On Weak Concircular Symmetries of (LCS) 2n+1 - Manifolds By D. Narain & S. Yadav D.D.U.Gorakhpur University, India

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Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume Issue 0 Versio.

(USA Olie ISSN: 49-466 & Prit ISSN: 0975-5896 O Weak Cocircular Symmetries of (LCS + - Maifolds By D. Narai & S. Yadav D.D.U.Gorakhpur Uiversity,

cocircular Ricci symmetric Loretzia cocircular structure maifolds.

symmetric ad Loretzia cocircular structure maifold.

1 Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume Issue 0 Versio.0 0 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA Olie ISSN: & Prit ISSN: O Weak Cocircular Symmetries of (LCS + - Maifolds By D. Narai & S. Yadav D.D.U.Gorakhpur Uiversity, Idia Abstract - The object of the preset paper is to study weakly cocircular symmetric, weakly cocircular Ricci symmetric ad special weakly cocircular Ricci symmetric Loretzia cocircular structure maifolds. Keywords : Weakly cocircular symmetric maifold, weakly cocircular Ricci symmetric maifold, cocircular Ricci tesor,special weakly cocircular Ricci symmetric ad Loretzia cocircular structure maifold. GJSFR-F Classificatio : MSC 00: 53C0, 53C5, 53C5 O Weak Cocircular Symmetries oflcs+-maifolds Strictly as per the compliace ad regulatios of : 0. D. Narai & S. Yadav. This is a research/review paper, distributed uder the terms of the Creative Commos Attributio- Nocommercial 3.0 Uported Licese permittig all o commercial use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.

Ref. O Weak Cocircular Symmetries of (LCS + - Maifolds 0 [8] Tamasy,L. ad Bih,T.Q.,O weakly symmetric ad weakly projective symmetric Riemaia maifolds,coll.math.soc.,j.bolyai,56(989,663-670. D.

Yadav Abstract - The object of the preset paper is to study weakly cocircular symmetric, weakly cocircular Ricci symmetric ad special weakly cocircular Ricci symmetric Loretzia cocircular structure

Keywords : Weakly cocircular symmetric maifold, weakly cocircular Ricci symmetric maifold, cocircular Ricci tesor,special weakly cocircular Ricci symmetric ad Loretzia cocircular structure maifold. I.

A o-flat Riemaia maifold ( M, g ( is called weakly symmetric maifold if its curvature tesor R of type (0,4 satisfies the coditio ( X R ( Y, Z, U, V A ( X R ( Y, Z, U, V B ( Y R ( X, Z, U, V H ( Z R (

for all vector fields X, Y, Z, U, V ( M, ( M beig the Lie-algebra of the smooth vector fields of M, where A, B, H, D ad E are forms (ot simultaeously zero ad deote the operator of the covariat

2 Ref. O Weak Cocircular Symmetries of (LCS + - Maifolds 0 [8] Tamasy,L. ad Bih,T.Q.,O weakly symmetric ad weakly projective symmetric Riemaia maifolds,coll.math.soc.,j.bolyai,56(989, D. Narai & S. Yadav Abstract - The object of the preset paper is to study weakly cocircular symmetric, weakly cocircular Ricci symmetric ad special weakly cocircular Ricci symmetric Loretzia cocircular structure maifolds. Keywords : Weakly cocircular symmetric maifold, weakly cocircular Ricci symmetric maifold, cocircular Ricci tesor,special weakly cocircular Ricci symmetric ad Loretzia cocircular structure maifold. I. Itroductio The otio of weakly symmetric maifolds was itroduced by Tamassy ad Bih [8].A o-flat Riemaia maifold ( M, g ( is called weakly symmetric maifold if its curvature tesor R of type (0,4 satisfies the coditio ( X R ( Y, Z, U, V A ( X R ( Y, Z, U, V B ( Y R ( X, Z, U, V H ( Z R ( Y, X, U, V ( (,,, ( (,,, D U R Y Z X V E V R Y Z U X, (. for all vector fields X, Y, Z, U, V ( M, ( M beig the Lie-algebra of the smooth vector fields of M, where A, B, H, D ad E are forms (ot simultaeously zero ad deote the operator of the covariat differetiatios with respect to Riemaia metric g. The forms are called the associated forms of the maifold ad dimesioal maifold of this kid is deoted by ( WS. I 999, De ad Badyopadhyay [] studied a ( WS ad prove that i such a maifold the associated forms B H ad D E.Hece from (. reduces to the followig: α ( X R ( Y, Z, U, V A ( X R ( Y, Z, U, V B ( Y R ( X, Z, U, V B ( Z R ( Y, X, U, V ( (,,, ( (,,,. D U R Y Z X V D V R Y Z U X, (. A trasformatio of dimesioal Riemaia maifold M, which trasform every geodesic circle of M ito a geodesic circle, is called a cocircular trasformatio [].The itersectig ivariat of a cocircular trasformatio is the cocircular curvature tesor C, which is defied by []. σ 85 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I Author α : Departmet of Mathematics & Statistics, D.D.U.Gorakhpur Uiversity, Gorakhpur Idia. profddubey@yahoo.co.i Author σ : Departmet of Mathematics & Statistics A.I.E.T.College, Matsya Idustrial Area, Alwar-30030, Rajastha Idia. prof_sky6@yahoo.com 0 Global Jourals Ic. (US

