On Classifications Of Almost Paracontact Metric Manifolds

As an odd dimensional analog of the well known paracomplex geometry,the geometry of the paracontact metric manifolds is of great significance both in theory mathematics and math-ematical physics.Such manifolds have been extensively studied by many geometers since they were introduced in 1970s.Based on some earlier results,applying the well known tensorial analysis methods in differential geometry,we aim to study the classification problem of almost paracontact metric manifolds.Over the past 15 years,the study of nullity conditions on different types of almost contact metric manifolds has been widely developed.Many authors started the study of some types of nullity conditions on paracontact geometry due to an unexpected find-ing that there are canonical paracontact metric structures on a contact metric manifold.In this thesis,we mainly study the Eisenhart problem on a paracontact metric(?)-manifold and an al-most ?-para-Kenmotsu(?)-manifold.We also obtain some local descriptions of 3-dimensional paracontact metric(?)-manifolds and almost ?-para-Kenmotsu(?)-manifolds.1.We mainly investigate the Eisenhart problem on paracontact metric(?)-manifold and almost ?-para-Kenmotsu(?)-manifold,we show that if there exists a second order symmetric parallel tensor p on the paracontact metric(?)-manifold(M2n+1,g),then either locally M is a product of a flat n+1 dimensional manifold and an n dimensional manifold of constant sectional curvature-4,or ? = cg,c is a constant.What is more,if there is a second order symmetric parallel tensor ? on the almost ?-para-Kenmotsu((?)?0,(?))-manifold,then ? = cg,c is a constant.2.We consider paracontact metric(?,?,?=const.)-manifold with (?).some properities and lemmas of 3 dimensional paracontact metric manifold and paracontact met-ric(?)-manifold are presented.We also give the local description of paracontact metric((?)=const.)-manifold with (?) under the condition that (?)>-1,and in case that (?)<-1,we get the nonexistence of such manifolds.In the last part of this chapter,we give the local description of the 3 dimensional generalized paracontact metric(?)-manifold with second order parallel ? and (?)>-1.3.We mainly study almost a-para-Kenmotsu(?)-manifold,we get the local description of almost ?-para-Kenmotsu(?)-manifold with dk^?= 0 in the cases that h is of (?)1 type and of (?)3 type respectively.We also give a method constructing almost a-para-Kenmotsu(?)-manifolds for h is of (?)1-type or h is of (?)3-type and d(?)^? = 0.In the last,we get the equivalent conditions for an almost paracontact metric structure to be an almost ?-para-Kenmotsu(?)-manifold.