����������� ������ 54 (2002), 219–226

UDK 514.763.4

���������� ������ ��� research paper

INFINITESIMAL DEFORMATIONS OF CURVATURE TENSORS AT NON-SYMMETRIC AFFINE CONNECTION SPACE

Ljubica S. Velimirović, Svetislav M. Minčić, Mića S. Stanković

Abstract. At the present work we consider infinitesimal deformations of geometric objects, especially of curvature tensors, at a space L_{N} of non-symmetric affine connection.

0. Introduction

We consider a space L_{N} of a non-symmetric affine connection L_{jk}^{i} with the torsion tensor T_{jk}^{i} = L_{jk}^{i} − L_{kj}^{i} at local coordinates x^{i} (i = 1,…, N). Basic information on infinitesimal deformations one can find at [4–11].

Let x^{i}, i = 1,…, N be local coordinates at L_{N} and let us define infinitesimal deformations of space L_{N} in the next manner.

Definition 0.1. A transformation f: L_{N} → L_{N}: x = (x^{1},…, x^{N}) ≡ (x^{i}) ↦ x̄ = (x̄^{1},…, x̄^{N}) ≡ (x̄^{i}), where

x̄ = x + z(x)ε,  (0.1)

or in local coordinates

x̄^{i} = x^{i} + z^{i}(x^{j})ε, i, j = 1,…, N,  (0.1′)

where ε is an infinitesimal, is called infinitesimal deformation of a space L_{N}, determined by the vector z = (z^{i}), which is called the field of infinitesimal deformation (0.1′).

Under deformation of a space according to (0.1′) some geometrical objects are deformed. The fact that geometric object is given at point x at local coordinates x^{i}, is denoted by A(i, x). Under deformation (0.1′) we get geometric object Ā, which can be considered at the same point and at the same coordinate system (see [6,7,10,11]).

AMS Subject Classification: 53 C 25, 53 A 45, 53 B 05

Keywords and phrases: Infinitesimal deformation, non-symmetric affine connection, Lie derivative, geometric object, curvature tensor.

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

Remark 0.1. In this study of infinitesimal deformations according to (0.1′), quantities of an order higher then the first with respect to ε are neglected.

Definition 0.2. The magnitude DA, the difference between deformed object Ā and initial object A at the same coordinate system and at the same point with respect to (0.1′), i.e.

DA = Ā(i, x) − A(i, x),  (0.2)

is called Lie difference (Lie differential), and the magnitude

L_{z}A = lim_{ε→0} \frac{DA}{ε} = lim_{ε→0} \frac{Ā(i, x) − A(i, x)}{ε}  (0.2′)

is Lie derivative of geometric object A(i, x) with respect to the vector field z = (z^{i}(x^{j})).

Using the relation (0.2), for deformed object Ā(i, x) we have

Ā(i, x) = A(i, x) + DA,  (0.2′′)

and thus we can express Ā, finding previously DA.

From (0.2′) we get

\frac{DA}{ε} = L_{z}A + ε_{1}, ε_{1} → 0, if ε → 0,

and DA = εL_{z}A + εε_{1}, i.e.

DA = εL_{z}A,  (0.3)

where εε_{1}, as an infinitesimal value of the higher order with respect to ε, is omitted.

As it is known (see [6,7]), for an arbitrary tensor t_{j_{1},...,j_{v}}^{i_{1},...,i_{u}} we have

\begin{aligned}Dt_{j_{1}...j_{v}}^{i_{1}...i_{u}} & = [t_{j_{1}...j_{v},p}^{i_{1}...i_{u}}z^{p} − ∑_{α=1}^{u} z_{,p}^{i_{α}} \binom{p}{i_{α}}t_{j_{1}...j_{v}}^{i_{1}...i_{u}} + ∑_{β=1}^{v} z_{,j_{β}}^{p} \binom{j_{β}}{p}t_{j_{1}...j_{v}}^{i_{1}...i_{u}}]ε \\ & = L_{z}t_{j_{1}...j_{v}}^{i_{1}...i_{u}}ε, \end{aligned}  (0.4)

