BULLETIN of the Bull. Malaysian Math. Sc. Soc. (Second Series) 26 (2003) 163−174 MALAYSIAN
MATHEMATICAL SCIENCES SOCIETY
Homogeneous 2 − π Metrical Structures on T
2M Manifold
ADRIAN SANDOVICI AND VICTOR BLANUTA
Department of Mathematics, University of Bacau, Str. Marasesti 157, 5500, Bacau, Romania e-mail: [email protected] and e-mail: [email protected]
Abstract. On the geometrical model determined by the second order prolongation of a Riemannian space, we introduce for the first time the homogeneous Sasaki lift notion. We define almost 2−π homogeneous structures on the fibred of second order acceleration and study the normal conditions of the mentioned structures. Finally we determine a class of distinguished connections compatible with 2 – π metrical homogeneous structure ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛(0) (0)
,F G .
Introduction
The term homogeneity has recently been discussed in Radu Miron's papers (see [10] – [11]). He introduces new geometrical models on Riemannian spaces and
Finsler ones, respectively.
This paper discusses the second order prolongation of a Riemannian space.
The basic concepts were introduced by Radu Miron in his monography [8]. On the mentioned geometrical models, the former author of this paper (see [15]) introduced and studied the notion (αβγ) – lift Sasaki of a Riemannian (M,γ) space to T2M and then determined (αβγ)– the corresponding metrical linear connection; for the canonical metrical connection he determined the local components of the tensor fields of curvature and torsion. He has also introduced and studied the notion of μ – almost 2−π structure on T2M and dealt with the linear connection compatible with such a structure, as well
as with the conditions necessary normality. More, it has been considered also the d – gauge linear connections on T2M, preparing the basis for the determination of the
second order generalized EYM equations, and the gravitational field equations as well.
1. Homogeneous Sasaki lift of a (M,γ) Riemannian space to M
T2 manifold
We will consider Rn =(M,γ) a Riemannian space generated by a real, differentiable, n-dimensional manifold M and by a Riemannian metric γ on M, given by the local components (γij(x)),x ∈U ⊂ M. We will extend γ to π−1(U)⊂E =T2M, defining:
(
γij D π)
(u) = γij(x), u ∈ π−1(U), π(u) = x (1) In this case γij Dπ are the local components of a tensor field on E. Usually, we willwrite these local components with γij as well, and with γijk(x) we will note the Christoffel symbols of the second species of γ metric. As we well know (see [8]), on E we can introduce a nonlinear connection determined only by γ metric. More, the coefficients of connection
i
j i
j
N N (0)
) 2 ( ) 0 (
) 1
( , are determined by the following relations (see also [14]):
⎪⎪
⎩
⎪⎪
⎨
⎧
⎟ +
⎟
⎠
⎞
⎜⎜
⎝
⎛ + ⋅
∂
= ∂
=
ij mj im p p ij
ij
x y y
y x N
y x N
i
j
i
j
0 0
0 ) 1 0 ( )
2 ( ) 1 ( ) 0 (
) 2 (
0 ) 1 ) ( 0 (
) 1 (
2 ) 1 , , (
) , (
γ γ
γ γ
γ
(2)
where ''0'' means the contraction by (y(1)) and '' 0 '' means the contraction by (y(2)). In the following section, we will partially avoid this particular nonlinear connection and we use one, more general, determined in:
Theorem 1.1. If
i
j i
j
N N (0)
) 2 ( ) 0 (
) 1
( , are the local components of the nonlinear connection determined only by Riemannian metric γ , and Xij,Yji are the local components of any d- tensor field of (1,1) type on E, then the functions:
Homogeneous 2−πMetrical Structures on T MManifol
( )
⎪⎪
⎩
⎪⎪
⎨
⎧
⋅
− +
=
+
=
mj mi mi i
ij i
X Y
N N
X N N
i
j j
i
j j
0 )
0 (
) 2 ( ) 2 (
) 0 (
) 1 ( ) 1 (
γ
(3)
are the local components of a nonlinear connection N on E.
