Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.yu/filomat
Filomat 21:2 (2007), 235–242
INFINITESIMAL DEFORMATIONS OF BASIC TENSOR IN GENERALIZED RIEMANNIAN SPACE L. S. Velimirovi´ c, S. M. Minˇ ci´ c and M. S. Stankovi´ c Abstract. At the beginning of the present work the basic facts on generalized Riemannian space (GRN ) in the sense of Eisenhart’s definition [Eis] and also on infinitesimal deformations of a space are given. We study the Lie derivatives and infinitesimal deformations of basic covariant and contravariant tensor at GRN .
1. Introduction 1.0. At the beginning we are giving basic information on generalized Riemannian spaces and on infinitesimal deformations of a space. 1.1. A generalized Riemannian space GRN at the sense of Eisenhart’s definition [Eis] is a differentiable manifold, endowed with nonsymmetric basic tensor gij (x1 , . . . , xN ), where xi are local coordinates. So, generally we have (1.1)
gij (x) 6= gji (x).
The symmetric, respectively antisymmetric part of the basic tensor are (1.2 a, b)
hij =
1 (gij + gji ), 2
kij =
1 (gij − gji ), 2
2000 Mathematics Subject Classification. 53C25, 53A45, 53B05. Key words and phrases. Infinitesimal deformation, non-symmetric affine connection, Lie derivative, geometric object. Received: July 1, 2007 Typeset by AMS-TEX
235
236
´ S. M. MINCI ˇ C, ´ M. S. STANKOVIC ´ L. S. VELIMIROVIC,
from where it follows that (1.3)
gij = hij + kij .
For the lowering and raising of indices in GRN one uses the tensor hij respectively hij , where (1.4)
(hij ) = (hij )−1
(det(hij ) 6= 0).
Christoffel symbols at GRN are (1.5 a, b)
Γi.jk =
1 (gji,k − gjk,i + gik,j ), 2
Γijk = hip Γp.jk ,
where, for example, gji,k = ∂gji /∂xk . Based on (1.1) the non-symmetry of Christoffel symbols with respect to j, k at (1.5) follows. The symbols Γijk are connection coefficients at GRN . By a reason of non-symmetry of the connection, one can use in GRN four kinds of covariant derivatives of a tensor. For example: (1.6a − d)
tij | m = tij,m + Γipm tpj − Γpjm tip 1 2 3 4
mp pm mp
mj mj jm
1.2. Basic facts on infinitesimal deformations and their expressions by Lie derivative one can find, e.g., at [Ya,Ya1,St,Sch,MVS,MVS1]. Definition 1.1. A transformation f : GRN → GRN : x = (x1 , . . . , xN ) ≡ (xi ) → x ¯ = (¯ x1 , . . . , x ¯N ) ≡ (¯ xi ), where (1.7)
x ¯i = xi + z i (xj )ε,
i, j = 1, . . . , N,
ε being an infinitesimal, is called infinitesimal deformation of a space GRN , determined by the vector field z = (z i ), which is called a field of infinitesimal deformations (1.7). We denote with (i) local coordinate system in which the point x is endowed with coordinates xi , and the point x ¯ with the coordinates x ¯i . We will 0 also introduce a new coordinate system (i ), corresponding to the point x = (xi ) new coordinates (1.8)
0
xi = x ¯i ,
INFINITESIMAL DEFORMATIONS OF BASIC TENSOR IN GENERALIZED ... 237 0
i.e. as new coordinates xi of the point x = (xi ) we choose old coordinates (at the system (i)) of the point x ¯ = (¯ xi ). Namely, at the system (i0 ) is i0 i x = (x ) = (¯ x ), where = denotes ”equal according to (1.8)”. (1.8)
(1.8)
Let us consider a geometric object A with respect to the system (i) at the point x = (xi ) ∈ GRN , denoting this with A(i, x). Definition 1.2. The point x ¯ is said to be deformed point of the point x, ¯ x) is deformed object A(i, x) with if (1.7) holds. Geometric object A(i, respect to deformation (1.7), if its value at system (i0 ), at the point x is equal to the value of the object A at the system (i) at the point x ¯, i.e. if (1.9)
¯ 0 , x) = A(i, x A(i ¯).
