ON RICCI TYPE IDENTITIES IN MANIFOLDS WITH NON-SYMMETRIC AFFINE CONNECTION
Svetislav M. Minčić
Abstract. In [18], using polylinear mappings, we obtained several curvature tensors in the spaceLNwith non-symmetric affine connection∇. By the same method, we here examine Ricci type identities.
1. Introduction
ConsiderN-dimensional differentiable manifoldMN on which a non-symmetric affine connection ∇1 is defined. IfX(MN) is a Lie algebra of smooth vector fields andX, Y ∈X(MN), then the mapping∇2:X(MN)×X(MN)→X(MN) given by
(1.1) ∇2XY =∇1YX+ [X, Y]
defines an other non-symmetric connection ∇2 on MN [14]. That means that we have
θ
∇Y1+Y2X =∇θY1X+∇θY2X, ∇θf YX =f∇θYX,
θ
∇Y(X1+X2) =∇θYX1+∇θYX2, ∇θY(f X) =Y f·X+f∇θYX, for θ= 1,2 andX, Y, X1, X2, Y1, Y2 ∈X(MN),f ∈ F(MN), whereF(MN) is an algebra of smooth real functions onMN. In that case we writeLN = (MN,∇,1 ∇)2 andLN call a space on non-symmetric connections∇,1 ∇.2
If we introduce local coordinates x1, . . . , xN and put ∂/∂xi = ∂i, in view of (1.1) it will be
(1.2) ∇2∂j∂k=∇1∂k∂j.
2010Mathematics Subject Classification: Primary 53C05; Secondary 53B05.
Key words and phrases: non-symmetric connection, curvature tensors, independent curvature tensors, Ricci type identities.
205
Denoting coefficients of the connection∇1 in the base∂1, . . . , ∂N withLijk, we have
∇1∂k∂j =Lijk∂i,∇2∂k∂j =
(1.2)
∇1∂j∂k =Likj∂i, where =
(1.2)denotes “equal with respect to (1.2)".
Further, if we take by definition
θ
T(X, Y) =∇θYX−∇θXY + [X, Y], θ∈ {1,2}, it follows
2
T(X, Y) =−T1(X, Y)≡ −T(X, Y), (T2(X, Y) =T1(X, Y))⇔(∇1 =∇2 =∇).
We proved in [18] how it is possible to obtain several curvature tensors inLN by polylinear mappings. It is proved that among these tensors there are 5 independent ones:
R(X1 ;Y, Z) =∇1Z
∇1YX−∇1Y
∇1ZX+∇1[Y,Z]X, (1.3)
2
R(X;Y, Z) =∇2Z 2
∇YX−∇2Y 2
∇ZX+∇2[Y,Z]X, (1.4)
R(X3 ;Y, Z) =∇2Z
∇1YX−∇1Y
∇2ZX+∇2 1
∇YZX−∇12
∇ZYX, (1.5)
R(X4 ;Y, Z) =∇2Z
∇1YX−∇1Y
∇2ZX+∇2 2
∇YZX−∇11
∇ZYX, (1.6)
R(X5 ;Y, Z) =1 2(∇1Z
∇1YX−∇2Y
∇1ZX+∇2Z
∇2YX−∇1Y
∇2ZX+∇1[Y,Z]X+∇2[Y,Z]X), (1.7)
while the rest can be expressed as linear combinations of these five tensors. For X =∂/∂xj≡∂j,Y =∂k,Z =∂l, one obtains
1
Rijkl=Lijk,l−Lijl,k+LpjkLipl−LpjlLipk, (1.8)
R2ijkl=Likj,l−Lilj,k+LpkjLilp−LpljLikp, (1.9)
R3ijkl=Lijk,l−Lilj,k+LpjkLilp−LpljLipk+Lplk(Lipj−Lijp), (1.10)
R4ijkl=Lijk,l−Lilj,k+LpjkLilp−LpljLipk+Lpkl(Lipj−Lijp), (1.11)
R5ijkl= 1
2(Lijk,l+Likj,l−Lijl,k−Lilj,k+LpjkLipl+LpkjLilk−LpjlLikp−LpljLipk) (1.12)
2. Identities for a vector and for a covector by both connections 2.1. Consider an expression
(2.1) (∇νZ
µ
∇YX−∇µY
∇νZX)(ω), µ, ν∈ {1,2}.
