AJETI, MUSA

TEOFILOVA, MARTA

English

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 1 (2012), no.1, 4550
ISSN 1857-8365, UDC: 514.754
ON TRIADS OF COMPOSITIONS IN AN EVENDIMENSIONAL
SPACE WITH A SYMMETRIC AFFINE CONNECTION
MUSA AJETI AND MARTA TEOFILOVA
Presented at the 8th International Symposium GEOMETRIC FUNCTION THEORY AND
APPLICATIONS, 27-31 August 2012, Ohrid, Republic of Macedonia.
Abstract. Let (v
1
; v
2
; : : : ; v
n
) be a net, dened by the independent vector elds v

( =
1; 2; : : : ; n), in a space with a symmetric ane connection An. An apparatus for
studying of more than one composition for which v

 are eigen-vectors of the matrices
of their anors is developed in [5], [8]. This apparatus is applied in A3n for studying
of a triad of compositions without common basic manifolds [7].
In the present paper two triads of compositions Xn  Xn, Xn  Yn , Xn  Zn
and Xn  Xn, Yn  Xn, Zn  Xn, such that each of them has one common basic
manifold, are introduced in A2n with the help of the independent vector elds v

( =
1; 2; : : : ; 2n). Characteristics of each of the triads of compositions are obtained in the
cases when they are cartesian or chebyshevian. The spaces admitting such triads of
compositions are characterized.
1. Preliminaries
Let AN be a space with a symmetric ane connection. The connection coecients are
denoted by   . In AN we consider a composition Xn Xm (n +m = N) of two basic
dierentiable manifolds. Two positions P (Xn) and P (Xm) of the basic manifolds pass
thought any point of the space AN [2, 3, 4].
It is known that a composition is completely dened by the eld of the anor a

,
satisfying the condition [2, 3]
(1.1) aa

 = 

:
The anor a

 is called the composition anor [3, 4]. The projecting anors
1
a

,
2
a


[4] are dened by,
1
a

 =
1
2 (

 + a

),
2
a

 =
1
2 (

   a) and they satisfy the conditions
1
a

 +
2
a

 = 

 and
1
a

   2a = a.
The following characteristics for some special types of compositions are given in [3]:
2010 Mathematics Subject Classication. 53A40, 53A55.
Key words and phrases. Anely connected spaces, spaces of compositions, anors of compositions.
45
46 M. AJETI AND M. TEOFILOVA
Composition of the type (c; ) or composition of the type ( ; c), for which the positions
P (Xn) or P (Xm) are parallelly translated along any line in the space AN , is characterized
by the condition
(1.2)
1
a r
1
a = 0 or
2
a r
2
a = 0:
Composition of the type (c; c), for which the positions P (Xn) and P (Xm) are both
parallelly translated along any line in the space AN , is characterized by both conditions
in (1.2) or by
(1.3) r a = 0:
Composition of the type (ch; ) or composition of the type ( ; ch), for which the
positions P (Xn) or P (Xm) are parallelly translated along any line of the manifold Xm
or Xn, is characterized by the condition
(1.4)
2
a
1
a r
1
a = 0 or
1
a
2
a r
2
a = 0:
Composition of the type (ch; ch), for which the positions P (Xn) and P (Xm) are paral-
lelly translated along any line of the manifolds Xm and Xn, respectively, is characterized
by both conditions in (1.4) or by
(1.5) r[ a] = 0:
Let us consider an even-dimensional space A2n equipped with a symmetric ane con-
nection. Let v

( = 1; 2; :::; 2n) be independent vector elds. The reciprocal covectors

v are dened by
(1.6) v

 v = 

 , v

 v = 

 :
We introduce the following notations
(1.7)
; ; ; ; ; ; ::: = 1; 2; :::; 2n;
i; j; s; k; ::: = 1; 2; :::; n; i; j; s; k; ::: = n+ 1; n+ 2; ::::; 2n:
Following [5], [8], we consider the anor
(1.8) a = v
i
 iv   v
i

i
v:
According to (1.6), it follows aa

 = 

 , i.e. the anor (1.8) denes a composition
Xn Xn in A2n. Denote by P (Xn) and P (Xn) the positions of this composition. The
projecting anors of the composition Xn Xn are of the form [5]:
(1.9)
1
a = v
i
 iv;
2
a = vi

i
v:
According to [6], the following derivative equations are valid
(1.10) r v

 =

T

 v

 ; r v =  

T



v :
The lines dened by the vector eld v

 are denoted by

v


, and the net dened by
the lines

v


( = 1; 2; :::; 2n) is denoted by

v
1
; v
2
; :::; v
2n

. If we choose

v
1
; v
2
; :::; v
2n

to
ON TRIADS OF COMPOSITIONS... 47
be the coordinate net, then
(1.11)
v
1
(1; 0; 0; :::; 0; 0); v
2
(0; 1; 0; :::; 0; 0); :::; v
2n
(0; 0; 0; :::; 0; 1);
1
v(1; 0; 0; :::; 0; 0);
2
v(0; 1; 0; :::; 0; 0); :::;
2n
v (0; 0; 0; :::; 0; 1):
In this case the net

v
1
; v
2
; :::; v
2n

denes a coordinate system which is adapted to the
composition Xn Xn in the meaning of [2].
In the parameters of the coordinate net the following equality holds [5], [8]
(1.12)   =