Definition 1.3. The magnitude DA, the difference between deformed object A¯ and initial object A at the same coordinate system and at the same point with respect to (1.7), i.e. (1.10)
¯ x) − A(i, x), DA = A(i,
is called Lie difference (Lie differential), and the magnitude (1.11)
¯ x) − A(i, x) DA A(i, = lim ε→0 ε ε→0 ε
Lz A = lim
is Lie derivative of geometric object A(i, x) with respect to the vector field z = (z i (xj )). ¯ x) we have Using the relation (1.10), for deformed object A(i, (1.12)
¯ x) = A(i, x) + DA, A(i,
¯ finding previously DA. The known main cases and thus we can express A, are: According to (1.10) we have Dxi = x ¯i − xi , i.e. for the coordinates we have (1.13)
Dxi = z i (xj )ε,
from where (1.130 )
Lz xi = z i (xj ).
´ S. M. MINCI ˇ C, ´ M. S. STANKOVIC ´ L. S. VELIMIROVIC,
238
For the scalar function ϕ(x) ≡ ϕ(x1 , . . . , xN ) we have Dϕ(x) = ϕ,p z p (x)ε = Lz ϕ(x)ε,
(1.14)
(ϕ,p = ∂ϕ/∂xp ),
i.e. Lie derivative of the scalar function is derivative of this function in direction of the vector field z. For a tensor of the kind (u, v) we get
(1.15)
u Dtij11...i ...jv
=
p u [tji11...i ...jv ,p z
−
u X α=1
=
iα z,p
µ ¶ µ ¶ v p i1 ...iu X p jβ i1 ...iu t + z,jβ t ]ε iα j1 ...jv p j1 ...jv β=1
u Lz tij11...i ...jv ε,
where we denoted µ ¶ p i1 ...iu i ...i piα+1 ...iu = tj11 ...jα−1 , (1.16) t v iα j1 ...jv
µ ¶ jβ i1 ...iu u = tij11...i t ...jβ−1 pjβ+1 ...jv . p j1 ...jv
For the vector dxi we have D(dxi ) = Lz (dxi ) = 0.
(1.17)
In the same way, as for the tensors, for the connection coefficients we have (1.18)
p i p i i i DLijk = (Lijk,p z p + z,jk − z,p Lpjk + z,j Lpk + z,k Ljp )ε = Lz Lijk ε.
From (1.11) we have (1.19)
DA = εLz A,
and, for study infinitesimal deformations of geometric objects, it is enough to study their Lie derivatives, and that is what we are doing in the present work.
INFINITESIMAL DEFORMATIONS OF BASIC TENSOR IN GENERALIZED ... 239
2. Lie derivative of the basic tensor 2.1. Based on the equations (2.12) at [VMS], for the Lie derivative of a u tensor tij11...i ...jv we have (2.1)
u Lz tji11...i ...jv
=
p u tji11...i ...jv | p z
−
λ
u X
z i|αp µ
α=1
µ ¶ µ ¶ v jβ i1 ...iu p i1 ...iu X p t , z | jβ tj1 ...jv + p j1 ...jv iα ν β=1
where (λ, µ, ν) ∈ {(1, 2, 2), (2, 1, 1), (3, 4, 3), (4, 3, 4)}. By applying to the tensor gij one obtains two cases Lz gij = gij | p z p + z p| i gpj + z p| j gip , 1
2
p
Lz gij = gij | p z + 2
z p| i gpj
2
+ z p| j gip ,
1
1
because the third case reduces to the second and the fourth case reduces to the first case. On account of (2.2)
hij | p = 0,
θ = 1, 2,
θ
the previous equations become (2.3a)
Lz gij = kij | p z p + zi| j + zj | i + z p| i kpj + z p| j kip , 1
(2.3b)
2
2
2
Lz gij = kij | p z p + zi| j + zj | i + z p| i kpj + z p| j kip . 2
1
1
1
In the case of Riemannian space RN (gij = gji = hij , the known equation (2.4)
2
1
kij = 0), we obtain
Lz gij = zi;j + zj;i ,
where by ; is denoted covariant derivative with respect to Christoffel symbols at RN . From here it follows that, forming at GRN by hij (symmetric part of gij ) Christoffel symbols Γi.jk , Γijk , with respect to (1.5), we obtain a 0
0
corresponding RN , and, based on (2.4): (2.5)
Lz hij = zi;j + zj;i ,
Based on exposed the following theorem is valid.
