REND.SEM.MAT.UNIV.PADOVA, Vol. 124 (2010)
On Equitorsion Geodesic Mappings of General Affine Connection Spaces
MICÂA S. STANKOVICÂ -SVETISLAV M. MINCÏICÂ -LJUBICA S. VELIMIROVICÂ MILAN LJ.ZLATANOVICÂ
ABSTRACT - In the papers [19], [20] several Ricci type identities are obtained by using non-symmetric affine connection. In these identities appear 12 curvature tensors, 5 of which being independent [21], while the rest can be expressed as linear combinations of the others. In the general case of a geodesic mapping f of two non-symmetric affine connection spaces GAN and GAN it is impossible to obtain a generalization of the Weyl projective curvature tensor. In the present paper we study the case when GAN and GAN have the same torsion in corresponding points. Such a mapping we name ``equitorsion mapping''. With respect to each of mentioned above curvature tensors we have obtained i quantities jmn (u 1; ; 5), that are generalizations of the Weyl tensor, i.e. Eu ÁÁÁ they are invariants based on f . Among only is a tensor. All these quantities Eu E5 are interesting in constructions of new mathematical and physical structures.
1. Introduction.
Geodesic mappings of Riemannian spaces and of affine spaces and their generalizations were investigated by many authors, for examle N. S. Sinjukov [34], J. MikesÏ [12]-[18], M. Prvanovic [25]-[28, 29], S. MincÏic [23], [24], [38]-[41], M. Stankovic [35]-[42] and many others. Consider two N-dimensional differentiable manifolds GAN and GAN and differentiable mapping f : GA GA . We can consider these N ! N manifolds in the common by this mapping system of local coordinates.
Indirizzo degli A.: Faculty of Sciences and Mathematics, University of NisÏ, VisÏegradska 33, 18000 NisÏ, Serbia E-mails: [email protected], [email protected], [email protected], [email protected] AMS Subj. Class.: 53B05. 78 M. S. Stankovic - S. M. MincÏic - L. S. Velimirovic - M. Lj. ZlatanovicÂ
Namely, if f : M GA M GA and if ( ; W) is local chart around the 2 N ! 2 N U point M it will be W(M) x (x1; ; xN) EN (Euclidean N-space). In ÁÁÁ 2 this case, we define for the coordinate mapping in the GAN the mapping 1 W W f À , and then 1 1 N N 1 W(M) (W f À )(f (M)) W(M) x (x ; ; x ) E ; ÁÁÁ 2 that is the points M and M f (M) have the same local coordinates. If i i connection coefficients Ljk(x) and Ljk(x) , for the connection introduced in GAN and GAN respectively, are non-symmetric in lower indices, we call GAN and GAN general affine connection spaces. Although the notion of non-symmetric affine connection is used in several works before A. Einstein, for example in [2] (Eisenhart, 1927), [8] (Hayden, 1932), the use of non-symmetric connection became especially actual after appearance the works of Einstein, relating to create the Unified Field Theory (UFT). Einstein was not satisfied with his General Theory of Relativity (GTR, 1916), and from 1923. to the end of his life (1955), he worked on various variants of UFT. This theory had the aim to unite the gravitation theory, to which is related GTR, and the theory of electromagnetism. One says that reciprocal one valued mapping f : GA GA is geo- N ! N desic, [23], [24] if geodesics of the space GAN pass to geodesics of the space GAN. In the corresponding points M(x) and M(x) we can put 2 Li (x) Li (x) Pi (x); (i; j; k 1; :::; N); jk jk jk i where Pjk(x) is the deformation tensor of the connection L of GAN ac- cording to the mapping f : GA GA . N ! N A necessary and sufficient condition that the mapping f be geodesic [23] i is that the deformation tensor Pjk from (2) has the form 3 Pi (x) di c (x) ji (x); jk ( j k) jk where 1 1 4 c Pp (Lp Lp ); i N 1 ip N 1 ip À ip 1 1 5 ji Pi (Pi Pi ) Pi jk jk 2 jk À kj 2 [ jk] _ where ij denotes a symmetrization, ij-antisymmetrization, [i ...j] denotes an antisymmetrization without division_ with respect to the indices i, j, and On Equitorsion Geodesic Mappings of General Affine Connection Spaces 79 also (i ...j) denotes a symmetrization without division with respect to in- dices i, j. In GAN one can define four kinds of covariant derivatives [19, 20]. For i example, for a tensor aj in GAN we have i i i p p i i i i p p i aj m aj;m Lpmaj Ljmap; aj m aj;m Lmpaj Lmjap; j À j À 1 2