where we denoted, for instance, z_{,p}^{i} = ∂z^{i}/∂x^{p}, and

\binom{p}{i_{α}}t_{j_{1}...j_{v}}^{i_{1}...i_{u}} = t_{j_{1}...j_{v}}^{i_{1}...i_{α−1}pi_{α+1}...i_{u}}, \binom{j_{β}}{p}t_{j_{1}...j_{v}}^{i_{1}...i_{u}} = t_{j_{1}...j_{β−1}pj_{β+1}...j_{v}}^{i_{1}...i_{u}}.

For the connection coefficients we have

DL_{jk}^{i} = (L_{jk,p}^{i}z^{p} + z_{,jk}^{i} − z_{,p}^{i}L_{jk}^{p} + z_{,j}^{p}L_{pk}^{i} + z_{,k}^{p}L_{jp}^{i})ε = L_{z}L_{jk}^{i}ε.  (0.5)

Because of non-symmetry of the connection, at L_{N} we can consider two types of covariant derivatives for a vector and four types for general tensor. So, denoting by |_{θ}(θ = 1,…, 4) the derivative of the type θ, we have ([1]–[3]):

t_{j_{1}...j_{v}}^{i_{1}...i_{u}}_{|_{1}_{2}_{3}_{4}m} = t_{j_{1}...j_{v},m}^{i_{1}...i_{u}} + ∑_{α=1}^{u} Li_{α} ; pm_{mp}_{pm}_{mp}\binom{p}{i_{α}}t_{j_{1}...j_{v}}^{i_{1}...i_{u}} − ∑_{β=1}^{v} Lp ; j_{β}m_{mj_{β}}_{mj_{β}}_{j_{β} m}\binom{j_{β}}{p}t_{j_{1}...j_{v}}^{i_{1}...i_{u}}. (0.6a − d)

By virtue of Ricci-type identities twelve curvature tensors in L_{N} are obtained [1,2,3]. Among these tensors only five of them are independent:

R_{1}_{jmn}^{i} = L_{jm,n}^{i} − L_{jn,m}^{i} + L_{jm}^{p}L_{pn}^{i} − L_{jn}^{p}L_{pm}^{i},  (0.7)
R_{2}_{jmn}^{i} = L_{mj,n}^{i} − L_{nj,m}^{i} + L_{mj}^{p}L_{np}^{i} − L_{nj}^{p}L_{mp}^{i},  (0.8)
R_{3}_{jmn}^{i} = L_{jm,n}^{i} − L_{nj,m}^{i} + L_{jm}^{p}L_{np}^{i} − L_{nj}^{p}L_{pm}^{i} + L_{nm}^{p}T_{pj}^{i},  (0.9)
R_{4}_{jmn}^{i} = L_{jm,n}^{i} − L_{nj,m}^{i} + L_{jm}^{p}L_{np}^{i} − L_{nj}^{p}L_{pm}^{i} + L_{mn}^{p}T_{pj}^{i},  (0.10)
R_{5}_{jmn}^{i} = \frac{1}{2}(L_{jm,n}^{i} + L_{mj,n}^{i} − L_{jn,m}^{i} − L_{nj,m}^{i}
+ L_{jm}^{p}L_{pn}^{i} + L_{mj}^{p}L_{np}^{i} − L_{jn}^{p}L_{mp}^{i} − L_{nj}^{p}L_{pm}^{i}).  (0.11)

If we define symmetric connection

L_{0}_{jk}^{i} = \frac{1}{2}(L_{jk}^{i} + L_{kj}^{i}),  (0.12)

its curvature tensor

R_{0}_{jmn}^{i} = L_{0}_{jm,n}^{i} − L_{0}_{jn,m}^{i} + L_{0}_{jm}^{p}L_{0}_{pn}^{i} − L_{0}_{jn}^{p}L_{0}_{pm}^{i}  (0.13)

is an ordinary Riemann-Christoffel’s curvature tensor of symmetric connection. Denoting by semicolon (;) covariant derivative with respect to symmetric connection then according to [3], we have