Let us quote the relations (for µ= 1,2) (
µ
∇YX)(ω) =Y[X(ω)]−X(
µ
∇Yω), (
µ
∇Yω)(X) =Y[ω(X)]−ω(
µ
∇YX).
and denote
(2.2) a)
µ
∇YX =X ∈X(MN), b)∇νZω=ω∈X∗(MN).
Then we have (∇νZ
µ
∇YX)(ω) =
(2a)(∇νZX)(ω) =
(2)Z[X(ω)]−X(∇νZω)
(2)= Z[(∇µYX)(ω)]−(∇µYX)(ω)
(2b)= Z{Y[X(ω)]−X(∇µYX)} − {Y[X(∇νZω)]−X(∇µY
∇νZω)}
(2.3)
=ZY[X(ω)]−Z[X(∇µYω)]−Y[X(∇νZω)] +X(∇µY ν
∇Zω), and one gets
(2.4) (∇νZ µ
∇YX−
µ
∇Y ν
∇ZX)(ω) = [Z, Y][X(ω)]−X(∇νZ µ
∇Yω−
µ
∇Y ν
∇Zω), i.e.,
(∇νZ µ
∇YX−
µ
∇Y ν
∇ZX)(ω) = [Z, Y][X(ω)]−(∇νZ µ
∇Yω−
µ
∇Y ν
∇Zω)(X), µ, ν∈ {1,2}.
From
(2.5) (∇ν[Z,Y]X)(ω) =∇ν[Z,Y][X(ω)]−X(∇ν[Z,Y]ω) = [Z, Y][X(ω)] + (∇ν[Y,Z]ω)(X), we find the first addend on the right side and substitute in (2.7). We obtain (2.6) (∇νZ
µ
∇YX−∇νY µ
∇ZX+∇ν[Y,Z]X)(ω)
=−(∇νZ µ
∇Yω−∇νY µ
∇Zω+∇ν[Y,Z]ω)(X), µ, ν∈ {1,2}
Definition 2.1. The equations (2.4) for µ, ν∈ {1,2} areRicci type identities for a vector inLN.
2.2. Takingµ=ν= 1, we obtain the corresponding identity for∇:1 (2.7) R(X1 ;Y, Z)(ω) =−(∇1Z
∇1Yω−∇1Y
∇1Zω+∇1[Y,Z]ω)(X).
Denoting (2.8)
1
R(ω;Y, Z) =∇1Z
∇1Yω−∇1Y
∇1Zω+∇1[Y,Z]ω, the equation (2.7) gives a relation
(2.9) R(X1 ;Y, Z)(ω) =−
1
R(ω;Y, Z)(X).
In order to write the equation (2.4) in local coordinates for µ =ν = 1, we take X =Xj∂j,Y =∂k,Z =∂l,ω=dxi. For the left side in (2.4) we obtain
L= (∇1∂l
∇1∂kX−∇1∂k
∇1∂lX)(dxi)
= [∇1∂l(X|j
1
k∂j)−∇1∂k(X|j
1
l∂j)](dxi)
= [(Xj|
1
k),l∂j+Xj|
1
kLpjl∂p−(Xj|
1
l),k∂j−X|j
1
lLpjk∂p](dxi)
= (X|j
1
k),lδij+X|j
1
kLpjlδip−(X|j
1
l),kδji−X|j
1
lLpjkδip
= (X|i
1
k),l+X|j
1
kLijl−(Xi|
1
l),k−Xj|
1
lLijk
=Xi|
1
k|
1
l+LpklXi|
1
p−Xi|
1
l|
1
k−LplkX|i
1
p
=Xi|
1
k|
1
l−X|i
1
k|
1
l+TklpX|i
1
p. For the right-hand side in (2.4) we obtain
R= [∂l, ∂k](X(dxi)) +X(∇1∂l
∇1∂kdxi−∇1∂k
∇1∂ldxi)
= 0 +X[∇1∂l(−Lipldxp)−∇1∂l(−Lipkdxp)]
=X(R
1 i
pkldxp) =R
1 i
pkldxp(X) =R
1 i pklXp, and from L=R, we have
(2.10) Xi|
1
kl−X|i
1
lk=R
1 i
pklXp−TklpX|i
1
p,
i.e., the known identity in local coordinates. So, we have proved the following theorem.
