Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 1(2012), Pages 72-82.
THRIAD GEODESIC COMPOSITIONS IN FOUR DIMENSIONAL
SPACE WITH AN AFFINE CONNECTEDNESS WITHOUT A
TORSION
(COMMUNICATED BY KRISHNAN LAL DUGAL)
MUSA AJETI
Abstract. Let A4 be an affinely connected space without a torsion. Followβ
β
β σ 2
β
ing [7] we introduce the affinors aβ
α , bα and c̃α = icα = −iaσ bα (i = −1)
which define the compositions X2 × X̄2 , Y2 × Ȳ2 and Z2 × Z̄2 , respectively.The
first two compositions are conjugate. The composition U2 × Ū2 generated by
β
β
β
the affinor dβ
α = aα + bα + cα is considered too. We have found necessary and
sufficient condition for any of the above compositions to be of the kind (g − g).
Characterisics of the spaces A4 that contain such compositions are obtained.
Conections between Richis tensor and fundamental density of Eq A4 are establish when the space is equiaffine and the compositions X2 × X̄2 , Y2 × Ȳ2 , Z2 × Z̄2 ,
are simultaneously of the kind (g − g).
1. Introduction
Let AN be a space with a symmetric affine connectedness without a torsion,
defined by Γγαβ . Let consider a composition Xn × Xm of two differentiable basic
manifolds Xn and Xm (n + m = N ) in the space AN . For every point of the space of
compositions AN (Xn ×Xm ) there are two positions of the basic manifolds, which we
denotes by P (Xn ) and P (Xm ) [3]. The defining of a composition in the space AN
is equivalent to defining of a field of an affinor aβα , that satisfies the condition [2], [3]
(1)
aβσ aσα = δαβ
The affinor aβα is called an affinor of the composition [2]. According to [3] and
[5] the condition for integrability of the structure is aσβ ∇[αaνσ ] − aσα ∇[βaνσ ] = 0 The
projective affinors n aσα and m aσα [3], [4], defined by the equalities n aσα = 12 (δαβ + aβα ),
m σ
aα = 21 (δαβ − aβα ) satisfy the conditions n aβα +m aβα = δαβ , n aβα +m aβα = aβα For
2000 Mathematics Subject Classification. 53A40, 53A55.
Key words and phrases. Affinely connected spaces, spaces of compositions, affinors of compositions, geodesic composition.
c
⃝2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted August 26, 2011. Published September 19, 2011.
72
THRIAD GEODESIC COMPOSITIONS IN FOUR DIMENSIONAL...
n
n
73
m
every vector v α ∈ AN (Xn × Xm ) we have v α = aβα v β +m aβα v β = Vα + Vα , where
n
n
m
V α = aβα v β ∈ P (Xn ), V α =m aβα v β ∈ P (Xm )[4].
The composition Xn × Xm ∈ AN (n + m = N ) for which the positions P (Xn )
and P (Xm ) are parallelly translated along any line of Xn and Xm , respectively is
called a composition of the kind (g − g)[3] or geodesic composition [6]. According
to [3] the geodesic composition is characterized with the equality
aσα ∇β aνσ + aσβ ∇σ aνα = 0
(2)
Let A4 be an space with affine connectedness without a torsion, defined by Γσαβ (α, β, σ =
1; 2; 3; 4). Let υ α , υ α , υ α , υ α are independent vector fields in A4 . Following [7]
1
2
3
σ
4