R_{1}_{jmn}^{i} = R_{o}_{jmn}^{i} + \frac{1}{2}T_{jm;n}^{i} − \frac{1}{2}T_{jn;m}^{i} + \frac{1}{4}T_{jm}^{p}T_{pn}^{i} − \frac{1}{4}T_{jn}^{p}T_{pm}^{i},  (0.14)
R_{2}_{jmn}^{i} = R_{0}_{jmn}^{i} − \frac{1}{2}T_{jm;n}^{i} + \frac{1}{2}T_{jn;m}^{i} + \frac{1}{4}T_{jm}^{p}T_{pn}^{i} − \frac{1}{4}T_{jn}^{p}T_{pm}^{i},  (0.15)
R_{3}_{jmn}^{i} = R_{0}_{jmn}^{i} + \frac{1}{2}T_{jm;n}^{i} + \frac{1}{2}T_{jn;m}^{i} − \frac{1}{4}T_{jm}^{p}T_{pn}^{i} + \frac{1}{4}T_{jn}^{p}T_{pm}^{i} − \frac{1}{2}T_{mn}^{p}T_{pj}^{i},(0.16)

R_{4}_{jmn}^{i} = R_{0}_{jmn}^{i} + \frac{1}{2}T_{jm;n}^{i} + \frac{1}{2}T_{jn;m}^{i} − \frac{1}{4}T_{jm}^{p}T_{pn}^{i} + \frac{1}{4}T_{jn}^{p}T_{pm}^{i} + \frac{1}{2}T_{mn}^{p}T_{pj}^{i} (0.17)

R_{5}_{jmn}^{i} = R_{0}_{jmn}^{i} + \frac{1}{4}T_{jm}^{p}T_{pn}^{i} + \frac{1}{4}T_{jn}^{p}T_{pm}^{i}.  (0.18)

At (0.14–18) all the addends at the right side are tensors.

At the present work we will consider infinitesimal deformations of cited five curvature tensors.

1. Infinitesimal deformations of curvature tensor R_{1}

1.1. According to the (0.7) for deformed first curvature tensor at L_{N}, i.e. for

R̄_{1}, we have

R̄_{1}_{jmn}^{i}(x) = L̄_{jm,n}^{i} − L̄_{jn,m}^{i} + L̄_{jm}^{p}L̄_{pn}^{i} − L̄_{jn}^{p}L̄_{pm}^{i},  (1.1)

and with respect to L̄_{jm}^{i}(x) = L_{jm}^{i}(x) + DL_{jm}^{i}, we have

R̄_{1}_{jmn}^{i} = (L_{jm}^{i} + DL_{jm}^{i})_{,n} − (L_{jn}^{i} + DL_{jn}^{i})_{,m}

+ (L_{jm}^{p} + DL_{jm}^{p})(L_{pn}^{i} + DL_{pn}^{i}) − (L_{jn}^{p} + DL_{jn}^{p})(L_{pm}^{i} + DL_{pm}^{i}).

Developing this and omitting the members of the form DL_{...}^{...} · DL_{...}^{...}, as they include (ε)^{2}, we have

R̄_{1}_{jmn}^{i} = L_{jm,n}^{i} + (DL_{jm}^{i})_{,n} − L_{jn,m}^{i} − (DL_{jn}^{i})_{,m} + L_{jm}^{p}L_{pn}^{i}

+ L_{jm}^{p}DL_{pn}^{i} + (DL_{jm}^{p})L_{pn}^{i} − L_{jn}^{p}L_{pm}^{i} − L_{jn}^{p}DL_{pm}^{i} − (DL_{jn}^{p})L_{pm}^{i}. (1.2)

