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On a Type of Kenmotsu Manifolds BEHNAM POOLADRAG b pooladrag@math iut ac ir Nov 16 2015 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mansour Aghasi m aghasi@cc iut ac irAdvisor Dr Azam Etemad ae110mat@cc iut ac ir2000 MSC 53C15 53C40Keywords Kenmotsu manifold W2 curvature tensor Einstein manifold projective curvature tensor concircular curvature tensor quasi conformal curvature tensor conformal curvature tensor Abstract This M Sc thesis is based on the following paperAhmet Yidiz Uday Chand De On a Type of Kenmotsu Manifolds Differential Geometry DynamicalSystems Vol 12 2010 pp 289 298 In 1970 S Tanno classified connected almost contact metric manifolds whose automorphism groups possess themaximum dimension For such a manifold M the sectional curvature of plane sections containing is a constant say c If c 0 M is homogeneous Sasakian manifold of constant sectional curvature If c 0 M is the productof a line or a circle with a Kaehler manifold of constant holomorphic sectional curvature If c 0 M is a warpedproduct space R Cn In 1971 Kenmotsu studied a class of contact Riemannian manifolds satisfying some specialconditions We call it Kenmotsu manifold Recently Kenmotsu manifolds have been studied by many authors such asJun De Pathak and Ozgur and many others On the other hand Pokhariyal and Mishra have introduced new tensorfields called W2 and E tensor fields in a Riemannian manifold and studied their properties Then Pokhariyalhas studied some properties of this tensor fields in a Sasakian manifold Recently Matsumoto Ianus and Mihaihave studied P Sasakian manifolds admitting W2 and E tensor fields and De and Sarkar have studied P Sasakianmanifolds admitting W2 tensor field The curvature tensor W2 is defined by W2 X Y U V R X Y U V n 1 g X U S Y V g Y U S X V where S is a Ricci tensor of type 0 2 1Let M n g be an n dimensional where n 2m 1 almost contact metric manifold where is a 1 1 tensor field is the structure vector field is a 1 form and g is the Riemannian metric It is well known thatthe g structure satisfies the conditions 1 0 X 0 1 2 2 X X X g X X 3 g X Y g X Y X Y for any vector fields X and Y on M n Ifmoreover 4 X Y g X Y Y X 5 X X X where denotes the Riemannianconnection of g hold then M n g is called a Kenmotsu manifold Kenmotsu proved that if in a Kenmotsu manifold the condition R X Y R 0 holds then the manifold is ofnegative curvature 1 where R is the curvature tensor of type 1 3 and R X Y denotes the derivation of thetensor algebra at each point of the tangent space A Riemannian manifold satisfying the condition R X Y R 0is called semi symmetric In analogous manner a Riemannian manifold is called Ricci semi symmetric respectively Weyl semi symmetric ifR X Y S 0 respectively R X Y C 0 where S is the Ricci tensor respectively C is the Weyl conformalcurvature tensor of type 1 3 Though R X Y R 0 implies R X Y S 0 but the converse is not true ingeneral So it is meaningful to undertake the study of Kenmotsu manifold satisfying the condition R X Y S 0 In the present thesis we have studied some curvature conditions on Kenmotsu manifolds Firstly we have studiedtheir geometric and relativistic properties in Kenmotsu manifolds satisfying W2 0 Then we have studied W2 semisymmetric Kenmotsu manifolds Also we have classified Kenmotsu manifolds which satisfy P W2 0

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تانسور انحناي W2 , خمينه ي انيشتين , تانسور انحناي تصويري , تانسور انحناي حلقوي - مخروط , تانسور انحناي شبه همديس