we define the covectors v α by the equalities
α
σ
υ β υ σ = δσβ ↔ υ β υ β = δασ
(3)
α
α
According to [6], [7] we can define the affinor
1
2
3
4
aβα = υ β υ α + υ β υ α − υ β υ α − υ β υ α
(4)
1
2
3
4
that satisfy the equalities (1). The affinor (4) defines a composition (Xn × Xm ) in
A4 . The projective affinors of the composition (Xn × Xm ) are [7]
1
1
2
2
3
4
aβα = υ β υ α + υ β υ α , aβα = υ β υ α + υ β υ α
(5)
1
2
3
4
Following [7] we choose the net (υ , υ , υ , υ ) for a coordinate one. Then we have
1 2 3 4
α
α
υ (1, 0, 0, 0) , υ (0, 1, 0, 0) , υ α (0, 0, 1, 0) , υ α (0, 0, 0, 1)
1
2
3
4
(6)
1
4
3
2
υ α (1, 0, 0, 0) , υ α (0, 1, 0, 0) , υ α (0, 0, 1, 0) , υ α (0, 0, 0, 1)
Let consider the vectors [7]
wα = υ α + υ α , wα = υ α + υ α , wα = υ α − υ α , wα = υ α − υ α
(7)
1
1
2
We define the covectors
2
α
wσ
2
4
3
1
2
4
2
4
by the equalities
β
α
wν wσ = δσν ⇔ wσ wσ = δαβ
(8)
α
α
From (3) and (8) follow
(9)
1
wα =
) 2
) 3
) 4
)
1 (2
1 (1
1 (2
1 (1
2
4
2
4
υα + υα , wα =
υ α + υ α , wα =
υ α − υ α , wα =
υα − υα
2
2
2
2
Let consider the affinor
74
MUSA AJETI
1
2
3
4
bβα = wβ wα + wβ wα −wβ wα − wβ wα ,
(10)
1
2
3
4
β
which according to [7] satisfies the equality bβα bα
σ = δσ . Therefore the affinor (10)
defines a composition Y2 × Y 2 in A4 . According to [7] the compositions X2 × X 2
and Y2 × Y 2 are conjugate. By (3), (7), (8) and (10) we obtain
3
1
4
2
bβα = υ β υ α + υ β υ α +υ β υ α + υ β υ α .
(11)
1
3
2
4
Following [7] let consider the affinor cβα = −aβα bα
σ , which satisfies the equality
cβσ cσα = −δαβ . With the help of (3), (4), (11) we establish
1
3
2
4
cβα = υ β υ α − υ β υ α +υ β υ α − υ β υ α .
(12)
3
1
4
2
∼
The affinor c βα = i cβα , where i2 = −1, defines a composition Z2 × Z 2 in A4 .
§2.
Geodesic compositions in spaces A4
According to [8] we have the following derivative equations
ν
α
α
ν
▽ σ υ β =T σ υ β , ▽ σ υ β = − T σ υ β .
(13)
α
α
ν
ν
Let consider the composition X2 × X 2 and let accept:
α, β, γ, σ, ν, τ ∈ {1, 2, 3, 4}; i, j, k, s ∈ {1, 2}; i, j, k, s ∈ {3, 4}.
Theorem 1.1. The composition X2 × X 2 is of the kind (g − g) if and only if the
coefficients of the derivative equations (13) satisfy the conditions
i
i
T α υ α = 0 , T α υ α = 0.
(14)
s
k
s
k
Proof: According to (4) and (13) we have
τ
1
1
τ
τ
2
2
τ
▽β aνβ = T β υ ν υ σ − T β υ ν υ σ + T β υ ν υ σ − T β υ ν υ σ
τ
1
τ
τ
τ
2
1
2
(15)
τ
3
3
τ
τ
4
4
τ
− T β υν υσ + T β υν υσ − T β υν υσ + T β υν υσ .
3
τ
τ
3
4
τ
τ
4
σ
Taking into account the independence of the covectors υ α and using (2), (3), (4),
(15) we find the equalities
3
3
4
4
(δβσ + aσβ )(T σ υ ν + T σ υ ν ) = 0 , (δβσ + aσβ )(T σ υ ν + T σ υ ν ) = 0 ,
1
3
1
1
4
3
1
4
(16)
1
2
1
2
(δβσ − aσβ )(T σ υ ν + T σ υ ν ) = 0 , (δβσ − aσβ )(T σ υ ν + T σ υ ν ) = 0 .
3
3
3
4
4
ν
3
4
4
Because of the independence of the vectors υ it follows an equivalence of (16)
α
to the following equalities
THRIAD GEODESIC COMPOSITIONS IN FOUR DIMENSIONAL...
