On Four Dimensional Semi-C-reducible. Landsberg Space

Transcription

1 Internatonal Matematcal Forum Vol no On Four Dmensonal Sem--reducble Landsberg Space V. K. aubey and *Rakes Kumar Dwved Department of Matematcs & Statstcs D.D.U. Gorakpur Unversty Gorakpur (U.P.) Inda vkcoct@gmal.com * Pundt Deen Dayal Inter ollege Pprac Gorakpur (U.P.) Inda Abstract. In te present paper we ave work out te -connecton vectors of fourdmensonal Landsberg and Berwald spaces four dmensonal sem -reducble Landsberg spaces and necessary and suffcent condton under wc te fourdmensonal sem--reducble Landsberg space to be a Berwald space. Matematcs Subject lassfcaton: 5B0 560 Keywords: Landsbeg space Berwald space Sem--reducble Fnsler space four-dmensonal Fnsler space 1. Introducton A teory of ntrnsc ortonormal frame feld on n-dmensonal Fnsler space as been studed by Matsumoto and Mron ([2] []) and s called Mron frame by Matsumoto. A four dmensonal Fnsler space wt Mron frame as been studed n [5] [6] and [7]. Let F be a four dmensonal Fnsler space wt fundamental functon L. Te metrc tensor g j and () v-torson tensor jk of F are defned by g L 2 y y j j 2 1 L. y y y jk j k

2 192 V. K. aubey and R. K. Dwved Let {e }; λ 1 2 ; be te Mron frame of F were λ s te normalzed supportng element e2) m p are constructed by ge j λ) eμ) λμ jk jkg e 1) l y L s te normalzed torson vector e) n and e) δ. Hence s te lengt of torson vector. Te greek letters λ μ ν vares from 1 to trougout te capter. Te () v-torson tensor jk of F s wrtten as [7] (1.1) L jk λμνeλ) eμ) j eν )k 222mm jmk 2 (jk)(mn jn k ) 2 (jk)(mp jp k) 22 (jk)(mm jn k) (nnjn k) (jk)(np jpk) (m m p ) (n n p ) (p p p ) 22 (jk) j k (jk) j k j k 2 (jk){(m( np j k nkpj)} were (jk) denote te cyclc permutaton of ndces I j k and summaton for nstance (jk)(ab j k) AB j k AB j k AkB j. If we put 222 H 2 I 2 K J J H I 2 K Ten we ave [7] H I K L 22 - (J J ) 22 - (H I ) And ence (1.1) may be wrtten as (1.2) L jk Hmm jmk Jnn jnk Hppjpk I (jk)(mnjn k ) K (jk)( m pjpk ) J (jk)( npjp k) (J J ) (n mm ) I ( n n p ) (jk) j k (jk) j k (H I ) (m mp ) K {( m (np n p )}. (jk) j k (jk) j k k j Te egt scalars H I J K H I J K are called te man scalars of a four dmensonal Fnsler space. Defnton 1.1. [] A Fnsler space F n s called a Landsberg space f te Berwald connecton BГ s -metrcal. Defnton 1.1. [] In terms of artan s connecton Г a Landsberg space s caracterzed by (1) te (v) v-torson tensor Pjk vanses dentcally namely P ( ) 0 jk jk 0

3 On four dmensonal sem--reducble Landsberg space 19 or (2) te v-curvature tensor P jk vanses dentcally were te suffx 0 ndcates te contracton by supportng element y. Defnton 1.2. [2] In a Fnsler space F n f te connecton coeffcent G jk of Berwald connecton BГ are functon of poston alone ten space s called a Berwald space. Teorem 1.2. [2] In terms of te artan connecton Г a Berwald space s caracterzed by 0. jk 2. Te -connecton vectors and te tensor jk Let us consder te Mron frame {e λ } of F. If a tensor Tj of (1 1) type s gven ten [2] Tj Tλμeλ) eμ)j. Now we denote te -covarant dervatve wt respect to artan * * connecton ( Γjk Γ 0k jk ) by n. Let H λμν ) be scalar components of -covarant dervatve e λ ) j of vector e λ ) belongng to te Mron frame {e λ )} of F.e. (2.1) eλ) j Hλ) μν eμ) eν) j. Te scalar H λ ) μν satsfy te followng condton ([7] [2]) H 1) μν 0 and Hλμν ) Hμ) λν. If we assume (2.2) H2)μ μ H2)μ jμ and H)μ Kμ ten from (.1) we obtan j (2.) l j 0 m j n j p jj n j mj pkj p j m jj n Kj were (2.) j λ e λ ) j j λ e λ ) K K λ e λ ). Defnton 2.1. Te vectors j and K defned n (2.) are called te - connecton vectors of a four dmensonal Fnsler space. Now f we denote te scalar components of T j k by T λμ ν tat s (2.5) Tj k Tλμ νeλ) eμ) j eν)k ten we obtan K (2.6) T ( δ T ) e T H T H were δ Γ &. k k λμ ν K λμ ν) δμ δ) λν λδ δ) μν *j 0k r