As DL_{jm}^{i} is a tensor, we can consider covariant derivative:

(DL_{jm}^{i})_{|_{1}n} = (DL_{jm}^{i})_{,n} + L_{pn}^{i}DL_{jm}^{p} − L_{jn}^{p}DL_{pm}^{i} − L_{mn}^{p}DL_{jp}^{i}

where

(DL_{jm}^{i})_{,n} + L_{pn}^{i}DL_{jm}^{p} = (DL_{jm}^{i})_{|_{1}n} + L_{jn}^{p}DL_{pm}^{i} + L_{mn}^{p}DL_{jp}^{i}  (1.3)

and in the same manner

(DL_{jn}^{i})_{,m} + L_{pm}^{i}DL_{jn}^{p} = (DL_{jn}^{i})_{|_{1}m} + L_{jm}^{p}DL_{pn}^{i} + L_{nm}^{p}DL_{jp}^{i}  (1.3′)

If we have in mind (0.7), (1.3, 3′), the equation (1.2) becomes

R̄_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + (DL_{jm}^{i})_{|_{1}n} + L_{jn}^{p}DL_{pm}^{i} + L_{mn}^{p}DL_{jp}^{i}

− (DL_{jn}^{i})_{|_{1}m} − L_{im}^{p}DL_{pn}^{i} − L_{nm}^{p}DL_{jp}^{i} + L_{jm}^{p}DL_{pn}^{i} − L_{jn}^{p}DL_{pm}^{i},

i.e.

R̄_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + (DL_{jm}^{i})_{|_{1}n} − (DL_{jn}^{i})_{|_{1}m} + T_{mn}^{p}DL_{jp}^{i}.  (1.4)

From here

DR_{1}_{jmn}^{i} = R̄_{1}_{jmn}^{i} − R_{1}_{jmn}^{i} = (DL_{jm}^{i})_{|_{1}n} − (DL_{jn}^{i})_{|_{1}m} + T_{mn}^{p}DL_{jp}^{i},  (1.4′)

and (dividing with ε):

L_{z}R_{1}_{jmn}^{i} = (L_{z}L_{jm}^{i})_{|_{1}n} − (L_{z}L_{jn}^{i})_{|_{1}m} + T_{mn}^{p}L_{z}L_{jp}^{i}.  (1.4′′)

Infinitesimal deformations of curvature tensors 223

1.2. We can also start from (0.14), and then we have

¯R_{1}_{jmn}^{i} = ¯ R_{0}_{jmn}^{i} + \frac{1}{2}T̄_{jm;n}^{i} − \frac{1}{2}T̄_{jn;m}^{i} + \frac{1}{4}T̄_{jm}^{p}T̄_{pn}^{i} − \frac{1}{4}T̄_{jn}^{p}T̄_{pm}^{i}

= R_{0}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{2}(T_{jm}^{i} + DT_{jm}^{i})_{;n} − \frac{1}{2}(T_{jn}^{i} + DT_{jn}^{i})_{;m}

+ \frac{1}{4}(T_{jm}^{p} + DT_{jm}^{p})(T_{pn}^{i} + DT_{pn}^{i}) − \frac{1}{4}(T_{jn}^{p} + DT_{jn}^{p})(T_{pm}^{i} + DT_{pm}^{i}).

Omitting the members containing DT · DT, we get

¯R_{1}_{jmn}^{i} = R_{0}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{2}T_{jm;n}^{i} + \frac{1}{2}(DT_{jm}^{i})_{;n} − \frac{1}{2}T_{jn;m}^{i} − \frac{1}{2}(DT_{jn}^{i})_{;m}

+ \frac{1}{4}(T_{jm}^{p}T_{pn}^{i} + T_{jm}^{p}DT_{pn}^{i} + DT_{jm}^{p}T_{pn}^{i} − T_{jn}^{p}T_{pm}^{i} − T_{jn}^{p}DT_{pm}^{i} − DT_{jn}^{p}T_{pm}^{i}).

