On para-kenmotsu manifolds satisfying certain conditions on the curvature tensors

Transcription

1 Available olie at Advaces i Applied ciece esearch :08-3 IN: CODEN A: AAFC O para-kemotsu maifolds satisfyi certai coditios o the curvature tesors K. L. ai rasad ad T. atyaarayaa 2 Departmet of Mathematics Gayatri idya arishad Collee of Eieeri for ome Madhurawada isakhapatam Idia 2 Departmet of Mathematics raati Eieeri Collee urampalem Near eddapuram Adhra radesh Idia ABTACT The objective of this paper is to study certai properties of para-kemotsu maifolds satisfyi the coditios ad. 0 where is the iemaia curvature tesor is the icci curvature tesor ad is the eyl projective curvature tesor. It is show that a p -Kemotsu maifold satisfyi the coditios. 0 is flat ad. 0 is a Eistei maifold. Fially we also proved that if a p -Kemotsu maifold satisfies the coditio. 0 the the structure vector is ormal to the taet space of the maifold. Mathematics ubject Classificatio: 53C25 Keywords: ara Kemotsu maifold Curvature tesor icci tesor calar curvature Eistei Maifold INTODCTION The otio of almost para cotact structure was itroduced by ato 8]. Later Adati ad Matsumoto ] defied ad studied p -asakia ad sp -asakia maifolds which are rearded as a special kid of a almost cotact iemaia maifolds. Before ato Kemotsu 6] defied a class of almost cotact iemaia maifolds. I 995 iha ad ai rasad 9] have defied a class of almost para cotact metric maifolds amely para- Kemotsu p - Kemotsu ad special para Kemotsu sp -Kemotsu maifolds as aaloues of p -asakia ad sp -asakia maifolds. A iemaia maifold M is locally symmetric if its curvature tesor satisfies 0 where is Levi-Civita coectio of the iemaia metric 4]. As a eeralizatio of locally symmetric spaces may eometers have cosidered semi-symmetric spaces ad i tur their eeralizatios. A iemaia maifold M is said to be semisymmetric if its curvature tesor satisfies. 0 where acts o as a derivatio 0]. Locally symmetric ad semi-symmetric p-asakia maifolds are widely studied by may eometers 2 5 7]. I this paper we cosider p -Kemotsu maifolds satisfyi the coditios o the iemaia curvature tesor the icci curvature tesor ad eyl projective curvature tesor ad studied their properties for the first time. 08

2 K. L. ai rasad ad T. atyaarayaa Adv. Appl. ci. es :08-3 ELIMINAIE Let M be a -dimesioal differetiable maifold equipped with structure tesors Φ where Φ is a tesor of type is a vector field is a -form such that 2. Φ 2 ; Φ The 2.2 M is called a almost para cotact maifold 8]. Let be the iemaia metric satisfyi such that for all vector fields ad o M Φ Φ 0 rak Φ 2.4 Φ Φ 2.5 The the maifold M 8] is said to admit a almost para cotact iemaia structure Φ. A maifold of dimesio with iemaia metric admitti a tesor field Φ of type a vector field ad a -form satisfyi alo with ] ] 2.7 Φ Φ Φ Φ 2.9 is called a para Kemotsu maifold or p -Kemotsu maifold 9]. A p -Kemotsu maifold admitti a -form satisfyi 2.0 ad ϕ whereϕ is a associate of Φ 2. is called as special p -Kemotsu maifold or a sp -Kemotsu maifold 9]. It is kow that 9] i a p -Kemotsu maifold the followi relatios hold: where Q 2.2 ] ]

3 K. L. ai rasad ad T. atyaarayaa Adv. Appl. ci. es : isorthooalto whe ; 2.5 where is the icci tesor r is the scalar curvature ad Q is the symmetric edomorphism of the taet space at each poit correspodi to the icci tesor ad is the iemaia curvature. If the icci curvature tesor is of the form b a 2.6 where a ad b are fuctios o M the M is called as a Eistei maifold ad if 0 b the it is a Eistei maifold. Moreover it is also kow that if a p -Kemotsu maifold is projectively flat the it is a Eistei maifold ad the scalar curvature has a eative costat value 9]. I this case 2.7 ad hece Φ Φ 2.8 Also if a p -Kemotsu maifold is of costat curvature we have ] 2.9 The above results will be used further i the ext sectios. ELT 3. p -Kemotsu Maifolds satisfyi 0. I this sectio we cosider semisymmetric p -Kemotsu maifolds i.e. p -Kemotsu maifolds satisfyi the coditios ad 0. where is cosidered as a derivatio of tesor alebra at each poit of the maifold for taet vectors ad. Now. 3. The usi 0. the eq 3. ca be writte as si 2.3 eq 3.2 ca be writte as utti i 3.3 ad o usi 2.3 we have

