l, we get
us us = 0.
(3.21)
From (3.20) and (3.21) we obtain uj ul = 0, ∀j, l ∈ {1, 2, ..., n}. Therefore w (g) = 0
λ
◦
and, from (2.3), it follows A = 0, so ∇ = ∇.
(v)=⇒(i) In local coordinates, the conformal Weyl connection has the components
¯ ¯
(3.22)
Γijk = ¯ijk ¯ + δji uk + δki uj − gjk ui ,
¯ ¯
¯ ¯
where ¯ijk ¯ are the Christoffel symbols of the second kind, constructed using the meµ
¶
λ
◦
◦
tric components gij . From (3.22) and from ∇ = ∇ + λ ∇ − ∇ , we obtain the
λ
components of the linear connection ∇:
λ
¯ ¯
Γijk = ¯ijk ¯ + δji ψk + δki ψj − gjk ψ i ,
(3.23)
λ
where ψi = λui . The curvature tensor of the connection ∇ has the components
λ
◦
Rijkl = Rijkl + δji (ψkl − ψlk ) + δki ψjl − δli ψjk − gjk ψli + gjl ψki ,
(3.24)
where we denoted
ψjl =
1
∂ψj ¯¯r ¯¯
+ jl ψr + ψj ψl − φ2 gjl ,
∂xl
2
ψli = g ij ψjl , φ2 = grs ψ r ψ s = ψi ψ i .
If we assign i = k and sum, from (3.24) we obtain
λ
(3.25)
◦
Rjl = Rjl + (n − 1) ψjl − ψlj + gjl ϕ,
λ
λ
◦
◦
where Rjl = Rijil , ϕ = ψii = g jl ψjl , Rjl = Rijil . Because the 1−form w (g) is exact, it
follows that ψjl = ψlj . From (3.25), we have
λ
(3.26)
◦
Rjl = Rjl + (n − 2) ψjl + gjl ϕ.
λ
◦
Using the hypothesis, Rjl = Rjl . Therefore, from (3.26) we get
(3.27)
(n − 2) ψjl + gjl ϕ = 0.
Multiplying (3.27) by g jl and summing with respect to j and l, it follows
(3.28)
ϕ = 0.
From (3.27) and (3.28), we have
(3.29)
ψjl = 0, ψli = 0.
104
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
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 : Tp M × Tp M × Tp M −→ Tp M 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).
µ
¶
λ
◦
Theorem 3.2. Let U M, ∇ − ∇ be the deformation algebra as above. We suppose
that the manifold M is connected and n > 2. The following assertions are equivalent:
λ
◦
(i) ∇ = ∇;
µ
¶
λ
◦
(ii) all elements of the deformation algebra U M, ∇ − ∇ are almost principal
fields;
(iii) all elements of the deformation algebra U
(iv) all elements of the deformation algebra U
µ
¶
λ
◦
M, ∇ − ∇ are principal fields;
µ
λ
◦
M, ∇ − ∇
¶
are almost special fields;
µ
¶
λ
◦
(v) all elements of the deformation algebra U M, ∇ − ∇ are special fields;
µ
¶
λ
◦
(vi) all elements of the deformation algebra U M, ∇ − ∇ are 2− nilpotent fields.
Proof. (i)=⇒(ii), (i)=⇒(iii), (i)=⇒(iv), (i)⇐⇒(v), (i)⇐⇒(vi) are obvious.
λ
(iv)=⇒(i). Because A (Z, X) = fX Z, ∀X, Z ∈ X (M ), it follows that there exists
λ
an 1−form θ on M such that θ (X) = fX . We have, therefore A (Z, X) = θ (X) Z,
λ
∀X, Z ∈ X (M ). Because A is symmetric, we get θ (X) Z = θ (Z) X, ∀X, Z ∈ X (M ).
In local coordinates, the last equality is written θi δkj = θk δij . If we make j = i and
λ
◦
sum, we obtain (n − 1) θk = 0, so θ = 0 and, finally, ∇ = ∇.