´ S. M. MINCI ˇ C, ´ M. S. STANKOVIC ´ L. S. VELIMIROVIC,
240
Theorem 2.1. If kij is antisymmetric part of the basic tensor gij of the space GRN and the covariant derivatives are defined by virtue of (1.6), then for Lie derivatives are in the force equations (2.3), where z(xi ) is the infinitesimal deformations vector field (1.7). For the symmetric part hij of the basic tensor is valid (2.5), where the covariant derivative is defined by virtue of hij . 2.2. We shall examine now the Lie derivative of the tensor g ij , where g ij = hip hjq gpq .
(2.6)
Let us firstly determine Lz hij . From (1.4) is hip hpj = δji ,
(2.7)
herefrom, using properties of the Lie derivatives: (Lz hip )hpj + hip Lz hpj = 0 ⇒ (Lz hip )hpj = −hip (Lz hpj ). From here, composing with hjq and applying (2.7): Lz hij = −hip hjq (Lz hpq ).
(2.8)
For the non symmetric tensor g ij based on (2.6), we obtain Lz g ij = (Lz hip )hjq gpq + hip (Lz hjq )gpq + hip hjq (Lz gpq ) = − hir hps (Lz hrs )hjq gpq − hip hjr hqs (Lz hrs )gpq + hip hjq (Lz gpq )
(2.8)
= − hir (Lz hrs )g sj − hjr g is (Lz hrs ) + hip hjq (Lz gpq ), where e.g., = signifies: equal on the base of (2.8). Finally, the previous (2.8)
equation can be written in the form: (2.9)
Lz g ij = −(hip g qj + hjp g iq )Lz hpq + hip hjq Lz gpq .
Thus, we have Theorem 2.2. The Lie derivative of the tensor g ij , defined with (2.6), is given by the equation (2.9), where hij is the symmetric part of the basic tensor gij of the generalized Riemannian space GRN , and hij is defined by
INFINITESIMAL DEFORMATIONS OF BASIC TENSOR IN GENERALIZED ... 241
(1.4). For a symmetric gij (gij = hij = hji ), the equation (2.9) reduce to (2.8). 2.3. We shall express Lz gij and Lz g ij by virtue of covariant derivatives formed with respect of Γijk . Summing the equations (2.3), we get 0
2Lz gij = (kij | p + kij | p )z p + (zi| j + zi| j ) + (zj | i + zj | i ) 1
2
1
2
1
2
+ (z p| i + z p| i )kpj + (z p| j + z p| j )kip . 1
2
1
2
Summing covariant derivatives of the first and the second kind, one obtains a covariant derivative of the same tensor in relation to the symmetric connection Γijk . So, from the previous equation we obtain the equation 0
(2.10)
p Lz gij = kij;p z p + zi;j + zj;i + z;ip kpj + z;j kip ,
which is of the form (2.3). By substitution at (2.9) with respect to (2.5) and (2.10), or by direct use of (2.1), we obtain (2.11)
ij p i pj j ip Lz g ij = k;p z − z;p g − z;p g .
Theorem 2.3. Using covariant derivatives with respect to symmetric part Γijk of the second kind Cristoffel symbols, Lie derivative of the basic tensor 0
gij is given by (2.10), and the Lie derivative of the tensor g ij , defined in (2.6), is given by (2.11), where kij is antisymmetric part of gij . References [Mi] [Mi1] [Mi2]
[St] [Sch]
Minˇ ci´ c, S. M., Ricci identities in the space of non-symmetric affine connexion, Matematiˇ cki vesnik 10(25)Sv. 2 (1973), 161-172. Minˇ ci´ c, S. M., New commutation formulas in the non-symmetric affine connexion space, Publ. Inst. Math. (Beograd)(N.S) 22(36) (1977), 189-199. Minˇ ci´ c, S. M., Independent curvature tensors and pseudotensors of spaces with non-symmetric affine connexion, Coll. math. soc. J´ anos Bolyai, 31. Dif. geom., Budapest (Hungary) (1979), 445-460. Stojanovi´ c, R, Osnovi diferencijalne geometrije, Gradjevinska knjiga, Beograd (1963). Schouten, J.A., Ricci calculus, Springer Verlag,Berlin-Gotingen-Heildelberg (1954).
242 [Ya] [Ya1] [IS] [IS1] [Eis] [MVS] [VMS]
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