Chen's Inequalities In A (?,?)-contact Space Form

To find the simple relationship between the extrinsic and intrinsic invariants of submanifold is one of the basic problems in submanifold theory.B.-Y.Chen established inequalities in this respect,called Chen inequalities.The main intrinsic invariants include Chen's ?-invariant,scalar curvature,Ricci curvature,K-Ricci curvature and so on.The main extrinsic invariants are squared mean curvature and shape operator.Chen inequalities include Chen's first inequality,Chen's Ricci inequalities and so on.Such invariants and inequalities have many nice applications to several areas in Mathematics.A class of contact metric manifold with contact metric structure in which the curvature tensor satisfies the specific condition is called(?,?)-contact metric manifold.If the contact metric manifold has constant ?-sectional curvature c,then it's called(?,?)-contact space form.Nowadays the problems of(?,?)-contact space form have aroused people's wide attention and research,especially Chen's inequalities.So the research on the Chen's inequalities of(?,?)-contact space form has a great significance in theory and application.In this dissertation,we study Chen inequalities for submanifolds in(?,?)-contact space form with a semi symmetric metric connection and semi-symmetric non-metric connection.In Chapter 1,we give a brief introduction about semi-symmetric metric connection,semi-symmetric non metric connection,Chen lemma and Ricci curvature.In Chapter 2,we mainly introduce the necessary knowledge about curvature tensor and algebraic lemmas.In Chapter 3,we obtain Chen's first inequality and the inequalities about Ricci and K-Ricci curvature for submanifolds in(?,?)-contact space form with semi symmetric metric connection and semi-symmetric non-metric connection using algebraic lemmas.We also obtain some corollaries for two special cases of(?,?)-contact space form,i.e Sasakian and non-Sasakian space form.