75
3
3
4
4
3
3
4
4
1
1
1
1
2
2
2
2
1
1
2
2
1
1
2
2
3
3
3
3
4
4
4
4
T β +aσβ T σ = 0, T β +aσβ T σ = 0, T β +aσβ T σ = 0, T β +aσβ T σ = 0,
(17)
T β −aσβ T σ = 0, T β −aσβ T σ = 0, T β −aσβ T σ = 0, T β −aσβ T σ = 0.
Now it is easy to see that the equalities (14) follow after contraction by υ β and
1
υ β for the first four equalities of (17) and by υ β and υ β for the last four equalities
2
3
4
of (17). Let?s note that the equalities (17) are proved in [6] by another approach.
(
)
Corollary 1.2. If the net υ , υ , υ , υ is chosen as a coordinate one then the com1 2 3 4
position X2 × X 2 from the kind (g − g) characterizes by the following equalities for:
(1.1) the coefficients of the derivative equations
i
(18)
i
Ts = 0, Ts = 0
k
k
1.2) the coefficients of the connectedness
Γisk , Γisk = 0.
(19)
(
Proof:
Let choose the net
)
υ, υ, υ, υ
1 2 3 4
for a coordinate one. Then by (6) and
ν
(14) we find (18). According to [1] and (13) we can write ∂σ υ β + Γβσν υ ν =T σ υ β
α
α
α
ν
from where using (6) we obtain
β
Γβσα =T σ .
(20)
α
The equalities (19) follow from (18) and (20). Let’s note that the equalities (19) are
obtained in [3], when the coordinates are adaptive with the composition X2 × X 2 .
This happens so, because the chosen coordinate net raises adaptive with the composition coordinates.
From (19) and Rαβσν. = 2∂ [α Γβ]σ − 2Γτσ [α Γβ]τ [1] we establish the validity of the
following statement:
Fact 1. When the composition
X
) 2 ×X 2 is of the kind (g −g) then in the parameters
(
of the coordinate net υ , υ , υ , υ the tensor of the curvature satisfy the conditions
1 2 3 4
Rijks. = 0 , Rijks. = 0.
Theorem 1.3. The composition Y2 × Y 2 is of the kind (g − g) if and only if the
coefficients of the derivative equations satisfy the conditions
(
)
(
)
(
)
(
)
1
3
1
3
1
3
1
3
σ
σ
σ
T σ − T σ υ + T σ − T σ υ = 0, T σ − T σ υ + T σ − T σ υ σ = 0,
1
3
1
3
1
3
1
3
3
3
1
1
76
MUSA AJETI
(
)
(
)
(
)
(
)
1
3
1
3
1
3
1
3
T σ − T σ υ σ + T σ − T σ υ σ = 0, T σ − T σ υ σ + T σ − T σ υ σ = 0,
1
3
2
3
1
4
1
3
1
3
1
2
)
(
)
(
)
(
)
(
2
4
2
4
2
4
2
4
T σ − T σ υ σ + T σ − T σ υ σ = 0, T σ − T σ υ σ + T σ − T σ υ σ = 0,
1
3
1
3
1
3
3
3
1
1
1
3
(
)
(
)
(
)
(
)
2
4
2
4
2
4
2
4
T σ − T σ υ σ + T σ − T σ υ σ = 0, T σ − T σ υ σ + T σ − T σ υ σ = 0,
1
(21)
3
2
3
1
4
1
(
)
(
)
1
3
1
3
σ
T σ − T σ υ + T σ − T σ υσ
2
4
1
4
2
3
(
)
(
)
1
3
1
3
T σ − T σ υσ + T σ − T σ υσ
2
4
2
4
2
4
(
)
(
)
2
4
2
4
T σ − T σ υσ + T σ − T σ υσ
2
4
1
4
2
3
(
)
(
)
2
4
2
4
T σ − T σ υσ + T σ − T σ υσ
2
4
2
4
2
4
= 0,
= 0,
= 0,
= 0,
3
4
3
1
2
(
)
(
)
1
3
1
3
σ
T σ − T σ υ + T σ − T σ υσ
2
4
3
4
2
1
(
)
(
)
1
3
1
3
T σ − T σ υσ + T σ − T σ υσ
2
4
1
4
2
2
(
)
(
)
2
4
2
4
T σ − T σ υσ + T σ − T σ υσ
2
4
3
4
2
1
(
)
(
)
2
4
2
4
T σ − T σ υσ + T σ − T σ υσ
2
4
4
4
2
2
= 0,
= 0,
= 0,
= 0.
