Differential Geometry Of Submanifolds In Four Dimensional Semi-Euclidean Space

This paper gives the diferential geometry properties and singularities of null curves,Lorenzian surfaces and some hypersurfaces in4dimensional semi-Euclidean space. In1850, French mathematician Bertrand J gave the defnition of Bertrand curve, since which,many mathematicians have studied the necessary and sufcient conditions of Bertrandcurves in diferent spaces[1,9,10,19,29,30,35,48,59,66,67,74,94]. However, the work about nullBertrand curves in semi-Euclidean space is rarely. This paper gives the existence condi-tions and the diferential geometry properties of null Bertrand curves in semi-Euclideanspace with index one and two. Meanwhile, the singularities of null curves in3-nullconeare also studied. As we all know, the index is more, the types of submanifolds are more.There are many types submanifolds in semi-Euclidean space with index two. The space-like surfaces and timelike surfaces have the similar properties with surfaces in Euclideanspace, for their normal planes have one type normal vector. Since the tangent plane andnormal plane for Lorenzian surfaces have one index respectively, there is lightlike Gaussmap in normal plane, by which, some diferential geometry properties and singularitiescan be obtained. The codimension of Lorenzian surface is two, so we can construct anew hypersurface along the lightlike vector, which is called lightlike hypersurface. Andsome intrinsic properties and singularities can be obtained by the unique lightlike normalvector.This paper contains fve chapters.The second chapter introduces the defnitions of some submanifolds and AW(k)-typecurves in semi-Euclidean space.The third chapter introduces the Frenet frame and diferential geometry propertiesof null Cartan Bertrand curves. Meanwhile, the singularities of regular binormal nullsurfaces of null curves in3-nullcone is also studied by Bruce’s singularity method.The fourth chapter studied the1-parameter Gauss indicatrices of Lorenzian hypersur- faces in semi-Euclidean3-spheres and the singularities of the surfaces along1-parameterGauss indicatrices. The singularities of generally Lorenzian surfaces in semi-Euclidean4-space are also given.The ffth chapter introduced the singularities of lightlike hypersurfaces, which gen-erated by Lorenzian hypersurfaces and Lorenzian surface. Meanwhile, the singularities ofB1type osculate hypersurfaces of partially null slant helices in semi-Euclidean4-spacewith index two.