Microsoft Word - Shah final (55-63) Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.55 (Online Publication: Dec., 2018) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal On some curvature tensors in N(k)-contact metric manifolds Riddhi Jung Shah Department of Mathematics, Janata Campus, Nepal Sanskrit University, Dang, Nepal Email: shahrjgeo@gmail.com Article history: Received 25 April, 2018; Accepted 14 September, 2018 DOI: http://dx.doi.org/10.3126/bibechana.v16i0.19674 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/ Abstract The purpose of this paper is to study 7W and 9W -curvature tensors on )(kN -contact metric manifolds. We prove that a )(kN -contact metric manifold satisfying the condition 0).,( 97 =WXW ξis η -Einstein manifold. We also obtain the Ricci tensor S of type (0, 2) for 9W−ϕ flat and 09 =divW conditions on )(kN -contact metric manifolds. Finally, we give an example of 3-dimensional )(kN -contact metric manifold. Keywords: Contact manifold; )(kN -contact metric manifold; η -Einstein manifold. 1. Introduction In 1988, Tanno [1] introduced the notion of k -nullity distribution of a contact metric manifold as a distribution such that the characteristic vector field or Reeb vector field (Reeb 1952) [2] ξ of the contact metric manifold belongs to the distribution. A contact metric manifold with ξ belonging to the k -nullity distribution is called )(kN -contact metric manifold. Generalizing this notion Blair, Koufogiorgos and Popantoniou [3] introduced the notion of a contact metric manifold with ξ belonging to the ),( µk -nullity distribution, where k and µ are real constants. In particular, if ,0=µ then the notion of ),( µk -nullity distribution reduces to the notion of k -nullity distribution. On the other hand, in 1982, Pokhariyal [4] defined ,7W 8W and 9W -curvature tensors with the help of the Weyl's projective curvature tensor defined by Eisenhart [5]. In [4] it is studied that distribution of vector field over the metric potentials and matter tensors plays an important role in shaping the various physical and geometrical properties of a tensor. In the same paper it is proved that 9W -curvature tensor satisfies the cyclic property. In this paper, we study some curvature conditions of 7W and 9W -curvature tensors in )(kN -contact metric manifolds and obtain some results. Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.56 (Online Publication: Dec., 2018) 2. Preliminaries A )12( +n -dimensional smooth manifold M is said to be a contact manifold, if it carries a global 1-form η such that 0)( ≠∧ ndηη everywhere. Given a contact form η , it is well known that there exists a unique vector field ξ , called the characteristic vector field or Reeb vector field (Reeb 1952) [2] of η , satisfying ,1)( =ξη 0),( =Xd ξη (2.1) for all vector fields .X A Riemannian metric g is said to be an associated metric if there exists a tensor field ϕ of type (1, 1) such that ),,(),( YXgYXd ϕη = ),,()( ξη XgX = ξηϕ )()(2 XXX +−= (2.2) for all vector fields YX , on .12 +nM From above conditions one can easily obtain ,0=ϕξ ,0=ϕη o ).