3 O Weak Cocircular Symmetries of LCS+- Maifolds C ( Y, Z, U, V R ( Y, Z, U, V g ( Z, U g ( Y, V g ( Y, U g ( Z, V (, (. where k is the scalar curvature of the maifold Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I Recetly Shaikh ad Hui [5] itroduced the otio of weakly cocircular symmetric maifolds. A Riemaia maifold is called weakly cocircular symmetric maifold if its cocircular curvature tesor C of type (0,4 is ot idetically zero ad satisfies the coditio (,,, ( (,,, ( (,,, ( X C Y Z U V A X C Y Z U V B Y C X Z U V H Z C ( Y, X, U, V D( U C ( Y, Z, X, V E( V C ( Y, Z, U, X, (.4 for all vector fields X, Y, Z, U, V ( M where A, B, H, D ad E are form (ot simultaeously zero a dimesioal maifold of this kid is deoted by ( WC S.Also it is kow that [5], i a ( WC S the associated forms B H ad D E, ad hece the defiig the coditio (.4 of a ( WC S reduces to the followig form: (,,, ( (,,, ( (,,, ( X C Y Z U V A X C Y Z U V B Y C X Z U V B Z C ( Y, X, U, V D( U C ( Y, Z, X, V D( V C ( Y, Z, U, X where ABad, D are forms (ot simultaeously zero., (.5 Agai Tamassy ad Bih [9] itroduced the otio of weakly Ricci symmetric maifolds. A Riemaia maifold ( M, g,( is called weakly Ricci symmetric maifold if its Ricci tesor S of type (0, is ot idetically zero ad satisfies the coditio: X S ( Y, Z A ( X S ( Y, Z B ( Y S ( X, Z D ( Z S ( Y, X, (.6 where AB, ad D are three o-zero forms called the associate -forms of the maifold, ad is the operator of covariat differetiatio with respect to metric g. Such dimesioal maifold is deoted by ( WRS. If A B D 0 the is called pseudo Ricci symmetric. Let ei : i,,... be a orthoormal basis of the taget space at each poit of the maifold ad let S Y V the from (.3, we have. (, C( Y, e, e, V i i i S ( Y, V S( Y, V g( Y, V, (.7 The tesor S is called the cocircular Ricci symmetric tesor [] which is symmetric tesor of type (0,. I [] De ad Ghose itroduced the otio of weakly cocircular Ricci symmetric maifolds. A Riemaia maifold ( M, g,( is called weakly cocircular Ricci symmetric maifolds [] if its cocircular Ricci tesor S of type (0, is ot idetically zero satisfies the coditio: Ref. [5] Shaikh,A.A. ad Hui,S.K.,O weakly cocircular symmetric maifolds,a.sti.ale Uiv.,Al.I.CUZA,Di Iasi,LV,f.(009, Global Jourals Ic. (US