λ
λ
(iii)=⇒(i). We have A (Z, X) = ω (Z) X, ∀X, Z ∈ X (M ). Because A (Z, X) =
λ
A (X, Z), ∀X, Z ∈ X (M ), we obtain ω (Z) X = ω (X) Z, ∀X, Z ∈ X (M ), which
λ
◦
implies ω = 0, i.e. ∇ = ∇.
(ii)=⇒(i). Using the hypothesis, we have
λ
A (Z, X) = fX Z + ω (Z) X, ∀X, Z ∈ X (M ) .
105
On some families of linear connections
From here, it follows that there exists an 1−form η, defined on M , such that η (X) =
λ
λ
fX . Therefore, we have A (Z, X) = η (X) Z + ω (Z) X, ∀X, Z ∈ X (M ). Because A 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 ) δkj − (ηk − ωk ) δij = 0.
If we assign j = k and sum, we get
(n − 1) (ηi − ωi ) = 0, ∀i ∈ {1, 2, ..., n} .
Therefore η = ω and we have
λ
◦
∇Z X = ∇Z X + η (X) Z + η (Z) X, ∀X, Z ∈ X (M ) .
λ
◦
The last equality shows that the symmetric linear connections ∇ and ∇ admit the
λ
◦
same geodesics. Using Theorem 3.1, we obtain ∇ = ∇.
¤
Remark 3.2. In the following, we will denote by C ∈ T31 (M ) the curvature conformal
i
Weyl tensor. Let Cjkl
the components of C in a system of local coordinates. Then,
we have ([11], [12])
i
Cjkl
◦
=
Rijkl
1
−
n−2
¶
µ
◦
◦
◦
◦
is
is
i
i
δk Rjl − δl Rjk gjk g Rsl + gjl g Rsk
◦
¡ i
¢
g rs Rrs
δk gjl − δli gjk .
+
(n − 1) (n − 2)
Theorem 3.3. With the above notations, we suppose that
(i) the 1−form w (g) is exact;
◦
(ii) Rp : Tp M × Tp M × Tp M −→ Tp M is surjective, ∀p ∈ M ;
(iii) the curvature conformal Weyl tensor is nowhere vanishing, i.e. Cp 6= 0, ∀p ∈ M .
λ
◦
If there exists a function f ∈ F (M ), f (p) 6= 0, ∀p ∈ M , such that R = f R, then
λ
◦
∇ = ∇.
λ
Proof. In local coordinates, the linear connection ∇ has the components
λ
¯ ¯
Γijk = ¯ijk ¯ + δji ψk + δki ψj − gjk ψ i ,
106
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
¯ ¯
¯ ¯
where ¯ijk ¯ are the Christoffel symbols of the second kind, constructed using the metric
components gij and ψi are the components of 1−form
w (g) is exact, the last equalities imply
λ
λ
2 w (g).
Because the 1−form
◦
Rijkl = Rijkl + δki ψjl − δli ψjk − gjk ψli + gjl ψki ,
(3.30)
where
(3.31)
ψjl =
∂ψj ¯¯r ¯¯
1
+ jl ψr + ψj ψl − gjl ψ s ψs = ψlj , ψki = g ij ψjk .
l
∂x
2
From (3.30), we have
λ
◦
Rjl = Rjl + (n − 2) ψjl + gjl g rs ψrs ,
(3.32)
λ
λ
λ
where Rjl = Rkjkl = Rlj . Multiplying (3.32) by g jl and summing with respect to j
and l, we obtain
λ
◦
g jl Rjl = g jl Rjl + 2 (n − 1) g rs ψrs .
(3.33)
Because n > 3, from (3.32) and (3.33) we deduce that
¶
µ
λ
◦
1
rs
(3.34)
g ψrs =
Rrs − Rrs g rs
2 (n − 1)
(3.35)
ψjl =
1
n−2
¶
µ
λ
◦
Rjl − Rjl −
µ
¶
λ
◦
gjl
g rs Rrs − Rrs .