The nonlinear connection N assures the existence of basis ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛ (1) (2) , , k k
k δ δ
δ adapted to the tangent space TuE. The vector fields of the adapted basis are defined with the help of the following relations:
i i
i k i
k k
k N y
N y
x (1) (1) (2) ∂ (2)
⋅ ∂
∂ −
⋅ ∂
∂ −
= ∂
δ (4)
k k i i
k k
k N y y
y (2)
) 2 ( ) 2 ) (
1 ) ( 1 ( ) 1 (
, ∂
= ∂
∂
⋅ ∂
∂ −
= ∂ δ
δ (5)
For further developments, we need the following result:
Theorem 1.2. Lie brackets of the vector fields of the adapted basis ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛ (1) (2) ,
, k k
k δ δ
δ are
given by:
[ ]
(2() )) 01 ( ) 1 ( ) 01
, ( i i
i jk i
k R jk R
j δ δ δ
δ = ⋅ + ⋅ (6)
) 2 (
) ) (
12 ( ) 1 ( ) 11 ( ) 1 (
, i i
i jk i
jk
k B B
j δ δ δ
δ ⎥ = ⋅ + ⋅
⎦
⎢ ⎤
⎣
⎡ (7)
) 2 (
) ) (
22 ( ) 1 ( ) 21 ( ) 2 (
, i i
i jk i
jk
k B B
j δ δ δ
δ ⎥ = ⋅ + ⋅
⎦
⎢ ⎤
⎣
⎡ (8)
) 2 (
) ) (
21 ( ) 2 ( ) 1 ( ) 2 ( ) 12 ( ) 1 ( ) 1 (
, ,
, i i
jk k
i j i
jk
k R B
j δ δ δ δ δ
δ ⎥ = ⋅
⎦
⎢ ⎤
⎣
⋅ ⎡
⎥ =
⎦
⎢ ⎤
⎣
⎡ (9)
where:
i jk i
jk i
jk ijk
i
jk i
jk ijk
i
jk i
jk
X R
R XY
R R
X R
R [ ]
) 1 ( ) 0 (
) 12 ( ) 12 ( )
0 (
) 02 ( ) 02 ( )
0 (
) 01 ( ) 01 (
, ) (
, = + = +
+
= (10)
i jk i
jk i
jk i
jk i
jk i
jk
XY B
B X B
B
) 12 ( ) 0 (
) 12 ( ) 12 ( ) 1 ( ) 0 (
) 11 ( ) 11
( = + , = + ( ) (11)
i jk i
jk i
jk i
jk i
jk i
jk
XY B
B X B
B
) 22 ( ) 0 (
) 22 ( ) 22 ( ) 2 ( ) 0 (
) 21 ( ) 21
( = + , = + ( ) (12)
with the following notations:
k ij i
k jk ij i
j jk ki k
ij ijk
y X X
y X X
x X x
x X
) 2 ( )
2 ( ) 1 ( )
1 (
,
, δ
δ δ
δ δ
δ δ
δ − = =
= (13)
mjk mi m
jk mi
mjk i
m
ijk N X X R X X
XY = ⋅ + ⋅ (0) + ⋅
) 01 ( )
0 (
) 1 (
) (
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ⋅
⋅ −
⎟ −
⎟
⎠
⎞
⎜⎜
⎝
⎛ ⋅
⋅ −
+ j
q k qi k
qj qi j
q k qi k
qj qi
x X x
X x
X Y x
X Y
δ γ δ δ
γ δ δ
δ δ
δ( ) ( ) ( 0 ) ( 0 )
(14)
j ki k
ji m
i jk m m
jk mi m
jk i
m i
jk y
Y y
X Y X B
X X N
XY (0) (1) (1) (1)
) 11 ( )
1 ( ) 0 (
) 1 ) (
12 (
)
( δ
δ δ
δ −
+
⋅ +
⋅ +
⋅
= (15)
k ji m
i jk m m
jk mi m
jk i
m i
jk y
X Y X B
X X N
XY (2)
) 2 ( )
0 (
) 21 ( )
2 ( ) 0 (
) 1 ) (
22 (
)
( δ
+ δ
⋅ +
⋅ +
⋅
= (16)
i kj i
jk i
jk X X
X[ ] (1) (1) )
1
( = − (17)
Theorem 1.3. ([8]) The pair Prol(2)R(n) =
(
T~2M,G)
where:j ij i
j ij i
j
ij x dxi dx x y y x y y
G=γ ( ) ⋅ ⊗ + γ ( ) ⋅δ (1) ⊗ δ (1) +γ ( ) ⋅δ (2) ⊗δ (2) (18) is a Riemannian space of 3n dimension, with G metric structure depending only on γ(x) Riemannian structure, apriori on Riemannian space R(n) = (M,γ).