T

:
2. Triads of compositions with one common basic manifold
Let the composition Xn  Xn be dened by the anor (1.8). Let us consider the
following anors
(2.1)
f

 = w
i
 iw   w
i

i
w; h

 = z
i
 iz   z
i

i
z;
F

 = x
i
 ix   x
i

i
x; H

 = y
i
 iy   y
i

i
y:
where
(2.2)
w
i
 = v
i
; w
i
 = 1p
2

v
i n
 + v
i


;
i
w =
i
v   n+iv ;
i
w =
p
2
i
v;
z
i
 = v
i
; z
i
 = 1p
2

v
i n
   v
i


;
i
z =
i
v +
n+i
v ;
i
z =  
p
2
i
v;
x
i
 = 1p
2

v
i
 + v
n+i


; x
i
 = v
i
;
i
x =
p
2
i
v;
i
x =
i
v  
i n
v ;
y
i
 = 1p
2

v
i
   v
n+i


; y
i
 = v
i
;
i
y =
p
2
i
v;
i
y =
i
v +
i n
v :
and
(2.3)

w w

 =  ;

z z

 =  ;

x x

 =  ;

y y

 =  :
From (2.1), (2.2) and (2.3) it follows that the anors f

 ; h

; F

 ; H

 dene compositions.
Let us denote with Xn  Yn; Xn  Zn; Yn  Xn; Zn  Xn; the compositions dene by
the anors f

 ; h

; F

 ; H

 , respectively. The triad of compositions Xn  Xn, Xn  Yn,
Xn  Zn have one common basic manifold Xn, and the triad of compositions Xn  Xn,
Yn  Xn, Zn  Xn have one common basic manifold Xn.
By (1.8), (2.1), (2.2), (2.3) we obtain
(2.4) f = a

   2d; h = a + 2d; F  = a + 2d
1

; H

 = a

   2d
1

;
where
(2.5) d = v
i
 n+iv ; d
1

 =   v
n+i
 iv:
48 M. AJETI AND M. TEOFILOVA
We denote by
1
f

,
2
f

 the projecting anors of the composition Xn  Yn, by
1
h

,
2
h


the projecting anors of the composition Xn  Zn, by
1
F

,
2
F

 the projecting anors
of the composition Yn  Xn and by
1
H

,
2
H

 the projecting anors of the composition
Zn Xn.
From (2.1), (2.2), (2.3), (2.4), (2.5) we obtain
(2.6)
1
f

 =
1
a

   d;
2
f

 =
2
a

 + d

;
1
h

 =
1
a

 + d

;
2
h

 =
2
a

   d
1
F

 =
1
a

 + d
1

;
2
F

 =
2
a

   d
1

;
1
H

 =
1
a

   d
1

;
2
H

 =
2
a

 + d
1

;
where
1
a

,
2
a

 are the projecting anors of the composition Xn Xn:
3. Special types of triads of compositions
Lemma 3.1. If the composition XnXn is of the type (c; c), the conditions r d = 0
and r d
1

 = 0 are equivalent.
Proof. Let the equalities
(3.1) r a = 0; r d = 0:
hold. According to (1.3), the rst one of the equalities (3.1) is a necessary and sucient
condition for the composition Xn Xn to be of the type (c; c). By (2.4), (2.6) and (3.1)
we obtain
(3.2) r f = 0; r h = 0;
i.e. the compositions XnYn and XnZn are also of the type (c; c). Hence, the positions
P (Xn), P (Xn), P (Yn) and P (Zn) are parallelly translated along any line in the space
A2n. From here it follows that the compositions Yn Xn and Zn Xn are of the type
(c; c), too. Then, according to (1.3), we have
(3.3) r F  = 0; r H = 0:
By (2.4), (3.1) and (3.3) we obtain r d
1