()(),(),( YXYXgYXg ηηϕϕ −= (2.3) The structure ),,,( gηξϕ is called a contact metric structure and a manifold 12 +nM with a contact metric structure ),,,( gηξϕ is said to be a contact metric manifold [6]. Now, we define the operators l and h by ,),( ξξXRlX = h = 21 Ëξ ϕ (2.4) where R and Ë denote the curvature tensor and Lie differentiation respectively. The (1, 1)-type tensors h and l are symmetric and satisfy ,0=ξh ,0=ξl ,0. =hTr 0. =ϕhTr and .hh ϕϕ −= (2.5) We also have the following relations for a contact metric manifold: hXXX ϕϕξ −−=∇ (2.6) 0=∇ ϕ ξ (2.7) 2.2),(. hTrnQglTr −== ξξ (2.8) )(2 22 hll +=− ϕϕϕ (2.9) 2hlh ϕϕϕ ξ −−=∇ (2.10) where Q is the Ricci operator defined by ),,(),( YXSYQXg = ∇ is the Levi-Civita connection of the Riemannian metric g and S the Ricci tensor [6,7]. An almost contact metric manifold is an odd dimensional smooth manifold 12 +nM equipped with an almost contact metric structure ),,,( gηξϕ satisfying the conditions (2.1) - (2.3) except the condition ),(),( YXgYXd ϕη = or .0),( =Xd ξη Sasaki and Hatakeyama [8] defined an almost complex structure J on RM n ×+12 by       −=      dtdXfXdtdfXJ )(,, ηξϕ (2.11) where f is a smooth real valued function on .12 RM n ×+ An almost contact metric structure is said to be normal if J is integrable. An almost contact metric manifold is Sasakian if and only if ( ) .)(),()( XYYXgYX ηξϕ −=∇ (2.12) A contact metric manifold is Sasakian if and only if YXXYYXR )()(),( ηηξ −= (2.13) for all vector fields X and Y [6]. Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.57 (Online Publication: Dec., 2018) A contact metric manifold for which ξ is a Killing vector is called a K -contact manifold. It is well known that a contact manifold is K -contact if and only if .0=h We note that a Sasakian manifold is K -contact, but the converse holds only if dim .3=M The k -nullity distribution [1] of a Riemannian manifold ),( gM for a real number k is a distribution ]}),(),([),(:)({)(:)( YZXgXZYgkZYXRMTZkNpkN pp −=∈=→ (2.14) for any ).(, MTYX p∈ If the characteristic vector field ),(kN∈ξ then the relation ])()([),( YXXYkYXR ηηξ −= (2.15) hods. A contact metric manifold satisfying the relation (2.15) is called a )(kN -contact metric manifold. From (2.13) and (2.15) it follows that a )(kN -contact metric manifold is a Sasakian manifold if and only if .1=k If 0=k i.e., ,0),( =ξYXR then )(kN -contact metric manifold is locally isometric to the Riemannian product )4()0(1 nn SE ×+ for 1>n and flat for .1=n If ,1n or M is η -Einstein manifold. Proof. Let us consider a )12( +n -dimensional )(kN -contact metric manifold M which satisfies the curvature condition ( ) 0),().,( 97 =ZVUWXW ξ for any vector fields ZVUX ,,, on .12 +nM By definition we have ( ) ( ) ( ) ZXWVUWZVXWUW ZVUXWWZVUWXWZVUWXW ),(),(),(, ,),(),(),(),().,( 7979 799797 ξξ ξξξ −− −= ( ) ( ) .),(),(),(, ,),(),(),(0 or, 7979 7997 ZXWVUWZVXWUW