i i i p p i i i i p p i aj m aj;m Lpmaj Lmjap; aj m aj;m Lmpaj Ljmap: j À j À 3 4 Denote by ; a covariant derivative of the kind u in GA and GA re- j j N N spectively. u u While at the Riemannian space (the space of GTR) the connection coefficients are expressed by virtue of symmetric basic tensor gij, at Einstein's works from UFT (1950-1955) the connection between these magnitudes is determined by equations @g 6 g g Gp g Gp g 0; (g ij ): ij ;m ij;m À im pj À mj ip ij;m @xm À The equation (6) signifies that the index i one treats in the sense of the first kind of derivative ( ), and j in the sense of the second one ( ). 1j 2j Proceeding at that sense, Einstein in [1], 1950, for covariant curvature tensor in his theory obtains a Bianchi-type identity: 7 R R R 0; iklm ;n ikmn ;l iknl ;m ÀÀ À ÀÀÀ where R g Rs : iklm si klm In the case of the space GAN we have five independent curvature tensors [21] (in [21] R is denoted by R~): 5 2 i i p i i i p i Rjmn Lj[m;n] Lj[mLpn]; Rjmn L[mj;n] L[mjLn]p; 1 2 i i i p i p i p i Rjmn Ljm;n Lnj;m LjmLnp LnjLpm LnmL[pj]; 3 À À 8 i i i p i p i p i Rjmn Ljm;n Lnj;m LjmLnp LnjLpm LmnL[pj]; 4 À À i 1 i i p i p i p i p i Rjmn (Lj[m;n] L[mj;n] LjmLpn LmjLnp LjnLmp LnjLpm): 5 2 À À i By virtue of the geodesic mapping f : GAN GAN we obtain tensors Rjmn, ! u (u 1; :::; 5), where for example i i p i 9 Rjmn Lj[m;n] Lj[mLpn]: 1 80 M. S. Stankovic - S. M. MincÏic - L. S. Velimirovic - M. Lj. ZlatanovicÂ
U. P. Singh ([31], 1968) for a special nonsymmetric connection of M. Prvanovic ([27], 1959) uses 2 kinds of covariant derivatives and obtains 3 curvature tensors by forming Ricci identities for a vector. F. Graif [4] gives geometric interpretations of two kinds parallel dis- placement, based on non-symmetric connection. M. Prvanovic [26], using non-symmetry of the connection, obtains geometric interpretations of of the tensors R; ...;R; given in (8), and S. MincÏic in [22] gives also corre- 1 4 sponding interpretations of R from (8). 5 An application of more kinds of covariant derivative at LN makes possible to express more concise some results. For example, let us consider infinitesimal deformations, defined by
10 xi xi ezi(x); x (x1; ...; xN); i 1; ...; N; where e is an infinitesimal parameter, zi(x) a vector field. As it is known, a deformed geometric object (x) (e. g. a tensor, a connection) of the object A (x) is A 11 e ; AA LzA i where z is the Lie derivative of in the direction of the field z (x): Then, L A ij A e. g. for a tensor tkl ([44] (K. Yano, 1949), [45](K. Yano, 1957) we have: 12 tij tij zp zi tpj zj tip zp tij zptij ; Lz kl kl;p À ;p kl À ;p kl ;k pl ;l kp Using the covariant derivatives of one kind, for example the first, instead of partial derivatives, we have
ij ij p i pj j ip 13 ztkl tkl pz z ptkl z ptkl L j À j À j 1 1 1
p ij p ij i sj j is s ij s ij p z ktpl z l tkp 2(Lpstkl L pstkl Lkptsl Llptks)z ; j j 1 1 _ _ _ _ from where we see that Lie derivative is a tensor. But, using more kinds of covariant derivative, we obtain in the considered case [43] (Lj. VelimirovicÂ, S. MincÏicÂ, M. StankovicÂ):
ij ij p i pj j ip p ij p ij 14 ztkl tkl pz z ptkl z ptkl z ktpl z ltkp; L j À j À j j j l m m n n where (l; m; n) (1; 2; 2); (2; 1; 1); (3; 4; 3); (4; 3; 4) i.e. in (14) we have 4 2f g manners of the presenting of Lie derivative, that are more concise than in (13). On Equitorsion Geodesic Mappings of General Affine Connection Spaces 81
In the case of geodesic mapping f : A A of the symmetric affine N ! N connection spaces AN and AN we have an invariant geometric object (Weyl projective tensor) 1 N 1 15 Wi Ri di R di R di R ; jmn jmn N 1 j [ mn ] N2 1 [m jn] N2 1 [m n] j À À i where Rjmn is Riemann-Cristoffel's curvature tensor of the space AN, and Rjm is Ricci tensor. i The object Wjmn is a tensor and it is called Weyl tensor, or a tensor of projective curvature [34]. Having a geodesic mapping of two general affine connection spaces, we can not find a generalization of Weyl tensor as an invariant of geodesic mapping in general case. For that reason we define a special geodesic mapping. A mapping f : GA GA is equitorsion geodesic mapping if the tor- N ! N sion tensor of the spaces GAN and GAN are equal. Then from (2), (3) and (5) 16 L h Lh jh (x) 0; ijÀ ij ij _ _ where ijdenotes an antisymmetrization with respect to i; j. _