4 O Weak Cocircular Symmetries of LCS+- Maifolds (, ( (, ( (, ( X S Y Z A X S Y Z B Y S X Z D Z S ( Y, X, (.8 where A, Bad D are three -form (ot simultaeously zero.if A B D 0 the M is called pseudo cocircular Ricci symmetric. A Riemaia maifold is called special weakly Ricci symmetric maifold if where A is a form ad is defied by X S ( Y, Z A ( X S ( Y, Z A ( Y S ( X, Z A ( Z S ( Y, X, (.9 A( X g( X,. (.0 0 where is the associated vector field. Motivated by above studied we defie special weakly cocircular Ricci symmetric maifold. A dimesioal Riemaia maifold is called special weakly cocircular Ricci symmetric maifolds. If where A is a form ad is defied by (.0. (, ( (, ( (, ( X S Y Z A X S Y Z A Y S X Z A Z S ( Y, X. (. A ( -dimesioal Loretzia maifold M is smooth coected Para cotact Hausdorff maifold with Loretzia metric g, that is, M admits a smooth symmetric tesor field g of type (0, such that for each poit p M, the tesor g : T M T M R is a o degeerate ier p p p product of sigature (,,... where TpMdeotes the taget space of M umber space. I a Loretzia maifold ( Mg, a vector field defied by g( X, A( X at p ad R is the real for ay vector field X ( M is said to be cocircular vector field [5 ( X A ( Y g ( X, Y ( X A ( Y where is a o zero scalar fuctio, A is a -form ad is a closed -form. Let M be a Loretzia maifold admittig a uit time like cocircular vector field characteristic vector field of the maifold. The we have, called the g (,, (. Sice is the uit cocircular vector field, there exist a o zero -form such that 87 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I the equatio(.3 of the followig form holds: g( X, ( X, (.3 ( ( (, ( ( ( 0 X Y g X Y X Y, (.4 0 Global Jourals Ic. (US

5 O Weak Cocircular Symmetries of LCS+- Maifolds for all vector field XY,, where deotes the operator of covariat differetiatio with respect to Loretzia metric g ad is a o zero scalar fuctio satisfyig ( ( ( X X X, (.5 where beig a scalar fuctio. If we put 0 88 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I The from (.4 ad (.6, we have from which it follows that X X, (.6 X X ( X, (.7 is a symmetric (, -tesor. Thus the Loretzia maifold M together with uit time like cocircular vector field, it s associate -form ad (, tesor field is said to be Loretzia cocircular structure maifolds (briefly ( LCS -maifold [6]. I particular if, the the maifold becomes LP-Sasakia structure of Matsumoto [3]. II. Loretzia Cocircular Structure Maifolds A differetiable maifold M of dimesio ( is called ( LCS -maifold if it admits a (, tesor, a cotravariet vector field, a covariat vector field ad a Loretzia metric g which satisfy the followig (, (. I, (. g( X, Y g( X, Y ( X ( Y, (.3 g( X, ( X, (.4 0, ( X 0, (.5 for all X, Y TM. Also i a ( LCS maifold the followig relatios are satisfied [7]. ( R( X, Y Z ( g( Y, Z ( X g( X, Z ( Y, (.6 R( X, Y ( ( Y X ( X Y, (.7 R(, X Y ( g( X, Y ( Y X, (.8 R(, X ( ( X X, (.9 ( ( (, ( ( ( X Y g X Y X Y Y X, (.0 Ref. [3] Matsumoto, K, O Loretzia paracotact maifolds,bull of Yamagata Uiv.Nat.Soci. ( Global Jourals Ic. (US

6 O Weak Cocircular Symmetries of LCS+- Maifolds S( X, ( ( X, (. S( X, Y S( X, Y ( ( X ( Y, (. A Loretzia cocircular structure maifold is said to be satisfies Eistei if the Ricci operator Q Q aid b, where a ad b are smooth fuctios o the maifolds, I particular if b 0, the M is a Eistei maifolds. 0 III. Mai Result Defiitio3..A Loretzia cocircular structure maifold ( M, g ( is said to be weakly cocircular symmetric if its cocircular curvature tesor C of type (0,4 satisfies (.5. Substitutig X Y V ei i (.5 ad takig summatio over i, i, we get d ( X S ( Z, U g( Z, U A( X S( Z, U g( Z, U B( Z S( X, U g( X, U D( U S( X, Z g( X, Z B( R( X, Z U D( R( X, U Z ( B( X D( X g( Z, U B( Z g( X, U D( U g( Z, X Agai settig X Z U i (3. ad usig (.7 ad (., we have This leads to the followig result. d ( ( A( B( D(. ( Theorem 3..I a weakly cocircular symmetric Loretzia cocircular structure maifold M g the relatio (3. holds. (, ( Corollary. I a weakly cocircular symmetric Loretzia cocircular structure maifold M g the sum of -forms ABad, D over the characteristics vector field is uity provide (, ( the scalar curvature Next, puttig X ad Z by of the maifold isvaishes. i (3. ad usig (.4(.7ad(. we obtai (3. (3. 89 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I X d ( S (, U ( U A( B( ( ( U ( D( U ( D( ( ( U. ( (3.3 0 Global Jourals Ic. (US