We will say that G is Sasaki lift of γ Riemannian structure. We define the homotety
∈ ∗
→ x ty t y t R
y y x
ht :( , (1), (2)) ( , (1), 2 (2)), on the fibres of T2M. We also mention that G is transformed in accordance with:
Homogeneous 2−πMetrical Structures on T MManifol
+
⊗
⋅
= ij i j
t x y y x dx dx
h
G D ( , (1), (2)) γ ( )
j ij i
j
ij x y i y t x y y
t2 ⋅ γ ( ) ⋅δ (1) ⊗δ (1) + 4 ⋅γ ( ) ⋅δ (2) ⊗δ (2) (19) The above remark makes us affirm that the Sasaki lift G is nonhomogeneous on the fibres of T2M.
In the following part we concentrate upon a new lift of Sasaki type, called homogeneous Sasaki lift and noted as :
) 0 (
G
+
⊗
⋅
= ij x dxi dxj
G ( )
) 0
( γ
j ij i
j
ij i x y y
y F y
F x
) 2 ( )
2 ( 4
) 1 ( )
1 (
2 1 ( )
)
1 ⋅ γ ( ⋅ δ ⊗ δ + ⋅ γ ⋅ δ ⊗ δ (20)
where F2 = λij(x) ⋅ y(1)i ⋅ y(1)j.
Theorem 1.4. The following properties holds:
(a) The pair ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛ 2 (0)
~ , G M T
T is a Riemannian space;
(b)
) 0 (
G depend only by the γ(x) Riemannian metric;
(c) The distributions N, V1,V2 are ortogonal with respect to .
) 0 (
G Definition 1.1. A D linear connection on T~2M
is call (0) - metrical connection with respect to (G0) if 0
) 0 (G=
D and D preserves by parallelism the N horizontal distribution.
With respect to adapted basis , , ,
) 2 ( ) 1 (
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
k k
k δ δ
δ any D linear connection on E can be representated as follows:
i i i jk i i jk i
jk H
j L L L
Dδkδ = ( ) ⋅δ + (1) ⋅δ(1) + (2) ⋅ (δ2) (21)
i i i jk i
jk V i i
jk
j L L L
Dδk (δ1) = (3) ⋅ δ + ( 1) ⋅δ(1) + (4) ⋅ (δ2) (22)
i i
jk v i i i jk i
j L jk L L
Dδk (δ2) = (5) ⋅ δ + (6) ⋅ δ(1) + ( 2) ⋅ (δ2) (23)
i i i jk i i jk i
jk H
j F F F
D
k
) 2 ( ) 2 ( ) 1 ( ) 1 ( )
(
) 1
( δ δ δ δ
δ = ⋅ + ⋅ + ⋅ (24)
i i i jk i
jk V i i
jk
j F F F
D
k
) 2 ( ) 4 ( ) 1 ( ) ( )
3 ( ) 1
( 1
) 1
( δ δ δ δ
δ = ⋅ + ⋅ + ⋅ (25)
i i
jk v i i i jk i
jk
j F F F
D
k
) 2 ) ( ) ( 1 ( ) 6 ( )
5 ( ) 2
( 2
) 1
( δ δ δ δ
δ = ⋅ + ⋅ + ⋅ (26)
i i i jk i i jk i
jk H
j C C C
D
k
) 2 ( ) 2 ( ) 1 ( ) 1 ( )
(
) 2
( δ δ δ δ
δ = ⋅ + ⋅ + ⋅ (27)
i i i jk i
jk v i i
jk
j C C C
D
k
) 2 ( ) 4 ( ) 1 ) ( ) (
3 ( ) 1
( 1
) 2
( δ δ δ δ
δ = ⋅ + ⋅ + ⋅ (28)
i i
jk v i i i jk i
jk
j C C C
D
k
) 2 ( ) ( ) 1 ( ) 6 ( )
5 ( ) 2
( 2
) 2
( δ δ δ δ
δ = ⋅ + ⋅ + ⋅ (29)
The set consisting of the functions
i jk i v
jk
H C
L) ( )
( 2
,
," represents the set of the