ZVUXWWZVUWXW ξξ ξξ −− −= (3.1) In view of (2.28) and (3.1) we get ( ) ( ) ( ) .),(),(21),()(),(),(2 ),(),(21),()(),(),(2 ),(),(21),()(),(),(2 ),(,21),(),(,20 999 999 999 999 ξηξ ξηξ ξηξ ξηξ VUWZXSnXVUWZkVUWZXkg ZUWVXSnZXUWVkZUWVXkg ZVWUXSnZVXWUkZVWUXkg ZVUWXSnXZVUWkZVUWXkg ++− ++− ++− −−= (3.2) Taking inner product on both sides of (3.2) by ξ and using (2.1) and (2.2) we obtain Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.59 (Online Publication: Dec., 2018) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).),(),(21),()(),(),(2 ),(),(21),()(),(),(2 ),(),(21),()(),(),(2 ),(,21)(),(),(,20 999 999 999 999 ξηηηξη ξηηηξη ξηηηξη ηη VUWZXSnXVUWZkVUWZXkg ZUWVXSnZXUWVkZUWVXkg ZVWUXSnZVXWUkZVWUXkg ZVUWXSnXZVUWkZVUWXkg ++− ++− ++− −−= (3.3) By the use of (2.30) - (2.33), (3.3) reduces to ).()(),(2),(),(41)()()},( ),({)}()(),(){,(2 )},(1 )()(){,()},(21)()(){,( )),(,(21)),(,(20 22 99 VUZXSnkVUSZXSnZVUXkg UXSkZUZUgVXgkVUSn VUkZXkgZUgZUVXSnk ZVUWXSnZVUWXkg ηηηη ηη ηηηη −+− +−+− +−+ −= (3.4) Putting ξ=U in (3.4) and using (2.1), (2.2), (2.17) and (2.29) we get )].()()()12()(),(1 )(),(21)(),(2)(),([0 ZXVknZVXSn VZXSnZVXkgVZXkgk ηηηη ηηη −++ −−= (3.5) From (3.5) it follows that either 0=k or .0)]()()()12()(),(1 )(),(21)(),(2)(),([ =−++ −− ZXVknZVXSn VZXSnZVXkgVZXkg ηηηη ηηη If ,0=k then we know that )(kN -contact metric manifold is locally isometric to the Riemannian product )4()0(1 nn SE ×+ for 1>n [9]. If ,0)]()()()12()(),(1 )(),(21)(),(2)(),([ =−++ −− ZXVknZVXSn VZXSnZVXkgVZXkg ηηηη ηηη then replacing Z by ξ we get ).()()12(),(2),( VXknnVXnkgVXS ηη−−= (3.6) From (3.6) we obtain ).()(),(),( 21 VXVXgVXS ηηλλ += (3.7) where nk21 =λ and .)12(2 knn −−=λ In view of (2.34) and (3.7) we conclude that )(kN -contact metric manifold 12 +nM is η -Einstein manifold. This completes the proof of the theorem. Theorem 3.2. In a ϕ - 9W flat )(kN -contact metric manifold 12 +nM the Ricci tensor S and the scalar curvature r are given by ),()1(4)()(})12(2{),()4(),( 2 ZhYgnZYrknnZYgknrZYS −+−++−= ηη Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.60 (Online Publication: Dec., 2018) and knnr )12(2 += respectively. Proof. Let us consider a )12( +n -dimensional )(kN -contact metric manifold M which is ϕ - 9Wflat. Then we have ( ) 0,),(9 =WZYXWg ϕϕϕϕ (3.8) for any vector fields ).(,,, MTWZYX p∈ In view of (2.27) we have ( ) ( ) )].,(),( ),(),([21,),(,),(9 ZYgWXS WZgYXSnWZYXRgWZYXWg ϕϕϕϕ ϕϕϕϕϕϕϕϕϕϕϕϕ − += (3.9) Using (2.3), (2.14), (2.22) and (3.8) in (3.9) we obtain )}]()(),()}{,()1(4)()(2),({ )}()(),()}{,()1(4)()(2),([{21 ),(),(),(),(0 ZYZYgWhXgnWXnkWXS ZWZWgYhXgnYXnkYXSn WYgZXkgWXgZYkg ηηηη ηηηη ϕϕϕϕϕϕϕϕ −−−−− −−−−+ −= ),(),()1(2),(),(21 )()(),()1(2)()(),(21 ),(),()1(2)()(),( ),(),(21)()(),()()(),( ),(),()()(),(),(),(0 or, WhXgZYgnnZYgWXSn ZWYhXgnnZWYXSn YhXgZWgnnYXZWkg ZWgYXSnWYZXkgZXWYkg WYgZXkgZYWXkgWXgZYkg − +− − +− − −− +++ −−= ηηηη ηη ηηηη ηη (3.10) ).