4 K. L. ai rasad ad T. atyaarayaa Adv. Appl. ci. es : For the above equatio becomes or which is othi but 0 i view of 2.3. This shows that either 0 i.e. either the maifold is flat or is ormal to the curvature tesor. Hece we have the followi. Theorem 3.: A semisymmetric p -Kemotsu maifold is either flat or the structure vector is ormal. roof: The proof follows immediately from p -Kemotsu Maifolds satisfyi 0. Now we suppose that p -Kemotsu maifold is icci-symmetric i.e. 0.. The we have utti i the above expressio ad usi 2.5 we et 0. ] 3.8 Aai o usi 2.2 we et si eq 2.3 we et Now putti i the above equatio ad o usi 2.7 we et or a - - which o cotractio ives the scalar curvature costat as b r - -. O usi 3.6 i 3.9 we et c - r/. Thus we have

5 K. L. ai rasad ad T. atyaarayaa Adv. Appl. ci. es : Theorem 3.2: A p -Kemotsu maifold bei icci symmetric is a Eistei maifold ad hece it is a sp - Kemotsu maifold. roof: The proof of the theorem is a immediate cosequece of the equatios 3. a 3. b ad 3. c because a p -Kemotsu maifold of costat scalar curvature is a sp -Kemotsu maifold. 3.3 p -Kemotsu Maifold satisfyi 0. Now we cosider p -Kemotsu maifolds satisfyi the the coditio 0. where deotes the eyl projective curvature tesor ] defied by ]. Q Q 3.2 Now we have si 3.3ad 2.2 we et 0 K K 3.4 where. The i view of 3.2 we have or 0. ] 0. ] Q Q Q Q K K b Q Q Q Q K K a 3.5 Now putti ad usi ad 2.3 we et 0. ] Q Q Q Q Q Q Q Q 3.6 O putti i 3.6 ad o usi 2.2 we have 0 Q Q Q Q 3.7 which o simplicatio implies 0 for > shows that is ormal to the taet space of M. Also for i 3.5 b we have 0. ] Q Q 3.8 For we have

6 K. L. ai rasad ad T. atyaarayaa Adv. Appl. ci. es :08-3 Q Q Q ] which o usi eq 2.2 reduces to Q or 2 Q Q 3.20 Hece we have the followi result. Theorem 3.3: I a p -Kemotsu maifold if the coditio 0 is satisfied the we have i either the structure tesor is ormal to T M or ii 2 Q Q Q Q for >. roof: The proof of the theorem is obvious i view of the results obtaied i 3.7 ad CONCLION I this paper we had studied three ew properties of para-kemotsu maifolds satisfyi the coditios ad. 0. The results obtaied here are similar to the fidis reported earlier for para-asakia maifolds. Ackowledemet The authors ackowlede Dr. B. atyaarayaa Assistat rofessor of Naarjua iversity for his valuable suestios i preparatio of the mauscript. The authors declare that there is o coflict of iterests reardi the publicatio of this paper. EFEENCE ] Adati T ad Matsumoto K T Math ] Adati T ad Miyazawa T Tesor N ] Bishop L ad Goldber I Caad. J. Math ] Carta E Bull. oc. Math. Frace ] De C ad Tarafdar D Math. Balkaica N ] Kemotsu K Tohoku Math. Joural ] adey N ad avita erma Idia J. ure Appl. Math ] ato I Tesor N ] iha B B ad ai rasad K L Bulleti of the Calcutta Mathematical ociety ] zabo I The local versio. J. Diff. Geom ] ao K ure ad Applied Mathematics. Marcel Dekker Ic. New ork