Proof: Because of the equalities (11) and (14) we have
ν
3
3
ν
ν
1
1
ν
▽σ bβα = T σ υ β υ α − T σ υ β υ α + T σ υ β υ α − T σ υ β υ α
ν
1
ν
ν
3
1
ν
3
(22)
ν
4
4
ν
ν
2
2
ν
+ T σ υβ υα − T σ υβ υα + T σ υβ υα − T σ υβ υα .
ν
2
ν
2
ν
4
ν
4
Transforming the condition bσα ▽β bνσ + bσβ ▽σ bνα = 0 with the help of (3), (11), (22)
σ
and using the independence of the covectors υ α we obtain the following equalities
(
)
(
)
1
3
1
3
2
4
2
4
T β − T β + bσβ T σ − T σ = 0 , T β − T β + bσβ T σ − T σ = 0,
1
(23)
3
3
1
1
3
3
1
)
(
)
(
4
2
4
2
3
1
3
1
T β − T β + bσβ T σ − T σ = 0 , T β − T β + bσβ T σ − T σ = 0.
4
2
2
4
4
2
4
2
α
Now after contraction by υ it is easy to see the equivalence of (23) to (21).
σ
(
)
Corollary 1.4. If the net υ , υ , υ , υ is chosen as a coordinate one then the com1 2 3 4
position Y2 × Y 2 from the kind (g − g) characterizes by the following equalities for:
2.1) the coefficients of the derivative equations
1
3
3
1
1
3
1
3
T α − T α =T α − T α
2
4
4
2
1
3
1
3
, T α − T α =T α − T α
(24)
1
3
3
1
2
4
2
4
T α − T α =T α − T α
2
4
4
2
2
4
2
4
, T α − T α =T α − T α ,
2.2) the coefficients of the connectedness
,
THRIAD GEODESIC COMPOSITIONS IN FOUR DIMENSIONAL...
77
α
α
α
α
α
α
α
α
α
Γα
11 +Γ33 = 2Γ13 , Γ22 +Γ44 = 2Γ24 , Γ12 +Γ34 = Γ14 +Γ23 ,
(25)
as when α accepts consecutively the values 1, 2, 3, 4 then α accepts the values 3, 4, 1, 2,
respectively.
(
Proof:
Let choose the net
)
υ, υ, υ, υ
1 2 3 4
for a coordinate net. With the help of
(6) and (21) we find (24).
Then by (20) and (24) we obtain (25).
Theorem 1.5. The composition Z2 × Z 2 is of the kind (g − g) if and only if the
coefficients of the derivative equations (13) satisfy the conditions
)
(
)
(
)
(
)
(
3
1
3
1
3
1
3
1
T σ − T σ υσ = T σ + T σ υσ , T σ − T σ υσ = T σ + T σ υσ ,
4
1
2
3
3
3
1
1
1
3
3
1
)
)
(
)
(
)
(
(
3
1
1
3
3
1
1
3
T σ − T σ υσ = T σ + T σ υσ , T σ − T σ υσ = T σ + T σ υσ ,
2
1
4
3
1
1
3
1
3
3
1
3
(
)
(
)
(
)
(
)
2
4
2
4
2
4
2
4
T σ − T σ υσ = T σ + T σ υσ , T σ − T σ υσ = T σ + T σ υσ ,
1
3
1
3
1
3
1
3
2
3
1
4
(
)
(
)
(
)
(
)
4
2
2
4
3
2
2
4
T σ − T σ υσ = T σ + T σ υσ , T σ − T σ υσ = T σ + T σ υσ ,
3
(26)
1
3
)
(
3
1
T σ − T σ υσ
4
(2
)1
3
1
T σ − T σ υσ
2
(4
)3
2
4
T σ − T σ υσ
4
)1
(2
2
4
T σ − T σ υσ
2
4
3
3
1
1
4
1
4
3
1
2
)
)
(
(
3
1
3
1
σ
= T σ − T σ υ , T σ − T σ υ = T σ + T σ υσ ,
2
4
2
(4
)3 ( 2
) 2
( 4
) 4
3
1
3
1
1
3
= T σ + T σ υσ , T σ − T σ υσ = T σ + T σ υσ ,
4
2
2
(2
) 1 (4
)4
(4
)2
2
4
2
4
2
4
σ
σ
= T σ + T σ υ , T σ − T σ υ = T σ + T σ υσ ,
2
4
2
)2
(4
)4
(4
) 3 (2
2
4
2
2
4
4
= T σ + T σ υσ , T σ − T σ υσ = T σ + T σ υσ .