coefficients of D linear connection. In what regards the notion of (0) - metrical connection, there can be proved the following result:
Theorem 1.5. There exist D (0) - metrical connections on T~2M, which depend only on the γ Riemannian tensor field. One of these has its coefficients given by:
0
) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1
( = = = = = =
i jk i
jk i
jk i
jk i
jk i
jk L L L L L
L (30)
0
) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1
( = = = = = =
i jk i
jk i
jk i
jk i
jk i
jk F F F F F
F (31)
0
) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1
( = = = = = =
i jk i
jk i
jk i
jk i
jk i
jk C C C C C
C (32)
ijk ijk
i jk i v i jk
jk i
jk i v
jk i
jk H
L F L F
L = γ = γ + ⋅θ = γ + ⋅θ
2 )
( 2
) ( )
( 2
1 ,
, 1 2 (33)
ijk i
jk i v
jk i
jk i v
jk H
F F F F
F = = − ⋅ Λ = − ⋅ Λ
2 )
( 2
) ) (
( 2
1 , ,
0 1 2 (34)
0
) ( ) ( )
( 1 2
=
=
=
i jk i v
jk i v
jk
H C C
C (35)
Homogeneous 2−πMetrical Structures on T MManifol
with the following notations:
(
is)
t(1)t jk i s
t k i j
t j i k
jk = X ⋅ δ + X ⋅ δ − X ⋅ γ ⋅ γ ⋅ y
θ (36)
jk i i j
k i k
i jk
jk = ⋅ y(1) + ⋅ y(1) − ⋅ y(1)
Λ δ δ γ (37)
Theorem 1.6. The set of all (0)-metrical connections is given by the coefficients ,
, ,
,*) ( ,*)
( 2 i
jk i v
jk
HL " C whose expression is show by the following relations:
2 , 1 ,
,
) ( ,*)
) ( ( )
( ,*)
( = + Ω ⋅ α + Ω ⋅ α α =
r hk ih v
rj i
jk r v
hk ih H rj i
jk i H
jk
HL L I L I (38)
2 , 1 , ,
) ) (
,*) ( ) (
( )
( ,*)
( = + Ω ⋅ α = α + Ω ⋅ α α =
r hk ih v rj i
jk i v
jk r v
hk ih H rj i
jk i H
jk
HF F J F F J (39)
2 , 1 , ,
) ) (
,*) ( ( ) ( )
( ,*)
( = + Ω ⋅ α = α + Ω ⋅ α α =
r hk ih v
rj i
jk i v
jk r v
hk ih H
rj i
jk i H
jk
HC C H C C H (40)
where:
(
h rj ih)
i j ih r
rj = ⋅ δ ⋅ δ − γ ⋅ γ
Ω 2
1 (41)
and
r hk r v hk r v hk r H hk r v hk r v hk r H hk r v hk r v
hk
HI) (I) (I) (J) (J) (J) (H) (H) (H)
( 1 2 1 2 1 2
, , , , , , ,
, are arbitrary
d - tensor fields.
2. 2–π structures on the fibred of second order acceleration We consider F(E)−linear operator : ( ) ( )
) 0 (
E X E X
F → defined on the adapted basis
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
k k k
) 2 ( ) 1 (
,
, δ δ
δ through:
i i i i
i F F F F
F δ λ δ δ δ ⎟⎟ = − λ ⋅ δ
⎠
⎜⎜ ⎞
⎝
= ⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⋅ ⎛
⋅
= 2
) 2 ( ) 0 ( )
1 ( ) 0 ( ) 2 2 ( )
0 (
, 0 ,
)
( (42)
where λ is an arbitrary nonzero complex number.