2. ET-projective parameter of the first kind.
Using (3), (9) and (16) we get a relation between the first kind of cur- vature tensors (8) of the spaces GAN and GAN
i i i i i p i Rjmn Rjmn dj c[mn] d[m cjn] 2 Lmncj 2Lmn cpdj ; 1 1 1 1 _ _ where we denote
17 cmn cm n cmcn; u 1; 2: j u u À Contracting with respect to i and n, from (2) one gets
q q q q q q p q Rjmq Rjmq dj c[mq] dm cjq dq cjm 2Lmq cj 2Lmq cpdj ; i:e: 1 1 1 1 À 1 _ _ q p R jm R jm c[mj] cjm Ncjm 2Lmq cj 2Lmj cp; 1 1 1 1 À 1 _ _ from where
q p 18 Rjm Rjm c[ jm ] (1 N)cjm 2Lmq cj 2Lmj cp: 1 1 À1 À 1 _ _ 82 M. S. Stankovic - S. M. MincÏic - L. S. Velimirovic - M. Lj. ZlatanovicÂ
Here Rjm and Rjm are the first kind Ricci tensors of the spaces GAN and 1 1 GAN respectively. From (18) we obtain q p 19 R[ jm ] R[ jm ] 2c[ jm ] (1 N)c[ jm ] 2L[mqcj] 4Lmj cp: 1 1 À 1 À 1 _ _ From (19) and (4) one obtains
2 q p p 20 R[ jm ] R[ jm ] (N 1)c[ jm ] Lmq(Ljp Ljp) 1 1 À 1 N 1 À _ 2 4 Lq (Lp Lp ) Lp (Lq Lq ); À N 1 jq mp À mp N 1 mj pq À pq _ _ from where
2 q p p 21 (N 1) c[ jm ] R[ jm ] R[ jm ] Lmq(Ljp Ljp) 1 1 À 1 N 1 À _ 2 4 Lq (Lp Lp ) Lp (Lq Lq ): À N 1 jq mp À mp N 1 mj pq À pq _ _ Because of (18), (4) and (21), we have
1 2 q p p Rjm Rjm R[ jm ] R[ jm ] Lmq(Ljp Ljp) 1 1 À N 1 1 À 1 N 1 À _ 2 Â 4 Lq (Lp Lp ) Lp (Lq Lq ) À N 1 jq mp À mp N 1 mj pq À pq _ _ Ã 2 q p p 2 p q q (N 1)cjm Lmq(Ljp Ljp) Lmj (Lpq Lpq); À À 1 N 1 À N 1 À _ _ that is
1 2 q p p (N 1)cjm Rjm Rjm R[ jm ] R[ jm ] Lmq(Ljp Ljp) À 1 1 À1 À N 1 1 À1 N 1 À _ 2 Â 4 Lq (Lp Lp ) Lp (Lq Lq ) À N 1 jq mp À mp N 1 mj pq À pq _ _ 2 2 Ã Lq (Lp Lp ) Lp (Lq Lq ); N 1 mq jp À jp N 1 mj pq À pq _ _ from where
2 2N q p p (N 1)cjm (NRjm Rmj) (NRjm Rmj) Lmq(Ljp Ljp) À 1 1 1 À 1 1 N 1 À 22 _ 2 N 1 Lq (Lp Lp ) 2Lp (Lq Lq ) À : N 1 jq mp À mp mj pq À pq N 1 _ _ On Equitorsion Geodesic Mappings of General Affine Connection Spaces 83
Substituting (22) at (2), by help of (16), it is easy to prove that i i 23 jmn jmn; E1 E1 where we have denoted N N 1 i Ri di R di R di R E jmn jmn N 1 j [ mn ] N2 1 [m jn] N2 1 [m n] j 1 1 1 À 1 À 1 2 2 Lp di Lq Lp di Lq (N 1)2(N 1) [mp n] jq À (N 1)2 [mp j n]q 24 À _ _ 2N p q L di L (N 1)(N2 1) jp [n m]q À _ 4 2 Lp Li Lq di Lp di Lp (N 1)di Lp À N 1 jp mnÀ (N 1)2 pq ( j nm) À n mj j mn _ _ _ _ i  à Obviously, the magnitude jmn is not a tensor and we call it the ET-pro- E1 jective parameter of the first kind for the mapping f : GA GA . So, N ! N we have
i THEOREM 1. The ET-projective parameter of the first kind jmn (24) is E1 an invariant of the equitorsion mapping f : GAN GAN. p !
