نامساوي هاي بهينه براي خميدگي هاي كازوراتي زير منيفلدها در فضا فرم هاي تعميم يافته ي مجهز به التصاق هاي متري نيمه متقارن

چكيده انگليسي :

Optimal inequalities for the Casorati curvatures of submanifolds of generalized space forms endowed with semi symmetric metric connections Sharzad Nickaein s nickaein@math iut ac ir December 30 2017 M Sc Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Assistant Professor Azam Etemad ae110mat@cc iut ac irAdvisor Associate Professor Mansour Aghasi m aghasi@cc iut ac ir2000 MSC 53C40 49K35 Keywords Casorati curvature real space form semi symmetric metric connection Abstract In this thesis that is based on 50 16 28 39 we use some concepts such as space forms semi symmetric metricconnection and the Casorati curvature to prove some optimal inequalities involving the intrinsic scalar curvatureand extrinsic Casorati curvature of submanifolds of generalized space forms endowed with a semi symmetric metricconnection Felice Casorati was an Italian mathematician He is best known for the Casorati Weierstrass theoremin complex analysis In 1889 he introduced a new concept for cuvature which defined as the average of squared ofthe principal curvatues This curvature was named Casorati curvature The Casorati curvature of surfaces is used atthis stage rather than for instance the Gauss curvature or the mean curvature which rightfully deserved most attentionin the traditional studies on surfaces in 3 dimensional Euclidean space till now beacauce the Casorati curvature forsurfaces in 3 dimensional Euclidean is zero if and only if the principal curvatues are zero at the same time In additioninstead of concentrating on the sectional curvature with the extrinsic squared mean curvature the Casorati curvatureof submanifold in a Riemannian manifold is also considered as a an extrinsic invariant defined as the normalizedsquare of the length of the second fundamental form of the submanifold The notation of Casorati curvature extendsthe concept of the principle direction of hypersurface of a Riemannian manifolds Recently several geometers foundmore geometrical meaning that shoe the importance of the Casorati curvature Therefore it is of great interest to obtainoptimal inequalities for the Casorati curvature of submanifolds in different ambient spaces As a natural prolongationof our research in this thesis we will study these inequalities for submanifolds in generalized space forms endowedwith semi symmetric metric connection The principal aim of this thesis is prove two main theorems about optimalinequalities between scalar curvature and Casorati curvature In detail one of them is about generalized copmlex spaceforms and the other is about generalized Sasakian space forms Let Mn be a submanifold of a generalized compelx inthe chapter 4 and Sasakian in the chapter 5 space forms with semi symmetric metric connection In chapter 4 and5 we state and prove two optimal inequalities between the normalized scalar curvature and the normalized Casorati curvature C n 1 and C n 1 that involves shape operators The equality case in two theorems implies thetM is an invariantly quasi umbilical submanifold that means the shape operators have only two eigenvalues withmultiplicity n 1 and 1 Furthermore there is a relation between two eigenvalues in theorems for the proof of thesetheorems we need to theorem of Cartan We name this theorem Cartan theorem In chapter 3 we aimed to provethe Cartan theorem By use of several definitions propositions and theorems this theorem have a very brief proof however mentioned tools themselves have a long processes At the end we summarized the main results of thesisin special cases in a table that contains inequalities in main theorems when ambient spaces have constant sectionalcurvature

نيك آئين، شهرزاد

نامساوي هاي بهينه براي خميدگي هاي كازوراتي زير منيفلدها در فضا فرم هاي تعميم يافته ي مجهز به التصاق هاي متري نيمه متقارن

خميدگي كازوراتي , فضا فرم حقيقي , التصاق متري نيمه متقارن