Theorem 2.1. (F0) operator has the following characteristies:
(a) It is global defined on ~; E
(b) It is a tensor field of (1,1) type on E~
and depends on γ Riemannian structure;
(c) 0;
) 0 2 ( )3 0
(F + λ ⋅ F =
; Im
, ker
(d) 0 2
) 0 ( 1 )
0 (
V N F N
F ⎟⎟⎠ = ⊕
⎜⎜ ⎞
⎝
= ⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛
(e) rank 2 .
) 0 (
n F⎟⎟ =
⎠
⎜⎜ ⎞
⎝
⎛
Definition 2.1.
(a) The tensor fields defined above is called 2−π homogeneous structure of second order on the fibred of second order acceleration.
(b) 2−π
) 0 (
F structure of second order is normal if:
( ) (
,)
0, , ( )) , (
1
) 1 ( 2
) 0
( X Y d y X Y X Y X E
N n
i i i
F + ⋅
∑
⋅ = ∀ ∈=
δ δ
λ (43)
where (0)
F
N represents Nijenhuis tensor associated to (F0) tensor field.
Definition 2.2. A D linear connection is compatible with 2−πhomogeneous structure of second order, if the following condition is achieved:
) ( ,
0
) 0 (
E X X F
Dx = ∀ ∈ (44)
We shall study in the first stage the linear connections compatible with 2−π homogeneous structure of second order. We can notice that
) 0 (
F can be written as:
i i
i i y
dx F F
F0) 2 (2) 2 (2)
( = λ ⋅ ⋅ δ ⊗ − λ ⋅ δ ⊗ δ (45)
Homogeneous 2−πMetrical Structures on T MManifol
There can be proved the following result:
Theorem 2.2. An arbitrary D linear connection is compatible with 2−π homogeneous structure of second order if, and only if the connection coefficients satisfy the following relations:
2 (1) 0
2 )
( )
( 2
=
⋅
⋅
⋅
−
− kp p ij
i jk i v
jk
H X y
L F
L δ (46)
1 (2) (5) 0
4 ⋅ + =
i jk i
jk L
F L (47)
0
) 6 ( ) 4 ( ) 3 ( ) 1
( = = = =
i jk i
jk i
jk i
jk L L L
L (48)
2 (1) 0
2 )
( )
( 2
=
⋅
⋅
−
− k ij
i jk i v
jk
H y
F F
F δ (49)
1 (2) (5) 0
4 ⋅ + =
i jk i
jk F
F F (50)
0
) 6 ( ) 4 ( ) 3 ( ) 1
( = = = =
i jk i
jk i
jk i
jk F F F
F (51)
0
) ( )
( 2
=
−
i jk i v
jk
H C
C (52)
1 (2) (5) 0
4 ⋅ + =
i jk i
jk C
F C (53)
0
) 6 ( ) 4 ( ) 3 ( ) 1
( = = = =
i jk i
jk i
jk i
jk C C C
C (54)
Using the result obtained above, there can proved that:
Theorem 2.3. A d-linear connection on E is compatible with (F0) structure if and only if there are achieved the following relations:
2 (1) 0
2 )
( )
( 2
=
⋅
⋅
⋅
−
− kp p ij
i jk i v
jk
H X y
L F
L δ (55)
2 (1) 0
2 )
( )
( 2
=
⋅
⋅
−
− k ij
i jk i v
jk
H y
F F
F δ (56)
0
) ) (
( 2
=
−
i jk i v
jk
H C
C (57)
In what follows we deal with the problem of normality of
) 0 (
F structure. In [16]
there was proved the following:
Theorem 2.4. (Fμ)μ−almost 2−π structure is normal if and only if d-tensor fields ,
, ji
ij Y
X which appear within the coefficients of N nonlinear connection, are solutions of the following system of equations with partial derivatives:
i
jk i
jk i
jk i
jk i
jk
ijk R X B X B
X (0)
) 21 ( )
2 ( ) 0 (
) 11 ( ) 1 ( ) 0 (
) 01 (
,
, = = −
−
= (58)
0 ) ( ) ( , )
( , )
(
) 22 ( )
22 ( ) 0 (
) 12 ) (
12 ( ) 0 (
) 02
( = − − =
−
= i
kj i
jk i
jk i
jk i
jk
ijk R XY B XY XY
XY (59)
,
) 0 (
) 12 ( ] [ ) 1
( i
jk i
jk R
X = − μ is a constant function (60)
Using this theorem, so that
) 0 (
F structure should be normal considering Definition 2.1., we conclude that a necessary condition, is that Finsler function associated to the initial Riemannian space should be constant.