3. ET-projective parameter of the second kind.
For the second kind of curvature tensors R and R (8) of the spaces GAN 2 2 and GAN respectively, we get the relation i i i i i p i 25 Rjmn Rjmn dj c[mn] d[m cjn] 2Lnmcj 2Lnmcpdj ; 2 2 2 2 _ _ Analogously to previous case, we get
i i 26 jmn jmn; E2 E2 where is N N 1 i Ri di R di R di R E jmn jmn N 1 j [ mn ] N2 1 [m jn] N2 1 [m n] j 2 2 1 À 2 À 2 2 2 Lp di Lq Lp di Lq (N 1)2(N 1) [mp n] qj À (N 1)2 [mp j qn] 27 À _ _ 2N p q L di L (N 1)(N2 1) jp [n qm] À _ 4 2 Lp Li Lq di Lp di Lp (N 1)di Lp À N 1 jp nmÀ (N 1)2 pq j mn (m jn) j nm _  _ _ _ à 84 M. S. Stankovic - S. M. MincÏic - L. S. Velimirovic - M. Lj. ZlatanovicÂ
i The magnitude jmn is not a tensor and we call it ET-projective parameter E2 of the second kind. In this case we have
i THEOREM 2. The ET-projective parameter of the second kind jmn (27) E2 is an invariant of the equitorsion mapping f : GAN GAN. p !
4. ET-projective parameter of the third kind.
In the case of the third kind of curvature tensors R and R (8) of the 3 3 spaces GAN and GAN we get
i i i i i 28 Rjmn Rjmn d(j cm)n d(j cn)m 2c(nLm):j 3 3 2 À 1 _ Since p cmn cmn 2Lmn cp; 2 1 _ from (28) one obtains
i i i i i p i 29 Rjmn Rjmn dj c[mn] d[m cjn] 2d( j Lm)ncp 2c(nLm) j : 3 3 1 1 _ _ Contracting in (29) over i; n, we obtain
q p 30 Rjm Rjm c[ jm ] (N 1)cjm 2cqLmj 2cmL pj: 3 3 À 1 À À 1 _ _ Alternating the last equation with respect to j; m, we have
q p R[ jm ] R[ jm ] 2 c[ jm ] (N 1)c[ jm ] 4 cqLmj 2c[mLpj]: 3 3 À 1 À À 1 _ _ By using (4), we get
4 p q q 31 (N 1) cjm R[ jm ] R[ jm ] Lmj (Lpq Lpq) 1 3 À 3 N 1 À _ 2 2 Lp (Lq Lq Lp (Lq Lq ); N 1 pj mq À mq À N 1 pm jq À jq _ _ On Equitorsion Geodesic Mappings of General Affine Connection Spaces 85
By virtue of (4), (31) and (30) we obtain 1 4 R R R R Lp (Lq Lq ) jm jm À N 1 [ jm ] À [ jm ] N 1 mj pq À pq 3 3 3 3 Â _ 2 p q q 2 p q q L pj(Lmq Lmq) Lpm(Ljq Ljq) (N 1) cjm N 1 À À N 1 À À À 1 _ _ 2 2 Ã Lp (Lq Lq ) Lp (Lq Lq ); N 1 mj pq À pq N 1 pj mq À mq _ _ i.e. 2 p q q N 1 (N 1) cjm (NR jm Rmj) (NRjm Rmj) 2Lmj (Lpq Lpq) À À 1 3 3 À 3 3 À N 1 32 _ N 2 2Lp (Lq Lq ) Lp (Lq Lq ): pj mq À mq N 1 N 1 pm jq À jq _ _ Substituting (32) into (29) using the condition (4) we obtain
i i 33 jmn jmn; E3 E3 where 1 N 1 i Ri di R di R di R E jmn jmn N 1 j [ mn ] N2 1 [m jn] N2 1 [m n] j 3 3 3 À 3 À 3 2 2 Lq di Lp Lq di Lp (N 1)(N 1)2 jq [n pm] (N 1)2 [mq j pn] 34 À _ 2N q p 2 q L di L L Li (N 1)2(N 1) [mq n] pj À (N 1) (mq n) j À _ _ 2 Lq [2(N 1)di Lp di Lp (N 1)di Lp ] À (N 1)2 pq À j nm [m n] j m jn _ _ _ i The magnitude jmn is not a tensor and we call it ET-projective parameter E3 of the third kind. In this case we have
i THEOREM 3. The ET-projective parameter of the third kind jmn (34) is E3 an invariant of the equitorsion mapping f : GAN GAN. p !