4 19 V. K. aubey and R. K. Dwved k Furter f S s a scalar feld of F ten S k S λ e λ)k were S λ (δ k S) e λ) and s called -scalar dervatve of S. Dfferentatng equaton (1.2) -covarantly and usng equaton (2.) we ave 1 2 (2.7) L Ammm Annn Appp A (mmn) jk j k j k j k (jk) j k were (2.7) A (mm p ) A (mn n ) A (mp p ) (jk) j k (jk) j k (jk) j k 8 9 A (jk)(nnp j k ) A (jk)(npp j k ) 10 A (jk){(m (n jpk nkp j)} A H (J J ) (H I ) j A J I IK A H Kj JK A (J J ) 2K j (H ) (H I )K A (H I ) 2K (H 2K)j (J J )K. A I (J 2J ) I j 2KK A K J (H )J 2KK A I 2K Ij (J 2J )K A J K 2K j (H 2J )K 10 A K (H ) (J 2J )j (I K)K We denote te scalar components of te vector r A by r A μ ; r e. (2.8) r A r A μ e μ) r for nstant te scalar component of 1 A are gven by (2.8) 1 A H (J J ) (H I )j. μ μ In te vew of equaton (2.6) (2.2) and (2.8) te explct form of αβγ δ can be wrtten as 1 5 A A A (2.9) 222 μ μ 6 2 μ Aμ 7 2 μ Aμ 10 2 μ Aμ. μ 22 μ μ 2 μ Aμ 9 μ Aμ μ 22 μ μ 8 μ Aμ μ Aμ

5 On four dmensonal sem--reducble Landsberg space 195 ontracton of equaton (2.7) by y gves 1 2 (2.10) P A mm m A n n n A pp p jk jk 0 1 j k 1 j k 1 j k A (mm n ) A (mm p ) A (mn n ) (jk) j k 1 (jk) j k 1 (jk) j k (jk) j k 1 (jk) j k 1 (jk) j k 10 A 1 (jk){(m (njpk nkp j)}. A (mp p ) A (n n p ) A (n p p ) Now we consder te four dmensonal Landsberg space. Snce te (v) vtorson tensor P jk of a Landsberg space vanses dentcally terefore we ave r A 1 0 for eac r Moreover jk j k s satsfed n a Landsberg space. So equaton (2.5) mples λμνδ λμδν. In te vew of equaton (2.9) ts can be explctely wrtten as 1 A A 2 1 A 5 A 2 6 A 2 A 6 A 2 A 2 (2.11) A 2 7 A A 9 A A 6 A 2 5 A 7 A 2 2 A 8 A 8 A 9 A A 5 A 10 A 2 6 A 8 A 2 10 A 7 A 9 A 2 10 A. In vew of equatons (2.10) and (2.11) equaton (2.7) gves (2.12) L 1 Ammmm 2 Annnn Appp p jk 2 j k j k j k B (mm mn) B (mmm p) 1 (jk) j k 2 (jk) j k B (m n n n ) B (n n np) (jk) j k (jk) j k B (mp p p ) B (n p p p ) 5 (jk) j k 6 (jk) j k B (m mn n ) B 7 (jk) j k 8 (jk) j k B (n npp ) B (m mp p ) (m mn p ) 9 (jk) j k 10 (jk) j k B (m nn p ) B (m np p ) 11 (jk) j k 12 (jk) j k were B 1 1 A A 2 B 2 1 A 5 A 2 B 6 A 2 A 2 B 2 A 8 A B 5 A 2 7 A B 6 A 9 A B 7 A 6 A 2 B 8 5 A 7 A 2 B 9 8 A 9 A B 10 A 5 A 10 A 2 B 11 6 A 8 A 2 10 A B 12 7 A 9 A 2 10 A. Te notaton (jk) n te expresson (2.12) stands for te possble permutaton of ndces j k and summaton for nstance (m mmn ) mmm n m mm n m mmn m mm n (jk) j k j k j k k j j k (m mn n ) mmn n m m n n m mnn m mn n mmnn m mnn (jk) j k j k j k k j j k k j j k (jk)(m mn j kp) mmn j kp mjmn k p mkmnp j mmnjpk mmn p m m np m m np m mn p j k j k k j k j