Theorem 2.1. In the space LN, with non-symmetric affine connection∇1 by equation (2.4)forµ=ν= 1 the first Ricci type identity for a vectoris given. That identity can be written in forms (2.5),(2.6),(2.9), here R1 is given by (1.3)and
1
R by (2.8). The corresponding identity in local coordinates is (2.10).
2.3. By using equation (2.4) and the conditionX(ω) =ω(X)∈ F(MN), we obtain the equation analogous to (2.4) (X andω have changed the roles):
(2.11) (∇νZ µ
∇Yω−∇µY
∇νZω)(X) = [Z, Y][ω(X)]−ω(∇νZ µ
∇YX−∇µY
∇νZX).
Definition 2.2. Equations (2.11) forµ, ν∈ {1,2}areRicci type identities for a covector in LN.
Equation (2.11) can be obtained also by consideration of the expression on the left-hand side in (2.11). The known Ricci identity for a covariant vector in local
coordinates can be obtained from (2.11) by substitutingω=ωjxj,X=∂i,Y =∂k, Z =∂l:
(2.12) ωj|
1
kl−ωj|
1
lk=−R
1 p
jklωp−Tklpωj|
1
p. So, the following theorem is valid.
Theorem 2.2. In the space LN, with non-symmetric affine connection ∇, by1 equation (2.11) for µ=ν = 1,the first Ricci type identity for a covectoris given.
The corresponding identity in local coordinates is (2.12).
2.4. Forµ=ν= 2 from (2.4) is obtained (2.13) (∇2Z
2
∇YX−∇2Y 2
∇ZX)(ω) = [Z, Y][X(ω)]−X(∇2Z 2
∇Yω−∇2Y 2
∇Zω).
From here
(2.14) R(X2 ;Y, Z)(ω) =−
2
R(ω;Y, Z)(X), where
2
Ris expressed by∇2 analogously to (2.8) andR2 is given in (1.3). Surpassing to local coordinates, from (2.13) one obtains
(2.15) Xi|
2
kl−X|i
2
lk=R
2 i
pklXp+TklpX|i
2
p, and also equations similar to (2.11), (2.12) (for a covector).
Thus, we state
Theorem 2.3. In the space LN with non-symmetric affine connection
2
∇, de- fined by(1.1),the second Ricci type identity for a vectoris given by equation(2.13).
The corresponding identity in local coordinates is (2.15).
3. Identities for a vector and covector obtained by combinations of both connections 3.1. Puttingµ= 1, ν= 2 into (2.4), we get the identity (3.1) (∇2Z
∇1YX−∇1Y
∇2ZX)(ω) = [Z, Y][X(ω)]−(∇2Z
∇1Yω−∇1Y
∇2Zω)(X).
and from (2.6) (3.2) (∇2Z
1
∇YX−∇1Y 2
∇ZX+∇2[Y,Z]X)(ω) =−(∇2Z 1
∇Yω−∇1Y 2
∇Zω+∇2[Y,Z]ω)(X) Analogously to (1.5), let us put
(3.3)
3
R(ω;Y, Z) =∇2Z 1
∇Yω−∇1Y 2
∇Zω+∇2 1
∇YZω−∇12
∇YZω∈X∗(MN)
and (3.2) becomes (3.4) (R(X3 ;Y, Z) +∇1 2
∇YZX−∇2 1
∇YZX+∇2[Y,Z]X)(ω)
=−(
3
R(ω;Y, Z) +∇12
∇ZYω−∇2 1
∇YZω+∇2[Y,Z]ω)(X).