(
1
3
4
2
)
σ
1
2
4
4
4
2
2
Proof: By the equalities (12) and (14) we obtain
ν
1
1
ν
ν
3
3
ν
▽σ cβα = T σ υ β υ α − T σ υ β υ α − T σ υ β υ α + T σ υ β υ α
3
ν
ν
3
1
ν
ν
1
(27)
ν
2
2
ν
ν
4
2
ν
+ T σ υβ υα − T σ υβ υα − T σ υβ υα − T σ υβ υα .
4
ν
ν
4
2
ν
ν
2
Transforming the condition cσα ▽β bνσ + cσβ ▽σ bνα = 0 with the help of (3), (12), (27)
σ
and using the independence of the covectors υ α we obtain the following equalities
(
)
(
)
3
1
1
3
4
2
2
4
T β − T β + cσβ T σ + T σ = 0 , T β − T β + cσβ T σ + T σ = 0,
3
(28)
1
3
1
3
1
3
1
78
MUSA AJETI
(
)
(
)
3
1
3
1
4
2
2
4
T β − T β + cσβ T σ + T σ = 0 , T β − T β + cσβ T σ + T σ = 0.
4
2
2
4
4
2
4
2
α
Now after contraction by υ it is easy to see the equivalence of (28) to (26).
σ
(
)
Corollary 1.6. If the net v, v, v, v, is chosen as a coordinate one then
1 2 3 4
the composition Z2 × Z 2 from the kind (g − g) characterizes by the following
equalities for: 3.1) the coefficients of the derivative equations
)
(
)
(
3
4
4
1
3
3
1
2
T α−T α =ϵ T α+T α , T α−T α =ϵ T α+T α
1
3
3
3 )
3 )
(1
(1
(29) 11
3
3
1
2
4
2
4
T α−T α =ϵ T α+T α , T α−T α =ϵ T α+T α
2
4
2
4
2
4
2
4
3.2) the coefficients of the connectedness
(30)
Γα11 − Γα33 = 2ϵΓα13 , Γα22 − Γα44 = 2ϵΓα24 , Γα12 − Γα34 = Γα14 + Γα23 ,
as when α accepts consecutively the values 1; 2; 3; 4 then α accepts the
values 3; 4; 1; 2; respectively and
ϵ = 1 for α = 1, 2, ϵ = −1 for α = 3, 4.
(
)
Proof. Let choose the net v, v, v, v, for a coordinate net. Then the equal1 2 3 4
ities (29) follow by (6) and (26), and the equalities (30) follow by (20) and
(29).
Using (19) (25) and (30) we establish the validity of the following statement:
Fact 2. If two of the compositions X2 × X 2 , Y2 × Y 2 , Z2 × Z 2 are from the
kind (g - g) then and the third composition is of the kind (g - g).
Since from (19), (25) and (30) it follows
(31)
α
Γαij = Γij
= 0, Γα13 = Γα44 = 0, Γα14 + Γα23 = 0
then we can formulate
Fact 3. When the compositions X2 × X 2 , Y2 × Y 2(, Z2 × Z 2 )
are of the kind
(g-g) then in the parameters of the coordinate net
v, v, v, v,
the tensor of
1 2 3 4
8
8
α
α
the curvature satisfy the conditions Rijks . = Rijks . = 0, R133 . = R244 . =
α
α
α
α
α
α
R311 . = R422 . = R143 . = R234 . = R321 . = R412 . = 0.
THRIAD GEODESIC COMPOSITIONS IN FOUR DIMENSIONAL...
79
Let consider the affinor
(32)
dβα = aβα + bβα + cβα
.