So, with respect to (0.14), one obtains

¯R_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} +DR_{0}_{jmn}^{i} + \frac{1}{2}(DT_{jm}^{i})_{;n} − \frac{1}{2}(DT_{jn}^{i})_{;m} + \frac{1}{4}D(T_{jm}^{p}T_{pn}^{i} −T_{jn}^{p}T_{pm}^{i}), (1.5)

wherefrom

DR_{1}_{jmn}^{i} = DR_{0}_{jmn}^{i} + \frac{1}{2}(DT_{jm}^{i})_{;n} − \frac{1}{2}(DT_{jn}^{i})_{;m} + \frac{1}{4}D(T_{jm}^{p}T_{pn}^{i} − T_{jn}^{p}T_{pm}^{i}) (1.5′)

where ; denotes covariant derivative with respect to L_{0} (0.12), and (dividing with ε):

L_{z}R_{1}_{jmn}^{i} = L_{z}R_{0}_{jmn}^{i} + \frac{1}{2}(L_{z}T_{jm}^{i})_{;n} − \frac{1}{2}(L_{z}T_{jn}^{i})_{;m} + \frac{1}{4}L_{z}(T_{jm}^{p}T_{pn}^{i} − T_{jn}^{p}T_{pm}^{i}). (1.5′′)

2. Infinitesimal deformations of curvature tensor R_{2} 2.1. In the similar manner as for R_{1}, using (0.8), we get

R̄_{2}_{jmn}^{i} = R_{2}_{jmn}^{i} + (DL_{mj}^{i})_{|_{2}n} − (DL_{nj}^{i})_{|_{2}m} + T_{nm}^{p}DL_{pj}^{i}.  (2.1)

and corresponding expressions for DR_{2}_{jmn}^{i} and L_{z}R_{2}_{jmn}^{i}.

2.1. Starting from (0.15), we obtain

R̄_{2}_{jmn}^{i} = R_{2}_{jmn}^{i} +DR_{0}_{jmn}^{i} − \frac{1}{2}(DT_{jm}^{i})_{;n} + \frac{1}{2}(DT_{jn}^{i})_{;m} + \frac{1}{4}D(T_{jm}^{p}T_{pn}^{i} −T_{jn}^{p}T_{pm}^{i}). (2.2)

3. Infinitesimal deformations of curvature tensor R_{3} 3.1. Starting from (0.9), we obtain

R̄_{3}_{jmn}^{i} = L_{jm,n}^{i} +(DL_{jm}^{i})_{,n} −L_{nj,m}^{i} −(DL_{nj}^{i})_{,m} +L_{jm}^{p}L_{np}^{i} +L_{jm}^{p}DL_{np}^{i} +DL_{jm}^{p}L_{np}^{i}

− L_{nj}^{p}L_{pm}^{i} − L_{nj}^{p}DL_{pm}^{i} − DL_{nj}^{p}L_{pm}^{i} + L_{nm}^{p}T_{pj}^{i} + L_{nm}^{p}DT_{pj}^{i} + DL_{nm}^{p}T_{pj}^{i}. (3.1)

224 Lj. S. Velimirović, S. M. Minčić, M. S. Stanković

In relation to

(DL_{jm}^{i})_{|_{4}n} = (DL_{jm}^{i})_{,n} + L_{np}^{i}DL_{jm}^{p} − L_{jn}^{p}DL_{pm}^{i} − L_{mn}^{p}DL_{jp}^{i},
(DL_{nj}^{i})_{|_{3}m} = (DL_{nj}^{i})_{,m} + L_{pm}^{i}DL_{nj}^{p} − L_{mn}^{p}DL_{pj}^{i} − L_{mj}^{p}DL_{np}^{i},

one obtains respectively

(DL_{jm}^{i})_{,n} + L_{np}^{i}DL_{jm}^{p} = (DL_{jm}^{i})_{|_{4}n} + L_{jn}^{p}DL_{pm}^{i} + L_{mn}^{p}DL_{jp}^{i},
(DL_{nj}^{i})_{,m} + L_{pm}^{i}DL_{nj}^{p} = (DL_{nj}^{i})_{|_{3}m} + L_{mn}^{p}DL_{pj}^{i} + L_{mj}^{p}DL_{np}^{i}.