()(),()1(2)()(),(21 ZYWhXgnnZYWXSn ηηηη − −+ Let 12,...,2,1},{ += niei be an orthonormal basis of the tangent space at any point of the manifold. Putting ieWX == in (3.10) and taking summation over ,121 , +≤≤ nii we get ),()1(4)()(})12(2{),()4(),( 2 ZhYgnZYrknnZYgknrZYS −+−++−= ηη (3.11) by the use of (2.5) and symmetry property of .h Again, taking an orthonormal frame field at any point of the manifold and contracting over Y and Z in (3.11) we obtain .)12(2 knnr += (3.12) In view of (3.11) and (3.12), the theorem is proved. Theorem 3.3. If a )12( +n -dimensional )(kN -contact metric manifold M satisfies the condition ,09 =divW then the Ricci tensor S is of the form ).,(),(12 )12(2),(2),( YhZShZYgn knnZYnkgZYS + + − += Proof. Let us consider a )12( +n -dimensional )(kN -contact metric manifold M which satisfies the condition 0),)(( 9 =ZYXdivW (3.13) where 'div' denotes the divergence. Taking inner product on both sides of (2.27) with V we get Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.61 (Online Publication: Dec., 2018) ( ) ( ) )].,(),(),(),([21,),(,),(9 ZYgVXSVZgYXSnVZYXRgVZYXWg −+= (3.14) Differentiating (3.14) covariantly along U we have )],(),)((),(),)([(21),,,)(~( ),,,)(~( 9 ZYgVXSVZgYXSnVZYXR VZYXW UUUU ∇−∇+∇= ∇ (3.15) where ),,,(~),),(( VZYXRVZYXRg = for all .,,, VZYX Let 12,...,2,1},{ += niei be an orthonormal basis of the tangent space at any point of the manifold. Putting ieVU == in (3.15) and taking summation over ,121 , +≤≤ nii we get )].,(2 )(),)([(21),)((),)(( 9 ZYgXdrYXSnZYXdivRZYXdivW Z −∇+= (3.16) From Bianchi second identity we also know that ).,)((),)((),)(( ZXSZYSZYXdivR YX ∇−∇= (3.17) In view of (3.13), (3.16) and (3.17) we have ).(),(41),)((21),)((),)((0 XdrZYgnYXSnZXSZYS ZYX −∇+∇−∇= (3.18) Replacing X by ξ in (3.18) we obtain 0),)((21),)((),)(( =∇+∇−∇ YSnZSZYS ZY ξξξ (3.19) since .0)( =ξdr From Ëξ 0=S we get ),(),(),()1(4 ),(),(),)(( YhZSZhYShZYgn YSZSZYS ZY ϕϕϕ ξξ ξ ++−= ∇−∇−=∇ by the use of (2.6) and (2.23). Again, ( ) ).,(),(),(2),(2 ),(),()(2 ),(),(),(),)(( ZhYSZYSZhYnkgZYnkg ZhYSZYSZnk ZSZSZSZS Y YYYY ϕϕϕϕ ϕϕη ξξξξ +++= ++∇= ∇−∇−∇=∇ by the use of (2.6), (2.17) and (2.25). Similarly, ).,(),(),(2),(2),)(( YhZSYZSYhZnkgYZnkgYSZ ϕϕϕϕξ +++=∇ Using above relations in (3.19) we get ).,(21),(),(),(2 ),(2),(2 )12(),(})1(4{0 YZSnYZkgZYSZhYnkg ZYnkgYhZSnnhZYgkn ϕϕϕϕ ϕϕϕ ++−− − + ++−= (3.20) Replacing Z by Zϕ in (3.20) and using (2.2), (2.3), (2.5), (2.17), (2.22) and symmetry property of h we obtain ),(21),(),(),(2 ),(2),(2 )12(),(})1(4{0 22 2YZSnZYkgZYSZhYnkg ZYnkgYZhSnnZhYgkn ϕϕϕϕϕϕ ϕϕϕϕϕ ++−− − + ++−= or, ).,(),(12 )12(2),(2),( YhZShZYgn knnZYnkgZYS + + − += (3.21) Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.62 (Online Publication: Dec., 2018) This completes the proof of the theorem. 