Because of (∇1 2
∇ZYX−∇2 1
∇YZX+∇2[Y,Z]X)(ω) = (∇1 2
∇ZYX−∇2 1
∇YZ+[Z,Y]X)(ω)
= (∇1 2
∇ZYX−∇2 2
∇ZYX)(ω) and
−(∇1 2
∇ZYω−∇2 1
∇YZω+∇2[Y,Z]ω)(X) =−(∇12
∇ZYω−∇21
∇YZ+[Z,Y]ω)(X)
= (∇22
∇ZYω−∇1 2
∇ZYω)(X)
=∇2 2
∇ZY[ω(X)]−ω(∇2 2
∇ZYX)−∇1 2
∇ZY[ω(X)] +ω(∇12
∇ZYX)
= (∇12
∇ZYX−∇22
∇ZYX)(ω),
we see that the right-hand sides of these equations are identical and from (3.4):
(3.5) R(X3 ;Y, Z)(ω) =−
3
R(ω;Y, Z)(X).
3.2. If we put X =Xj∂j, Y =∂k, Z =∂l, ω =dxi, equation (3.1) will be written in local coordinates as follows. For the left-hand sideL we have
L= (∇2∂l
∇1∂kX−∇1∂k
∇2∂lX)(dxi)
= [∇2∂l(Xj|
1
k∂j)−∇1∂k(Xj|
2
l∂j)](dxi)
= [(Xj|
1
k),l∂j+X|j
1
kLplj∂p−(X|j
2
l),k∂j−Xj|
2
lLpjk∂p](dxi) (3.6)
= (Xi|
1
k),l+Xj|
1
kLilj−(Xi|
2
l),k−X|j
2
lLijk
=X|i
1
k|
2
l−X|i
2
l|
1
k−Lplk(X|i
1
p−Xi|
2
p) =Xi|
1
k|
2
l−Xi|
2
l|
1
k−LplkTspi Xs, where
X|i
1
k|
2
l= (X|i
1
k),l+Xp|
1
kLilp−Xi|
1
pLplk, X|i
2
l|
1
k = (X|i
2
l),k+Xp|
2
lLipk−Xi|
2
pLplk.
For the right-hand side one obtains R=−(∇2∂l
1
∇∂kdxi−∇1∂k
2
∇∂ldxi)(X) (3.7)
= (Lipk,ldxp−LipkLplsdxs+Lilp,kdxp+LilpLpskdxs)(X)
= (Lipk,l−LiskLslp−Lilp,k+LilsLspk)Xp. By virtue of (3.6) and (3.7), from L=Rit is
(3.8) Xi|
1
k|
2
l−Xi|
2
l|
1
k=R
3 i pklXp,
and, analogously to that exposed above, for a covariant vectorω it is obtained
(3.9) ωj|
1
k|
2
l−ωj|
2
l|
1
k =−R
3 p jklωp.
3.3. IntroducingR(X;4 Y, Z) into (3.2) by virtue of (1.6) and defining
4
R(ω;Y, Z) according to
(3.10)
4
R(ω;Y, Z) =∇2Z
∇1Yω−∇1Y
∇2Zω+∇2 2
∇YZω−∇1 1
∇YZω∈X∗(MN), equation (3.2) gives
(R(X;4 Y, Z) +∇11
∇ZYX−∇22
∇YZX+∇2[Y,Z]X)(ω)
=−(
4
R(ω;Y, Z) +∇1 1
∇ZYω−∇2 2
∇YZω+∇2[Y,Z]ω)(X).
As in the case of
3
R, we obtain
(3.11) R(X4 ;Y, Z)(ω) =−
4
R(ω;Y, Z)(X).
PuttingX =∂j,Y =∂k,Z =∂l,ω=dxi and taking into consideration (3.10), we get
R4pjkl∂p(dxi) =−[∇2∂l(−Lipkdxp)−∇2∂k(−Lilpdxp) +∇2Lp
kl∂pdxi−∇1Lp
kl∂pdxi](∂j), from where for R4 the value (1.11) is obtained. In view of (1.10), (1.11) it is
R4pjkl−R3pjkl=Tpji Tklp. and, using (3.8) and (3.9), we obtain
Xi|
1
k|
2
l−Xi|
2
l|
1
k =R
4 i
pklXp+Tpsi TklsXp, (3.12)
ωj|
1
k|
2
l−ωj|
2
l|
1
k =−R
4 p
jklωp+TsjpTklsωp. (3.13)
Now, we can state the following theorem
Theorem3.1. In the spaceLN with two non-symmetric affine connections
1 2 ∇,
∇, linked by equation (1.1), the third Ricci type identity for a vectoris given by equation (3.1). This identity can be written also in forms (3.2), (3.5)and (3.11).