 = 0.
The statement that from r a = 0 and r d
1

 = 0 it follows r d = 0 is proved
analogously. 
Theorem 3.1. If the compositions XnXn, XnYn, XnZn, YnXn and ZnXn
are of the type (c; c), then the space A2n is ane.
Proof. According to Lemma 3.1, the compositions XnXn, XnYn, XnZn, YnXn
and Zn  Xn are of the type (c; c) i the conditions (3.1) hold. Having in mind (1.8),
(2.5) and (3.2), the equalities (3.1) can be written in the form
(3.4) r

v
i
 iv   v
i

i
v

= 0; r

v
i
 n+iv 

= 0:
Taking into account (1.10), the equalities (3.4) take the form
k
T
j
 = 0;
k
T
j
 = 0;
k
T
j
 
n+k
T
n+j
 =
0:
ON TRIADS OF COMPOSITIONS... 49
We choose

v
1
; v
2
; :::; v
2n

for the coordinate net. According to (1.12), the last equalities
are equivalent to  
k
j = 0,  
k
j
= 0,  kj    n+k n+j = 0; from where we obtain   = 0.
Hence, A2n is an ane space [1]. 
The following statement is obvious
Theorem 3.2. If the compositions Xn Xn, Xn  Yn and Xn  Zn are of the type
(ch; ch), then they are also of the type (c; ch). If the compositions XnXn, YnXn
and Zn Xn are of the type (ch; ch), then they are also of the type (ch; c).
Theorem 3.3. The following hold:
(i) If the compositions Xn  Xn, Xn  Yn and Xn  Zn are of the type (c; ch),
the curvature tensor satises R i: = 0.
(ii) If the compositions Xn Xn, Yn Xn and Zn  Xn are of the type (ch; c),
the curvature tensor satises R 
i:
= 0.
Proof. (i) Let the compositions Xn Xn, Xn  Yn and Xn  Zn be of the type (c; ch).
Then, the position P (Xn) is parallelly translated along any line in A2n, and the positions
P (Xn), P (Yn) and P (Zn) are parallelly translated along any line of the manifold Xn.
According to (1.2) and (1.4), the three compositions are of the type (c; ch) i the following
conditions
(3.5)
1
a r
1
a = 0;
1
a
2
a r
2
a = 0;
1
f
2
f r
2
f = 0;
1
h
2
h r
2
h = 0
hold.
By (1.8), (1.10) and (2.6) the equalities (3.5) take the form
i
T
j
 = 0;
i
T
j
 v
s
 = 0;
i
T
j
 v
s
 +
i
T
j n
 v
s
  
n+i
T
j
 v
s
  
n+i
T
j n
 v
s
 = 0;
i
T
j
 v
s
  
i
T
j n
 v
s
 +
n+i
T
j
 v
s
  
n+i
T
j n
 v
s
 = 0:
Let the net

v
1
; v
2
; :::; v
2n

be chosen for the coordinate net. Then, according to (1.11),
the last equalities can be written in the form
(3.6)
i
T
j
 = 0;
i
T
j
s = 0;
i
T
j
s +
i
T
j n
s  
n+i
T
j
s  
n+i
T
j n
s = 0;
i
T
j
s  
i
T
j n
s +
n+i
T
j
s  
n+i
T
j n
s = 0:
By (1.12) and (3.6) we obtain  
i
j = 0,  
i
sj
= 0,  i
sj
   n+i
s j n = 0,  
i
s j n    
n+i
sj
= 0,
from where it follows that  i = 0. Then, for the curvature tensor we get R

i: = 0.
Condition (ii) is proved analogously. 
Acknowledgment
The authors would like to express their gratitude to Prof. Georgi Zlatanov, the Ph.D.
instructor of the rst author.
50 M. AJETI AND M. TEOFILOVA
References
[1] A. P. Norden: Anely connected spaces, Moskow, Nauka, 1976.
[2] A. P. Norden: Spaces of Cartesian composition, Izv. Vyssh. Uchebn. Zaved., Math. 4 (1963),
117128.
[3] A. P. Norden, G. N. Timofeev: Invariant criteria for special compositions of multidimensional
spaces, Izv. Vyssh. Uchebn. Zaved., Mat. 8 (1972), 8189.
[4] G. N. Timofeev: Invariant criteria of special compositions in Weyl spaces, Izv. Vyssh. Uchebn.
Zaved., Mat. 1 (1976), 8799.
[5] G. Zlatanov: Compositions generated by special nets in anely connected spaces, Serdica Math.
J. 28 (2002), 10011012.
[6] G. Zlatanov, B. Tsareva: Geometry of the nets in equane spaces, J. Geom. 55 (1996), 192
201.
[7] G. Zlatanov, B. Tsareva: Triad compositions in anely connected spaces A3m, Plovdiv Univ.
Sci. Works  Math. 36,(3) (2009), 141156.
[8] G. Zlatanov: Special compositions in anely connected spaces without a torsion, Serdica, Math.
J. 37 (2011), 211220.
Presevo, Serbia
E-mail address: m-ajeti@hotmail.com
University of Plovdiv
Faculty of Mathematics and Informatics
24 "Tsar Assen" str., 4000
Plovdiv, Bulgaria
E-mail address: marta@uni-plovdiv.bg


ON TRIADS OF COMPOSITIONS IN AN EVEN-DIMENSIONAL SPACE WITH A SYMMETRIC AFFINE CONNECTION

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ON TRIADS OF COMPOSITIONS IN AN EVEN-DIMENSIONAL SPACE WITH A SYMMETRIC AFFINE CONNECTION

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