2 (n − 1) (n − 2)
Introducing (3.34), (3.35) in (3.30), we find
¶
µ
λ
λ
λ
λ
λ
1
δki Rjl − δli Rjk + gjk g is Rsl + gjl g is Rsk
Rijkl −
n−2
λ
(3.36)
+
¡ i
¢
g rs Rrs
i
δk gjl − δli gjk = Cjkl
,
(n − 1) (n − 2)
where
i
Cjkl
◦
=
Rijkl
1
−
n−2
¶
µ
◦
◦
◦
◦
is
is
i
i
δk Rjl − δl Rjk − gjk g Rsl + gjl g Rsk
◦
¢
¡ i
g rs Rrs
δk gjl − δli gjk
+
(n − 1) (n − 2)
are the components of the curvature conformal Weyl tensor C. On the other hand,
λ
◦
R = f R implies that
λ
◦
λ
◦
λ
◦
Rijkl = f Rijkl , Rjl = f Rjl , g rs Rrs = f g rs Rrs ,
107
On some families of linear connections
λ
Rijkl −
1
n−2
µ
¶
λ
λ
λ
λ
δki Rjl − δli Rjk − gjk g is Rsl + gjl g is Rsk
λ
(3.37)
+
¡ i
¢
g rs Rrs
i
δk gjl − δli gjk = f Cjkl
.
(n − 1) (n − 2)
i
Because Cp 6= 0, ∀p ∈ M , it follows that the functions Cjkl
are nowhere vanishing.
Taking this into the account, the relations (3.36) and (3.37) show that we have f = 1,
λ
λ
◦
◦
so R = R. Using Theorem (3.1), we obtain ∇ = ∇.
¤
Theorem 3.4. Let (M, g) be a connected n−dimensional semi-Riemannian manifold,
with n ≥ 3. We consider two Weyl structures w and w′ on the conformal manifold
(M, gb). Let ∇ (resp. ∇′ ) be the symmetric conformal Weyl connection, compatible
with the Weyl structure w (resp. w′ ). Define the family of linear connections
e {∇ + λ (∇′ − ∇) | λ ∈ R} .
C=
λ
e The following
For λ ∈ R, we consider the linear connection ∇ = ∇ + λ (∇′ − ∇) ∈ C.
assertions are equivalent:
λ
(i) ∇ = ∇;
(ii) the deformation algebra U
µ
¶
λ
M, ∇ − ∇ is associative;
λ
(iii) ∇ and ∇ admit the same geodesics;
(iv) all the elements of the deformation algebra U
µ
¶
λ
M, ∇ − ∇ are almost principal
fields.
Proof. (i)=⇒(ii), (i)=⇒(iii),µ(i)=⇒(iv) ¶
are obvious.
µ
¶
λ
λ
(ii)=⇒(i) The algebra U M, ∇ − ∇ is commutative. It follows that U M, ∇ − ∇
is associative if and only if we have
µ
¶
µ
¶
λ
λ
λ
λ
(3.38)
A X, A (Y, Z) − A Y, A (X, Z) = 0, ∀X, Y, Z ∈ X (M ) ,
λ
λ
λ
where A = ∇ − ∇ = λA′ , A′ = ∇ − ∇.
For λ = 0 the assertion is trivial. We suppose λ 6= 0. The linear connection ∇′ is
defined by
2g(∇′X Y, 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), ∀X, Y, Z ∈ X (M ) .
108
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
From here, it follows that A′ 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).
′
Let A′i
jk (resp. ϕi ) the components of A (resp. ϕ) in a system of local coordinates.
Then we can rewrite (3.39) as
i
i
i
A′i
jk = δj ϕk + δk ϕj − gjk ϕ ,
(3.40)
where ϕi = g ik ϕk . In local coordinates, (3.38) can be written
¡ i
¢
¡
¢
δk ϕl − δli ϕk ϕj + δli gjk − δki gjl ϕs ϕs + (gjl ϕk − gjk ϕl ) ϕi = 0.
For i = k, we sum and obtain
(n − 2) (ϕj ϕl − gjl ϕs ϕs ) = 0.
(3.41)
Because n ≥ 3, from (3.41) we get
ϕj ϕl − gjl ϕs ϕs = 0.