Using these equations and (0.9), the equation (3.1) becomes

R̄_{3}_{jmn}^{i} = R_{3}_{jmn}^{i} + (DL_{jm}^{i})_{|_{4}n} + L_{jn}^{p}DL_{pm}^{i} + L_{mn}^{p}DL_{jp}^{i} − (DL_{nj}^{i})_{|_{3}m}

− L_{mn}^{p}DL_{pj}^{i} − L_{mj}^{p}DL_{np}^{i} + L_{jm}^{p}DL_{np}^{i} − L_{nj}^{p}DL_{pm}^{i} + L_{nm}^{p}DT_{pj}^{i} + T_{pj}^{i}DL_{nm}^{p}, that is

R̄_{3}_{jmn}^{i} = R_{3}_{jmn}^{i} + (DL_{jm}^{i})_{|_{4}n} − (DL_{nj}^{i})_{|_{3}m}

+ T_{jn}^{p}DL_{pm}^{i} + T_{jm}^{p}DL_{np}^{i} + L_{mn}^{p}DT_{jp}^{i} + D(T_{pj}^{i}L_{nm}^{p}). (3.2)

3.2. According to (0.16), we get

R̄_{3}_{jmn}^{i} = R_{0}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{2}T_{jm;n}^{i} + \frac{1}{2}(DT_{jm}^{i})_{;n} + \frac{1}{2}T_{jn;m}^{i} + \frac{1}{2}(DT_{jn}^{i})_{;m}

− \frac{1}{4}T_{jm}^{p}T_{pn}^{i} − \frac{1}{4}T_{jm}^{p}DT_{pn}^{i} − \frac{1}{4}DT_{jm}^{p}T_{pn}^{i} + \frac{1}{4}T_{jn}^{p}T_{pm}^{i} + \frac{1}{4}T_{jn}^{p}DT_{pm}^{i}

+ \frac{1}{4}DT_{jn}^{p}T_{pm}^{i} − \frac{1}{2}T_{mn}^{p}T_{pj}^{i} − \frac{1}{2}T_{mn}^{p}DT_{pj}^{i} − \frac{1}{2}DT_{mn}^{p}T_{pj}^{i}

i.e.

R̄_{3}_{jmn}^{i} = R_{3}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{2}(DT_{jm}^{i})_{;n} + \frac{1}{2}(DT_{jn}^{i})_{;m}

+ \frac{1}{4}D(T_{pm}^{i}T_{jn}^{p} − T_{pn}^{i}T_{jm}^{p} − 2T_{pj}^{i}T_{mn}^{p}). (3.3)

4. Infinitesimal deformations of curvature tensor R_{4} 4.1. Starting from (0.10) and using the procedure which is similar to that used

for R_{3}, we obtain

R̄_{4}_{jmn}^{i} = R_{4}_{jmn}^{i} + (DL_{jm}^{i})_{|_{4}n} − (DL_{nj}^{i})_{|_{3}m}

+ T_{jm}^{p}DL_{np}^{i} + T_{jn}^{p}DL_{pm}^{i} + L_{mn}^{p}DT_{jp}^{i} + D(T_{pj}^{i}L_{mn}^{p}) (4.1)

Infinitesimal deformations of curvature tensors 225

4.2. Using (0.17), we obtain for R_{4}

R̄_{4}_{jmn}^{i} = R_{4}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{2}(DT_{jm}^{i})_{;n} + \frac{1}{2}(DT_{jn}^{i})_{;m}

+ \frac{1}{4}D(T_{pm}^{i}T_{jn}^{p} − T_{pn}^{i}T_{jm}^{p} + 2T_{pj}^{i}T_{mn}^{p}). (4.2)