According to (3), (4), (10) and (12) we have
(33)
aβα bβα + bβα aβα = 0, bβα cβα + cβα bβα = 0, cβα aβα + aβα cβα = 0.
From (32) and (33) it follows dβα = aβα aβα +bβα bβα + cβα cβα = δαβ + δαβ − δαβ = δαβ
, which means that the affinor dβα defines a composition U2 × U 2 with the
positions P(U2) and P (U 2 ):
Theorem 1.7. The composition U2 × U 2 is of the kind (g-g) if and only if
the coefficients of the derivative equations (13) satisfy the conditions
(34)
s
s
T
− dσβ T
β
k
(35)
s
T
k
s
β
+ T
k+2
s−2
β
− T
k
s−2
β
σ
=0
k
−2 T
k+2
β
(
s
s
+ dσβ T σ + T
k
k+2
)
s−2
σ − T
k
=0
σ
Proof According to (2) the composition U2 × U 2 will be of the kind (g g) if and only if
(36)
dσα ▽β dνσ + dσβ ▽σ dνα = 0
With the help of (4), (10), (12) and (32) we find
(37)
1
2
i
i
dνσ = ασν + 2(v ν vν +v ν vν = v ν vν −v ν vν +2 v
3
4
i
i
2+i
ν i
vσ
.
Then (36) can be written in the form
(38)
δ
i
i
δ
δ
i
i
δ
δ
dσα (T β v ν vσ − T β v ν vσ − T β v ν vσ + T β v ν vσ +2 T
i
δ
δ
δ
i
i
i
i
δ
δ
δ
δ
i
2+i
i
i
δ
δ
δ
i
i
δ
δ
i
2+i
i
v ν vσ −2 T
σ
v ν vα −2 T
δ
dσβ (T σ v ν vα − T σ v ν vα − T σ v ν vα + T σ v ν vα +2 T
i
i
β
δ
δ
i
δ
β
i
δ
σ
v
ν δ
vσ )+
v
ν δ
vα )
2+i
2+i
α
=0
The equalities received from (38) after contraction by v α and v are conk
s
k
s
tracted once again by v ν and v ν . As a result of these operations we reach
to (34) and (35).
(
)
Corollary 4
If the net v , v , v , v is chosen as a coordinate one then
1 2 3 4
the composition U2 × U 2 from the kind (g - g) characterizes by the following
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MUSA AJETI
equalities for:
4.1) the coefficients of the derivative equations
s
Ti=0
k
s
s
Ti+ T
k
k+2
s−2
i
− T
k
s−2
i
− T
k+2
s
i
+T
k
s
i+2
+ T
k+2
s−2
i+2
− T
k
i+2
=0
4.2) the coefficients of the connectedness
(40)
Γsk i = 0,
s−2
s−2
s
s
Γsi k + Γsi k+2 − Γs−2
k − Γi
k+2 + Γi+2 k + Γi+2 k+2 − Γi+2 k = 0
i
(
)
Proof: Let choose the net v , v , v , v as a coordinate one. Then taking
1 2 3 4
into account (4), (6) and (37) we find the following presentation of the affinor
dβα
(41)


1 0 2 0
 0 1 0 2 

(dβα ) = 
 0 0 -1 0 
0 0 0 -1
From (34), (35) and (41) we obtain the equalities (39), from where according to (20) follow (40) From [2] and the first equation of (39) it follows the
validity of the statement:
Fact 4 If the composition U2 × U 2 is of the kind (g - g) , then the composition X2 × X 2 is of the kind (X3 − g) , i.e. the positions P (X 2 ) are
parallelly translated along any line of X 2 .