6 196 V. K. aubey and R. K. Dwved mmnp k j mmnp k j mjmnp k mjmn kp. Proposton 2.1. Te -covarant dervatve of () v-torson vector jk of a four dmensonal Landsberg space can be wrtten n te form (2.12). Now we consder a four dmensonal Berwald space. From teorem (1.2) suc a space s caracterzed by αβγδ 0. In vew of equaton (2.7) we ave 22 δ δ δ (H I K) δ 0 wc mples δ 0. Smlarly 22 δ δ δ (H I K) j δ 0 wc mples j δ 0. Hence from 222 δ 0 we get H δ 0 and also J δ I K I δ 2K k δ K δ - 2K k δ ' ' ' H δ - J k δ I δ - (J - 2J )k δ J δ (H )k δ K δ (I - K) k δ ' wc gves (H I K) δ (L) δ 0. Summarzng above results we ave: Teorem 2.1. In a four dmensonal Berwald space te -connecton vectors and j vans dentcally. Also man scalar H and te unfed man scalar L are - covarantly constant. Furtermore f -connecton vector K vanses ten all te man scalars are -covarantly constant. To solve te -connecton vectors explctely n terms of man scalar n te next secton we consder a four dmensonal sem--reducble Landsberg space F.. Sem--reducble Landsberg space F Defnton.1. [] A Fnsler space F n (n ) wt te non-zero lengt of te torson vector s called sem--reducble f te () v-torson tensor jk s of te form p q ( ) n 1 jk j k jk k j 2 j k were p and q ( 1 p) do not vans. It as been seen ([6]) tat a four dmensonal Fnsler space F s sem-reducble f and only f (.1) H I K J J 0. Tus for a four dmensonal sem--reducble Landsberg space equaton (2.8) (2.11) and (.1) gves H (H ) 2 H (H 2K) j 2 I 2 (H ) I I 2 (.2) I Ij 2 (I K) K K 2 (H 2K) j K K 2 (I K) K K Kj 2 I Ij (H ) (I K) K 2 (H 2K) j K Ij K Kj.

7 On four dmensonal sem--reducble Landsberg space 197 Now we assume tat non of man scalars H I K vans. To solve te connecton vectors we dvde te four dmensonal sem--reducble Fnsler space n followng fve classes (I) H 2K (II) H 2K (III) H 2K (IV) H 2K (V) H 2K. For a sem--reducble Landsberg space equatons (2.10) and (2.8) gve H 1 I 1 K 1 1 j 1 0 and K 1 0 provded K I. Under te restrcton of class (I) usng te fact H I K L t follows from relaton (.2) tat 2 j 2 H I 2 H I I K (L) K L 0 I I K (L) K L and te dentty j 0 j K 2 0 K K 2 2K Ij 2 I K K K 2 I K KI 2 IK 2 2K Holds. Terefore we ave Teorem.1. Te -connecton vectors and -covarant dervatve of () vtorson tensor jk of a four dmensonal sem--reducble Landsberg space of class (I) are gven 2 m I 2 n j 1 j 2 m K 2 2K I K [Ij 2 n K 2 p ] I K II 2 KK 2 and L jk H 2 m m j m k m n n j n k n p p j p k p 2K H (jk)(m m j m k n ) H (jk) (m m j m k p ) I (jk) (m n j n k n ) K (jk)(m p j p k p ) I 2 (jk) (m m j n k n ) K 2 (jk) (m m j p k p ) p