5. Infinitesimal deformations of the tensor R_{5}

5.1 According to (0.11) we have

2R̄_{5}_{jmn}^{i} = L_{jm,n}^{i} + (DL_{jm}^{i}), n + L_{mj,n}^{i} + (DL_{mj}^{i}), n − L_{jn,m}^{i} − (DL_{jn}^{i}), m

− L_{nj,m}^{i} − (DL_{nj}^{i}), m + L_{jm}^{p}L_{pn}^{i} + L_{jm}^{p}DL_{pn}^{i} + (DL_{jm}^{p})L_{pn}^{i} + L_{mj}^{p}L_{np}^{i}

+ L_{mj}^{p}DL_{np}^{i} + (DL_{mj}^{p})L_{np}^{i} − L_{jn}^{p}L_{mp}^{i} − L_{jn}^{p}DL_{mp}^{i}

− (DL_{jn}^{p})L_{mp}^{i} − L_{nj}^{p}L_{pm}^{i} − L_{nj}^{p}DL_{pm}^{i} − (DL_{nj}^{p})L_{pm}^{i}. (5.1)

By differentiating the Lie differences one obtains

(DL_{jm}^{i})_{|_{3}n} = (DL_{jm}^{i})_{,n} + L_{pn}^{i}DL_{jm}^{p} − L_{nj}^{p}DL_{pm}^{i} − L_{nm}^{p}DL_{jp}^{i},
(DL_{mj}^{i})_{|_{4}n} = (DL_{mj}^{i})_{,n} + L_{np}^{i}DL_{mj}^{p} − L_{mn}^{p}DL_{pj}^{i} − L_{jn}^{p}DL_{mp}^{i},
(DL_{nj}^{i})_{|_{3}m} = (DL_{nj}^{i})_{,m} + L_{pm}^{i}DL_{nj}^{p} − L_{mn}^{p}DL_{pj}^{i} − L_{mj}^{p}DL_{np}^{i},
(DL_{jn}^{i})_{|_{4}} = (DL_{jn}^{i})_{,m} + L_{mp}^{i}DL_{jn}^{p} − L_{jm}^{p}DL_{pn}^{i} − L_{nm}^{p}DL_{jp}^{i}.

If we substitute at (5.1) three members from the right sides of previous four equations, we get

¯R_{5}_{jmn}^{i} = R_{5}_{jmn}^{i} + \frac{1}{2}[(DL_{jm}^{i})_{|_{3}n} − (DL_{nj}^{i})_{|_{3}m} + (DL_{mj}^{i})_{|_{4}n} − (DL_{jn}^{i})_{|_{4}m}].  (5.2)

5.2. If we start from (0.18), we have

R̄_{5}_{jmn}^{i} = R_{0}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{4}[T_{jm}^{p}T_{pn}^{i} + T_{jm}^{p}DT_{pn}^{i}

+ (DT_{jm}^{p})T_{pn}^{i} + T_{jn}^{p}T_{pm}^{i} + T_{jn}^{p}DT_{pm}^{i} + (DT_{jn}^{p})T_{pm}^{i}].

According to (0.18) the previous equation becomes:

R̄_{5}_{jmn}^{i} = R_{5}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{4}[T_{jm}^{p}DT_{pn}^{i} + (DT_{jm}^{p})T_{pn}^{i} + T_{jn}^{p}DT_{pm}^{i} + (DT_{jn}^{p})T_{pm}^{i}],

i.e.

R̄_{5}_{jmn}^{i} = R_{5}_{jmn}^{i} + DR_{0}_{jmn}^{i} + \frac{1}{4}D(T_{jm}^{p}T_{pn}^{i} + T_{jn}^{p}T_{pm}^{i}).  (5.3)

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(received 23.12.2002)

Faculty of Science and Mathematics, Višegradska 33, University of Niš, 18000 Niš, Yugoslavia