4. Example of 3-dimensional )(kN -contact metric manifold Consider a 3-dimensional manifold }0:),,{( 3 ≠∈= xRzyxM where ),,( zyx are the standard coordinates in .3R Let },,{ 321 eee be linearly independent global frame on M defined by zxyyxzxeyxe ∂ ∂ + ∂ ∂ − ∂ ∂ = ∂ ∂ = 42 ,2 21 and .3 ze ∂ ∂ = Let g be the Riemannian metric defined by ,0),(),(),( 213231 === eegeegeeg .1),(),(),( 332211 === eegeegeeg Let η be the 1-form defined by ),()( 3eVgV =η for any ),(MV χ∈ the set of vector fields. Let ϕ be the (1, 1) tensor field defined by .0,, 31221 =−== eeeee ϕϕϕ Then using the linearity of ϕ and g we have 323 )( ,1)( eVVVe ηϕη +−== and )()(),(),( VUVUgVUg ηηϕϕ −= for any ).(, MVU χ∈ Moreover, .0 , , 32211 ==−= heeheehe Thus for ,3 ξ=e the structure ),,,( gηξϕ defines a contact metric structure on .M Now, from definition of Lie bracket we have .22 24 242422],[ 31221 eex zyx yxzxyyxzxzxyyxzxyxee += ∂ ∂ + ∂ ∂ =       ∂ ∂       ∂ ∂ + ∂ ∂ − ∂ ∂ −      ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ = Similarly, we obtain 0],[ 31 =ee and .2],[ 132 eee = Now, we have Koszul's formula ( ) ( ) ( ) ( ).],[,],[, ],[,),(),(),(,2 YXZgZXYg ZYXgYXZgXZYgZYXgZYg X +− −−+=∇ Taking ξ=3e and using Koszul formula we obtain ( ) ),,(4,2 11121 eegxeeg e =∇ ( ) ),(40,2 21221 eegxeeg e ==∇ and ( ) ).,(40,2 31321 eegxeeg e ==∇ From above it follows that ( ) ).,(2, 121 XegxXeg e =∇ Hence .2 121 exee =∇ Similarly we obtain other results and finally we have ,2 211 exee −=∇ ,2 121 exee =∇ ,2 312 eee −=∇ ,2 132 eee =∇ .03323132231 =∇=∇=∇=∇=∇ eeeee eeeee From above one can easily seen that the conditions for )(kN -contact metric manifold are satisfied for ξ=3e and .0,4 ≠−= xxk Riddhi Jung Shah./ BIBECHANA 16 (2019) 55-63: RCOST p.63 (Online Publication: Dec., 2018) 5. Conclusion In this paper we have studied 7W and 9W curvature tensors on )(kN -contact metric manifolds. It is proved that a )12( +n -dimensional )(kN -contact metric manifold satisfying the condition 0).,( 97 =WXW ξ is either locally isometric to the Riemannian product )4()0(1 nn SE ×+ or is η -Einstein. It is also investigated that a )(kN -contact metric manifold satisfying the conditions 9W−ϕ flat and 09 =divW has the Ricci tensor S of different forms. References [1] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J. 40 (1988) 441-448. [2] G. Reeb, Sur certaines proprietes topologiques des trajectoires des systemes dynamiques, Memoires de 1Acad. Roy. de Beligique, Sci. Ser. 2 (1952) 1-62. [3] D. E. Blair, T. Koufogiorgos, B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. of Math. 19 (1995) 189-214. [4] G. P. Pokhariyal, Relativistic significance of curvature tensors, Internat. J. Math. and Math. Sci. 5(1) (1982) 133-139. [5] L. P. Eisenhart, Riemannian geometry, Princeton University Press, 1950. [6] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, 509, Springer-Verlag, Berlin, 1976. [7] D. E. Blair, J. N. Patnaik, Contact manifolds with characteristic vector field annihilated by the curvature, Bull. Inst. Math. Acad. Sinica 9 (1981) 533-545. [8] S. Sasaki, Y. Hatakeyama, On differentiable manifolds with certain structures which are closely related to almost contact structure II, Tohoku Math. J. 13 (1961) 281-294. [9] C. Baikoussis, T. Koufogiorgos, On a type of contact manifolds, J. of Geometry 46 (1993) 1-9.