(3.42)
Multiplying (3.42) by g jl and summing with respect to j and l, we have ϕs ϕs = 0.
Taking into account this, from (3.42) follows ϕj ϕl = 0, ∀j, l ∈ {1, 2, ..., n}. We obtain
λ
λ
ϕ = 0, i.e. w = w′ . From (3.39) we find A′ = 0, so A = 0, i.e. ∇ = ∇.
λ
(iii)=⇒(i). Because the linear connections ∇ and ∇ are symmetric, it follows that
they admit the same geodesics if and only if there exists an 1−form σ, defined on M ,
such that
(3.43)
λ
∇X Y = ∇X Y + 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 ) .
λ
For λ = 0, we obtain σ = 0 and from (3.34) we have ∇ = ∇. We shall consider the
case for λ 6= 0. For Y = 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 ) .
Because n ≥ 3, for any p ∈ M and any Zp ∈ Tp M − {0}, there exists Xp ∈ Tp M − {0}
such that we have
(3.47)
gp (Xp , Zp ) = 0, gp (Xp , Xp ) 6= 0.
109
On some families of linear connections
From (3.46) and (3.47) we obtain ϕp (Zp ) = 0, for any Zp ∈ Tp M − {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
λ
we find A′ (X, Y ) = 0, ∀X, Y ∈ X (M ), so A′ = 0. In conclusion, A = λA′ = 0, i.e.
λ
∇ = ∇.
(iv)=⇒(i). Because all elements of the algebra U
µ
¶
λ
M, ∇ − ∇ are almost principal
fields, it follows that for any X ∈ X (M ) there exists a function fX ∈ F (M ) and an
1−form ω ∈ Λ1 (M ) such that
λ
(3.48)
A (Z, X) = fX Z + ω (Z) X, ∀Z, X ∈ X (M ) .
From (3.48), there exists an 1−form η ∈ Λ1 (M ) such that η (X) = fX . From (3.48),
we obtain
λ
(3.49)
A (Z, X) = η (X) Z + ω (Z) X, ∀Z, X ∈ X (M ) .
µ
¶
λ
Because the algebra U M, ∇ − ∇ is abelian, from the last equality follows η = ω
and from (3.49) we obtain
λ
∇Z X = ∇Z X + η (X) Z + η (Z) X, ∀Z, X ∈ X (M ) .
λ
The last equality shows that the symmetric and linear connections ∇ and ∇ admit
λ
the same geodesics. From here, ∇ = ∇.
¤
References
[1] B. Alexandrov and S. Ivanov, Weyl structures with positive Ricci tensor, Diff.
Geom. Appl. 18, 3 (2003), 343-350.
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J.Geom. Appl., 11, no.1 (2006), 23-38.
[3] E. Cartan, Les problèmes d’équivalence, Oeuvres complètes, vol.III, C.N.R.S.
Paris, 1984, 1311-1334.
[4] R. Gardner, The method of equivalence and its applications, SIAM, Philadelphia,
1989.
[5] I. E. Hirică and L. Nicolescu, Some results on Weyl manifolds, Ann. Univ.
Bucharest, Mathematics, LVI (2007), 247-252.
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J.Geom. Appl., 13, no.2 (2008), 43-49.
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type (1, 2), Ann. Univ. Bucharest, Mathematics, LIII (2004), 63-72.
[8] L. Nicolescu, G.T. Pripoae and R. Gogu, On some families of semi-symmetric
metric connections, Ann. Univ. Bucharest, Mathematics, LVI (2007), 303-312.
110
Liviu Nicolescu, Gabriel-Teodor Pripoae and Virgil Damian
[9] T. Oprea, On the Geometry of Weyl Manifolds, Ann. Univ. Bucharest, Mathematics, LIV (2005), 123-126.
[10] D.K. Sen and J.R. Vanstone, On Weyl and Lyra Manifolds, J. Math Phys., 13
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Bucharest, 1964.
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
Politehnica University of Bucharest, Faculty of Applied Sciences,
Splaiul Independentei 313, RO-060042, Bucharest, Romania.
E-mail: virgil.damian@gmail.com