7 O Weak Cocircular Symmetries of LCS+- Maifolds Also from (., we have S (, U 0. (3.4 I view of (3.(3.4,equatio (3.3 reduces to ( ( ( ( ( D( U ( U ( U D( ( ( ( (. (3.5 0 Next settig X U i (3. ad proceedig i the similar maer as above, we have 90 ( ( B( ( B( B( Z ( Z B( ( ( ( ( ( ( (, (3.6 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I Agai, substitutig Z U (3., we obtai X d ( X S (, A( X ( ( B( X D( X ( B( D( ( X ( Also, we have ( B( D( ( X which yield by usig (.6 ad (. that. I view of (3.7 ad (3.8, we get S (, XS(, S( X,, (3.7 S (, (, (3.8 d ( X ( ( ( ( ( ( ( ( ( ( B( X D( X AX d A( ( X ( This leads to the followig result. d ( ( ( A( ( X., (3.9 Theorem3.. I a weakly cocircular symmetric Loretzia cocircular structure maifold M g the associated -forms DBad, A are give by (3.5 (3.6 ad (3.9 respectively. (, ( Agai Takig U (3., we obtai X d ( X S ( Z, ( Z A( X S( Z, ( Z B( Z S( X, ( X 0 Global Jourals Ic. (US

8 O Weak Cocircular Symmetries of LCS+- Maifolds D( S( X, Z g( X, Z B( R( X, Z D( R( X, Z, (3.0 ( B( X D( X ( Z B( Z ( X D( g( Z, X I view of (3.4 ad (3.0, we get d ( X ( Z A( X S( Z, ( Z B( Z S( X, ( X D( S( X, Z g( X, Z B( R( X, Z D( R( X, Z ( B( X D( X ( Z B( Z ( X D( g( Z, X Settig X Z i (3. ad the usig (., we have A( B( D( d ( ( (3. which show that if k is costat the A( B( D( 0. Coversely, we will show that A B D 0. holds for all vector field o (, This leads to the followig result. M g which is obvious. Theoem3.3.I a weakly cocircular symmetric Loretzia cocircular structure maifold M g the sum of -forms DBad, A is zero everywhere provided the scalar curvature of (, ( the maifold is costat Defiitio 3. A Loretzia cocircular structure maifold ( M, g ( is said to be weakly cocircular Ricci symmetric if its cocircular Ricci tesor of type (0, satisfies (.9. I view of (.8 ad (.9 yield X d ( X S ( Y, Z g( Y, Z A( X S( Y, Z g( Y, Z B( Y S( X, Z g( X, Z D( D S( X, Y g( X, Y (3. Settig X Y Z i above we get the relatio (3. ad hece we ca state the followig Theorem3.4. I a weakly cocircular Ricci symmetric Loretzia cocircular structure maifold ( M, g ( the relatios (3. holds. 0 9 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I Corollary. I a weakly cocircular Ricci symmetric Loretzia cocircular structure maifold M g the sum of -forms AB, ad D is zero if the scalar curvature of the maifold is (, ( costat. 0 Global Jourals Ic. (US

9 O Weak Cocircular Symmetries of LCS+- Maifolds Now, takig X ad Y by i (3., we have d ( X S (, Z ( Z { A( B( } D( Z S(, (3.3 I view of (3. (3.4, equatio (3.3 yields. ( B( { ( } D( Z ( Z (,for all Z. ( Agai puttig X Z i (3. ad proceedig i a similar maer as above, we get Ad ( B( { ( } B( Y ( Y (, (3.6 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I ( A( { ( } A( X ( X (, (3.7 This leads to the followig result. Theorem3.5. I a weakly cocircular Ricci symmetric Loretzia cocircular structure maifold ( M, g ( the associated -form DB, ad A are give by (3.5 (3.6 ad (3.7 respectively Addig (3.5(3,6ad(3.7,we obtai 6 ( A( X B( X D( X A( B( D( ( X (, (3.8 This leads to the followig result. Theorem3.6. I a weakly cocircular Ricci symmetric Loretzia cocircular structure maifold M g the sum of the associated -form ABad, D is give by (3.8 (, ( Corollary3. There is o weakly cocircular Ricci symmetric Loretzia cocircular structure maifold ( M, g ( uless the sum of the -forms is everywhere zero if 6 ( A( B( D( (. Also takig cyclic sum of (., we get S ( Y, Z S ( Z, X S ( X, Y 4{ A( X S ( Y, Z A( Y S Z, X A( Z S ( Y, X }, (3.9 X Y Z 0 Global Jourals Ic. (US