3. Geodesic compositions in spaces EqA4
Let A4 by an equiaffine space with a fundamental 4-vector eαβγδ . The quantity eαβγδ v α v β v γ v δ is called an extent, defined by the vector fields v σ It is
α
1 2 3 4
known that the extent V preserves when the vectors v σ are parallelly transα
lated along any line in EqA4 [1]. We denote by e = e1234 the fundamental
density of the equiaffine space EqA4 . The following characteristics for the
equiaffine spaces are known: Γσασ = ∂ − α ln e, ▽ν eαβγδ = 0, Rαβ = Rβα [1]
Proposition 1 If in the equiaffine space EqA4 with a fundamental 4-vector
1
2 3 4
σ
eαβγδ = v[α v β v γ v δ ], where v α are defined by (3), the compositions X2 ×
X 2 , Y2 × Y 2 are the kind of (g − g), then the space EqA4 is affine
1
2
3
4
Proof: From ▽eαβγδ = ▽ v[α v β v γ v δ ] = 0 and derivative equations (13)
THRIAD GEODESIC COMPOSITIONS IN FOUR DIMENSIONAL...
81
σ
we obtain T ν = 0 which, according to (20), is equivalent to Γδνδ = 0. Since
σ
the compositions X2 × X 2 , Y2 × Y 2 are the kind of (g-g), then on basis of
Fact 2, (31) and Γδνδ = ∂ ν ln e = 0 we establish Γδνδ = 0, i.e. the space EqA4
is affine.
Proposition 2 If in the equiaffine EqA4 with a fundamental density e the
compositions X2 × X 2 , Y2 × Y 2 are the kind of (g − g) hen the components
of the Richi?s tensor of the space EqA4 have the following presentation:
(42)
R13 = −σ13 ln e, R14 = −σ14 ln e + Γα14 ∂α ln e + ∂α Γα14 ,
R24 = −σ24 ln e, R23 = −σ23 ln e − Γα23 ∂α ln e − ∂α Γα23 ,
(43)
Rij = −δij ln e − ϵΓi14 Γj14 , Rij = −δij ln e − ϵΓi14 Γj14
where ϵ = 1 for i = j or i = j, ϵ = −1 for i ̸= j or i ̸= j
and for the indexes are fulfilled: 1 ↔ 4, 2 ↔ 3.
Proof. According to [1] for the tensor of Richi in the space EqA4 we have
α
Rβυ = Rαβυ . = δα Γαβυ − δβυ ln e + Γρβυ δρ ln e − Γαβρ Γραυ : Then applying (31)
we find
(44)
8 Γk
Rij = −σij ln e − Γ8ik Γk8j ,
Rij = −σij ln e − Γik
8j
Rij = σα Γαij − σij ln e + Γαij σα ln e
Now using (31) and (44) we obtain
R11 = −δ11 ln e − (Γ414 )2 , R33 = −δ33 ln e − (Γ214 )2 ,
R22 = −δ22 ln e − (Γ314 )2 , R44 = −δ44 ln e − (Γ114 )2 ,
R12 = −δ12 ln e + Γ414 Γ314 , R34 = −δ34 ln e + Γ114 Γ214 ,
and (42). It is obviously that R14 + R23 = −δ14 ln e − δ23 ln e.
References
[1] Norden, A.: Affinely connected spaces, Moskow,1976
[2] Norden, A.: Spaces with Cartesian compositions, Izv. Vyssh. Uchebn. Zaved. Math., 4 (1963),
117-128. (in Russian)
[3] Norden, A., Timofeev,G.: Invariant tests of special compositions in many-dimensional spaces,
Izv. Vyssh. Uchebn. Zaved. Math., 8 (1972), 81-89. (in Russian)
[4] Timofeev,G.: Invariant tests of special compositions in Weyl spaces, Izv. Vyssh. Uchebn.
Zaved. Math., 1 (1976), 87-99. (in Russian)
[5] Willmore, T.: Connexions for systems of parallel distributions, Quart. J., Math., v.7, 28,
(1956), 269-276.
[6] Zlatanov, G.: Compositions generated by special nets in affinely connected spaces, Serdika
Math. J., 28, (2002), 1001-1012.
[7] Zlatanov, G., Tsareva B. Conjugated Compositions in evendimensional affinely conncected
spaces without a torsion, REMIA 2010, 1012 December, 2010, Plovdiv, Bulgaria, 225231.
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MUSA AJETI
[8] Zlatanov, G., Tsareva B. Geometry of the Nets in Equiaffine Spaces, J. Geometry, 55 (1996),
192201.
Musa Ajeti, Preshevë, Serbia
E-mail address: m-ajeti@hotmail.com