On Hypersurfaces Of The Homogeneous Nearly K(?)hler S~3×S~3

Nearly K(?)hler manifolds are an important class of geometric objects,which are widely studied by many geometers.6 dimensional nearly K(?)hler manifolds are an important part of the nearly Kahler manifolds,it is very natural and meaningful to study their hypersurfaces(submanifolds of real codimension 1).In this dissertation,we study the hypersurfaces of the 6 dimensional homogeneous nearly K(?)hler S3×S3,and obtain the following main results:(1)We study the classification problem of Hopf hypersurfaces of the homogeneous nearly Kahler S3×S3.By using the established fundamental lemma of Hopf hypersurfaces(see Lemma 2.2.2),and combining with the structure equations of hypersurfaces,we give nonexistence theorem or classification theorem on Hopf hypersurfaces which satisfy different special conditions,respectively(see Theorem 1.1.7 to Theorem 1.1.10).Finally,we classify the Hopf hypersurfaces with constant principal curvatures of the homogeneous nearly K(?)hler S3×S3(see Theorem 1.1.11).(2)We study the existence problems of locally conformally flat hypersurfaces and Einstein hypersurfaces of the homogeneous neally K(?)hler S3 × S3.Firstly,by using the approach which called Tsinghua principle,we establish the relationships between the Gauss-Codazzi equations and the Ricci identity,and we prove the nonexistence of locally conformally flat hypersurfaces(see Theorem 1.1.13).Secondly,under the Hopf condition,we use the relevant conclusions,and prove the nonexistence of Einstein Hopf hypersurfaces(see Theorem 1.1.14).(3)We study two kinds of commutativity relationships between the shape operator A and the almost contact structure tensor φ of hypersurfaces of the homogeneous nearly K(?)hler S3×S3.Firstly,we derive a rigidity theorem for the compact minimal hypersurfaces of the homogeneous nearly K(?)hler S3×S3,we establish a Simons’ type integral inequality for such hypersurfaces,and show that the equality sign occurs if and only if the hypersurfaces satisfy the condition Aφ=φA(see Theorem 1.1.16).This fact gives a new characterization of the hypersurfaces which satisfy the commutativity condition.Secondly,we prove that the homogeneous nearly K(?)ler S3×S3 admits no hypersurface satisfying the anti-commutativity condition Aφ+φA=0(see Theorem 1.1.17).