From (2.8), for µ = 1, ν = 2, one obtains the third Ricci type identity for a covector. The corresponding identities in coordinates are (3.8), (3.9), (3.12) and (3.13).
3.4. In order to obtain an identity in whichR5 appears, let us start from the expression which appears in (1.7). So,
(∇1Z
∇1YX+∇2Z
∇2YX−∇1Y
∇2ZX−∇2Y
∇1ZX)(ω)
(2.3)
=
ZY[X(ω)]−Z[X(∇1Yω)]−Y[X(∇1Zω)] +X(∇1Y 1
∇Zω) +ZY[X(ω)]−Z[X(∇2Yω)]−Y[X(∇2Zω)] +X(∇2Y
∇2Zω)
−Y Z[X(ω)] +Y[X(∇2Zω)] +Z[X(∇1Yω)]−X(∇2Z 1
∇Yω)
−Y Z[X(ω)] +Y[X(∇1Zω)] +Z[X(∇2Yω)]−X(∇1Z
∇2Yω) (ω), that is
(3.14) 1 2(∇1Z
1
∇YX+∇2Z 2
∇YX−∇1Y 2
∇ZX−∇2Y 1
∇ZX)(ω)
= [Z, Y][X(ω)] +1 2X(∇1Y
1
∇Zω+∇2Y 2
∇Zω−∇1Z 2
∇Yω−∇2Z 1
∇Yω).
Definition 3.1. Equation (3.14) we call the combined Ricci type identity for a vector inLN.
Using (1.7), from (3.14) it is obtained (3.15) R(X5 ;Y, Z)(ω)+(∇0[Z,Y]X)(ω)
= [Z, Y][ω(X)]+1 2(∇1Y
1
∇Zω+∇2Y 2
∇Zω−∇1Z 2
∇Yω−∇2Z 1
∇Yω)(X).
From
(∇0[Z,Y]X)(ω) = [Z, Y][X(ω)]−X(∇0[Z,Y]ω) = [Z, Y][ω(X)] + (∇0[Y,Z]ω)(X), we find the first addend of the right-hand side and substitute into (3.15). So,
R(X;5 Y, Z)(ω) + (∇0[Z,Y]X)(ω)
= (∇0[Z,Y]X)(ω) + (∇0[Z,Y]ω)(X) +1
2(∇1Y
∇1Zω+∇2Y
∇2Zω−∇1Z
∇2Yω−∇2Z
∇1Yω)(X),
where R5 is given in (1.7). Denoting (3.16)
5
R(ω;Y, Z) = 1 2(∇1Z
1
∇Yω+∇2Z 2
∇Yω−∇1Y 2
∇Zω−∇2Y 1
∇Zω+∇1[Y,Z]ω+∇2[Y,Z]ω) the previous equation gives
(3.17) R(X5 ;Y, Z)(ω) =
5
R(ω;Z, Y)(X).
Substituting here X =∂j, Y =∂k,Z =∂l,ω =dxi and taking into consideration (3.16), for R5ijkl (1.12) is obtained.
Remark3.1. We see that relation betweenR5 and
5
Ris not of the form relating to R,θ
θ
R,θ= 1,2,3,4. In fact using the corresponding values from [22]
5
R=R(X0 ;Y, Z) +τ(τ(X, Y), Z) +τ(τ(X, Z), Y), R8 =1
2(∇1Z
∇2YX+∇2Z
∇1YX−∇1Y
∇1ZX−∇2Y
∇2ZX+∇1[Y,Z]X+∇2[Y,Z]X)
=R(X0 ;Y, Z)−τ(τ(X, Y), Z)−τ(τ(X, Z), Y).
We conclude thatR5 −2R0 =R8 and
5
R(X;Y, Z)(ω) =
5
R(ω;Z, Y)(X) =−
8
R(ω;Y, Z)(X).
So, we have
Theorem3.2. In the spaceLN with two non-symmetric affine connections∇,1
2
∇, linked according to (1.1), by equation (3.14) the combined Ricci type identity for a vector is given. Some other forms of (3.14) are (3.15)–(3.17). From (3.15) combined Ricci type identity for a covector is obtained:
(3.18) 1 2(∇1Y
∇1Zω+∇2Y
∇2Zω−∇1Z
∇2Yω−∇2Z
∇1Yω)(X)
= 1 2(∇1Z
∇1YX+∇2Z
∇2YX−∇1Y
∇2ZX−∇2Y
∇1ZX)(ω) + [Z, Y][ω(X)].