10 O Weak Cocircular Symmetries of LCS+- Maifolds Let M admits a cyclic Ricci tesor. The (3.9 reduces to A( X S ( Y, Z A( Y S Z, X A( Z S ( Y, X 0. Takig Z i above ad the usig (.7 (.0 ad (., we obtai Agai takig Z ( A( X ( Y A( Y ( X ( ( X, Y 0. i (3.0, we get (3.0 ( ( X A( X (3. 0 Takig X i (3. ad usig (.7, we yields 93 I view of (3. ad (3., we get A( X 0, X. This leads to the followig result. ( 0. (3. Theorem3.8.If a special weakly cocircular Ricci symmetric Loretzia cocircular structure maifold ( M, g ( admits a cyclic Ricci tesor the the -form A must vaishes. Corollary3.If a weakly cocircular Ricci symmetric Loretzia cocircular structure maifold M g is a Eistei maifolds the the sum of -form vaishes if the scalar curvature is (, ( costat. Fially for Eistei maifold X S ( Y, Z 0 ad S( Y, Z ag ( Y, Z.The (.7 ad (., we get d ( X g ( Y, Z A ( X a g ( Y, Z A ( Y a g ( X, Z A ( Z a g ( X, Y Pluggig Z X Y i (3.3, we obtai that 4 ( ( a d ( which implies that if is costat the ( 0,that is A( Y 0, Y. the we state the results, (3.3 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I Theorem3.9. A special weakly cocircular Ricci symmetric Loretzia cocircular structure maifold ( M, g ( ca ot Eistei maifold if the -form A 0or the scalar curvature of the maifold is costat. 0 Global Jourals Ic. (US

11 O Weak Cocircular Symmetries of LCS+- Maifolds 0 94 Global Joural of Sciece Frotier Research F Volume XII Issue X V ersio I REFERENCES RÉFÉRENCES REFERENCIAS [] De,U.C. ad Ghose,G.C.,O weakly cocircular Ricci symmetric maifolds,south East Assia J.Math.ad Math.Sci.,3((005,9-5. [] De, U.C. ad Badyopadhya, S., O weakly symmetric Riemaia spaces, Publ.Math.Debrece, 54/3-4(999, [3] Matsumoto, K, O Loretzia paracotact maifolds,bull of Yamagata Uiv.Nat.Soci. ( [4] Narai, Dhruwa ad Yadav, Suil, O weakly symmetric ad Weakly Ricci symmetric LP- Sasakia maifolds. Africa Joural of Mathematics &compute scieces Research, 4(0, (0, [5] Shaikh,A.A. ad Hui,S.K.,O weakly cocircular symmetric maifolds,a.sti.ale Uiv.,Al.I.CUZA,Di Iasi,LV,f.(009, [6] Shaikh, A.A., Loretzia almost paracotact maifolds with structure of cocircular type, Kyugpook Math.J.43 (003, [7] Shaikh, A.A., Basu, T. ad Eyasmi,S.,O the existece of Extracta Mathematicae, 3,(008, recurret (LCS - maifolds, [8] Tamasy,L. ad Bih,T.Q.,O weakly symmetric ad weakly projective symmetric Riemaia maifolds,coll.math.soc.,j.bolyai,56(989, [9] Tamasy,L. ad Bih,T.Q.,O weakly symmetries of Eistei ad Sasakia maifolds,tesor N.S.,53(993, [0] Yadav,S. ad Suthar,D.L., O a quarter symmetric o-metric coectios i a geeralized co-symplectic maifolds, Global Joural of Sciece Frotier Research,0(9,(0,5-57,Ope Associatio of Research Socety,Uited States. [] Yao, K., Cocircular geometry I, cocirculartrasformathios, Proc.Imp.Acad.Tokyo, 6(940, [] Yadav,Suil, Dwivedi,P.K. ad Suthar,Dayalal, O ( LC S - Maifolds Satisfyig Certai Coditios o the Cocircular Curvature Tesor, Thi Joural of Mathematics,(9(0, ,Thailad. [3] Yadav, S., Suthar, D.L. ad Srivastava, A.K, Some Results o M( f, f, f Maifolds. Iteratioal Joural of Pure &Applied Mathematics, 70 (3 (0, Sofia, Ruse. 0 Global Jourals Ic. (US

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