5. ET-projective parameter of the fourth kind.
For the curvature tensors of the fourth kind R and R (8) of the space 4 4 GAN and GAN respectively, we get
i i i i i 35 Rjmn Rjmn d( j cm)n d( j cn)m 2 c(mLn) j; 4 4 2 À 1 _ 86 M. S. Stankovic - S. M. MincÏic - L. S. Velimirovic - M. Lj. Zlatanovic from where
i i 36 jmn jmn; E4 E4 where 1 N 1 i Ri di R di R ] di R E jmn jmn N 1 j [ mn ] N2 1 [m jn N2 1 [m n] j 4 4 4 À 4 À 4 2 2 Lq di Lp Lq di Lp (N 1)(N 1)2 jq [n pm] À (N 1)2 (mq j pn) 37 À _ 2N q p 2 q L di L L Li (N 1)2(N 1) (mq n) pj À (N 1) [mq n] j À _ _ 2 Lq [2(N 1)di Lp di Lp (N 1)di Lp ] À (N 1)2 pq À j nm [m n] j m jn _ _ _ i The magnitude jmn is not a tensor and we call it ET-projective para- E4 meter of the fourth kind. In this case we have
i THEOREM 4. The ET-projective parameter of the fourth kind jmn (37) is E4 an invariant of the equitorsion mapping f : GAN GAN. p !
6. ET-projective curvature tensor.
For the curvature tensors R and R (8) of the space GAN and GAN re- 5 5 spectively, we find the relation
i i 1 1 i 1 i 38 Rjmn Rjmn (c[mn] c[mn]) d[m cjn] d[m cjn]: 5 5 2 1 2 2 1 2 2 Denoting 1 39 cjn (cjn cjn); 12 2 1 2 we can write (38) in the form
i i i i 40 Rjmn Rjmn dj c[mn] d[m cjn]: 5 5 12 12
Eliminating cmn from (40) by analogous procedure as in the previous 12 cases, one obtains
i i 41 jmn jmn; E5 E5 On Equitorsion Geodesic Mappings of General Affine Connection Spaces 87 where 1 N 1 42 i Ri di R di R di R : E jmn jmn N 1 j [ mn ] N2 1 [m jn] N2 1 [m n] j 5 5 5 À 5 À 5 i In contrast to the previous cases, when jmn (u 1; :::; 4) are not Eu i tensors, the magnitude jmn is a tensor. We call it ET-projective curva- E5 ture tensor. We see that the following is valid
i THEOREM 5. The ET-projective tensor jmn (42) is an invariant of the E5 equitorsion mapping f : GAN GAN. p !
7. Conclusion.
In the case of generalized Riemannian space (GRN) the connection coefficients are defined by means of a non-symmetric basic tensor [3], [19], i [20], [32] and they are non-symmetric too. ET-projective parameters Ejmn u i and the projective tensor Ejmn, obtained as invariants of a map 5 f : GRN GRN are particular cases of obtained here parameters i ! i jmn (u 1; 4) and of the tensor jmn. For example, Eu ÁÁÁ E5 N N 1 Ei Ri di R di R di R jmn jmn N 1 j [ mn ] N2 1 [m jn] N2 1 [m n] j 1 1 1 À 1 À 1 4 2 Lp Li Lq di Lp di Lp (N 1)di Lp À N 1 jp mn À (N 1)2 pq j nm [m n] j j mn _  _ _ _ à When GAN (GRN) reduces to the Riemannian space, the magnitudes (E), Eu u (u 1; 5) reduce to the Weyl tensor [34] ÁÁÁ 1 Wi Ri di R : jmn jmn N 1 [m jn] À Acknowledgments. The authors are grateful to the anonymous refer- ees for comments and suggestions, which were helpful in improving the manuscript. The authors gratefully acknowledge support from the research projects 144032 of the Serbian Ministry of Science. 88 M. S. Stankovic - S. M. MincÏic - L. S. Velimirovic - M. Lj. ZlatanovicÂ
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