From (3.14) and (3.18) we obtain the corresponding combined Ricci type iden- tities for a vector and covector respectivelylocal coordinates:
1 2(X|i
1
kl+X|i
2
kl−X|i
1
l|
2
k−X|i
2
l|
1
k) =R5ipklXp, 1
2(ωj|
1
kl+ωj|
2
kl−ωj|
1
l|
2
k−ωj|
2
l|
1
k) = (R5 −2R)0 pjklωp,
where R0ijkl is defined byL0ijk=12(Lijk+Lijk), i.e., by symmetric connection coeffi- cients.
Definition3.2. The objects
θ
R, (θ= 1, . . . ,5), defined by
θ
R:X×X×X→X∗ onMN, we calldual curvature tensors in relation toR.θ
4. Identities for a tensor field t of the type (r, s)
4.1. Let us consider a tensor field of the type (r, s), which will be denoted
rt
s≡t, i.e., consider a mappingrt
s: (X∗)r×(X)s7→ F(MN). So,
rt
s(ω1, . . . , ωr, X1, . . . , Xs)∈ F(MN), is a differentiable function onMN.
As known, a covariant derivative ∇Y rt
sis also of a type (r, s). As in (2.1), one can consider the expression (∇νZ
µ
∇Y rt
s−∇µY
∇νZ rt
s)(ω1, . . . , ωr, X1, . . . , Xs).
4.2. Let us examine nearer the case (r, s) = (2,1), i.e.,2t
1≡t. We have (∇µY
2t
1)(ω1, ω2;X) =∇µY[t(ω1, ω2;X]−t(∇µYω1, ω2;X)
−t(ω1,∇µYω2;X)−t(ω1, ω2;∇µYX).
Denoting∇µYt=t, we have (∇νZ
µ
∇Yt)(ω1, ω2;X) = (∇νZt)(ω1, ω2;X)
=
(4) ν
∇Z[t(ω1, ω2;X]−t(∇νZω1, ω2;X)−t(ω1,
ν
∇Zω2;X)−t(ω1, ω2;∇νZX)
=
ν
∇Z[(
µ
∇Yt)(ω1, ω2;X]−(
µ
∇Yt)(
ν
∇Zω1, ω2;X)−(
µ
∇Yt)(ω1,
ν
∇Zω2;X)−(
µ
∇Yt)(ω1, ω2;
ν
∇ZX)
(4)=
ν
∇Z{
µ
∇Y[t(ω1, ω2;X]−t(
µ
∇Yω1, ω2;X)−t(ω1,
µ
∇Yω2;X)−t(ω1, ω2;
µ
∇YX)}
− {∇µY[t(∇νZω1, ω2;X]−t(∇µY ν
∇Zω1, ω2;X)−t(∇νZω1,
µ
∇Yω2;X)−t(∇νZω1, ω2;∇µYX)}
− {∇µY[t(ω1,
ν
∇Zω2;X]−t(∇µYω1,
ν
∇Zω2;X)−t(ω1,
µ
∇Y ν
∇Zω2;X)−t(ω1,
ν
∇Zω2;∇µYX)}
− {
µ
∇Y[t(ω1, ω2;
ν
∇ZX]−t(
µ
∇Yω1, ω2;
ν
∇ZX)−t(ω1,
µ
∇Yω2;
ν
∇ZX)−t(ω1, ω2;
µ
∇Y ν
∇ZX)}.
wherefrom (∇νZ
µ
∇Y 2t
1−∇µY
∇νZ 2t
1)(ω1, ω2;X) = [Z, Y][2t
1(ω1, ω2;X)]
(4.1)
=−2t
1(∇νZ µ
∇Yω1−∇µY
∇νZω1, ω2;X)
−2t
1(ω1,∇νZ µ
∇Yω2−
µ
∇Y ν
∇Zω2;X)
−2t
1(ω1, ω2;∇νZ µ
∇YX−∇µY ν
∇ZX).