8 198 V. K. aubey and R. K. Dwved A (jk) (n n j p k p ) I (jk) (m n j n k p ) K (jk) (m n j p k p ) were H I K (L) 2 I K L H I K (L) j2 I K L KI 2 IK 2 and A. H H 2K By te smlar process for classes (II) (III) and (IV) of sem--reducble Landsberg space F we obtan Teorem.2. Te -connecton vectors and -covarant dervatve of () vtorson tensor jk of a four dmensonal sem--reducble Landsberg space of class (II) are gven by I 2 n j 1 p K s arbtrary and II 2 L jk H 2 m m j m k m (n n j n k n p p j p k p ) I 2 I 2 (jk)[(m m j (n k n p k p )] II 2 (jk) (n n j p k p ). Teorem.. Te -connecton vectors and -covarant dervatve of () vtorson tensor jk of four dmensonal sem--reducble Landsberg space of class (III) are gven by IK2 K2 n j 1 p 2K(I K) 2( I K) K 0 and 2 II2 K I2 L jk n n j n k n p p j p k p 2(K I) ( K I) KI2 I 2 (jk) m m j n k n (jk) (n n j p k p ). 2( K I) Furter we consder class (V). In ts case equaton (.2) gves I 2 I I 0 2 j 2 j 0 and j. Tus we ave Teorem.. If te () v-torson tensor of a four dmensonal Fnsler space F of class (V) s wrtten n te form L jk m m j m k I (jk)[(m (n j n k p j p k )] ten te man scalar I s -covarant constant. Te -connecton vectors and - covarant dervatve of () v-torson tensor jk for suc space are gven by

9 On four dmensonal sem--reducble Landsberg space 199 Dn j Dp j K s arbtrary vector and L jk DI [n n j n k n p p j p k p (jk) (n n j p k p )] were D s arbtrary scalar. Fnally we sall fnd a condton for a sem--reducble Landsberg space to be Berwald space. From teorem (1.2) equaton (2.7) and (2.8) we ave for a four dmensonal sem--reducble Berwald space (.) H I K 0. Furter f te condton (.) satsfes n a four dmensonal sem-reducble Landsberg space ten usng (.2) and (2.8) n equaton (2.7) we ave jk 0 provded H 2K does not old. Tus we ave Teorem.5. A necessary and suffcent condton for a four dmensonal sem- -reducble Landsberg space to be Berwald space s tat all te non-zero man scalars H I K are -covarant constant provded H 2K does not old. Snce H 1 0 s satsfed n sem--reducble Landsberg space from Rcc dentty t t H j H j - H t j - H t P j we ave H - H 0. Smlarly we obtan I - I 0 and K - K 0. Tus n vew of teorem (.5) we ave Teorem.6. In a four dmensonal sem--reducble Landsberg space F te followng condton are equvalent to eac oter (1) F s a Berwald space. (2) All te man scalar H I K are -covarant constant. () Man scalar H I K satsfy te relatons H 0 I 0 K 0 0 provded H 2K does not old. Acknowledgement: Frst autor s very muc tankful to NBHM-DAE for ter fnancal assstance as a Postdoctoral fellowsp. References [1] Ikeda F.: On some propertes of tree dmensonal Fnsler spaces Tensor N. S. 55 (199) [2] Matsumoto M.: Foundaton of Fnsler geometry and specal Fnsler spaces Kasesa Press Sakawa Otsu 520 Japan [] Matsumoto M. and Mron R.: On an nvarant teory of Fnsler spaces Perod Mat. Hungar 8 (1977) [] Matsumoto M. and Sbata.: On sem--reducblty T- Tensor 0 and S-lkeness of Fnsler spaces J. Mat. Kyoto Unv. 19-2(1979) 01-1.

10 200 V. K. aubey and R. K. Dwved [5] Prasad B. N. Pandey T. N. and Twar B.: On relatons of man scalars between four dmensonal Fnsler space and ts ypersurfaces. ommuncated. [6] Prasad B. N. Pandey T. N. and Jaswal A. K.: P-reducble and P- symmetrc four dmensonal Fnsler spaces accepted J. Nat. Acad. Mat. [7] Pandey T. N. and Dved D. K.: A teory of four dmensonal Fnsler spaces n terms of scalars Nat. Acad. Mat. 11 (1997) Receved: September 2012