3. 2−x homogeneous metrical structures on T2M
For the beginning, we mention that these can be easily show that the pair ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛(0) (0) ,F G is a π
−
2 metrical structure considering the definition in [16]. The fact that it is determined by a 2−π homogeneous structure we can call it 2−π homogeneous metrical structure of the prolongation of second order of Riemannian space. In the following part we like to determine the distinguished linear connections compatible with 2−π homogeneous metrical structure ⎟⎟
⎠
⎜⎜ ⎞
⎝
⎛(0) (0) ,F
G . For this target we shall suppose that the following systems of equations have solutions in the set of distinguished tensor fields of second order:
ij i k
r jk ih hk
rj ⋅ X = θ + X ⋅ δ
Ω 0 (61)
ij i k
r jk ih hk
rj ⋅ Y = Λ + y ⋅ δ
Ω 1 (62)
Homogeneous 2−πMetrical Structures on T MManifol
Theorem 3.1. The set of the distinguished linear connections compatible with
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛(0) (0) ,F
G 2−π homogeneous metrical structure is determined by the following relations:
r hk ih v i rj i jk
jk i
jk r v
hk ih H i rj
jk i
jk
H I
L F I
L,0) ( ) ( ,0) 2 ( )
( 1 1 1
, = + ⋅ +Ω ⋅
⋅ Ω +
= γ γ θ (63)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ − ⋅
⋅ Ω +
⋅ +
=
r hk r
hk ih H
i rj i jk
jk i
jk
v X
I F
L,0) F2 ( ) 2 (0)
( 2 γ 2 θ 2 (64)
r hk ih v i rj
jk i
jk r v
hk ih H
rj i
jk
H J
F F J
F,0) ( ) ( ,0) 2 ( )
( 1 1 1
, = − ⋅ Λ + Ω ⋅
⋅ Ω
= (65)
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ − ⋅
⋅ Ω + Λ
⋅
−
=
r hk r
hk ih H
i rj jk i
jk
v Y
J F
F,0) F2 ( ) 2 (0)
( 2 2 2
(66)
r ih hk rj i
jk r v
hk ih v rj i
jk r v
ih hk rj i
jk
HC,0) (H0) (C,0) (H) (C,0) (H0)
( 1 1 2
,
, = Ω ⋅ = Ω ⋅
⋅ Ω
= (67)
where
r hk r v hk r hk r v hk r H hk r v hk
HI) (I) (J) (J) (H0) (H)
( 1 1 1
, , , ,
, are arbitrary d-tensor fields, and
r hk r
hk Y
X0) (0)
(
, are the solutions of the systems of equations (61) and (62).
Open Problems and Comments
A. There is still open the problem of normality of (F0) structure, in the meaning of the determination of the basic Riemannian space and of the distinguished fields which enter the composition of nonlinear connection of the prolongation of second order of the mentioned Riemannian space.
B. The local components of the curvature and torsion fields of D(0)−metrical connection from Theorem 1.6. can be used for the developement of same theories of field on the geometrical model offered by the prolongation of second order of Riemannian space provided for ⎟⎟ −π
⎠
⎜⎜ ⎞
⎝
⎛ , 2
) 0 ( ) 0 (
F
G homogeneous metrical structure introduced in the last section. The same idea can be applied to the previous geometrical model provided for the distinguished connection determined through Theorem 3.1.
