Pseudo-Spherical Darboux Surfaces Of Three Kinds Of Curves On A Timelike Hypersurface In Minkowski 4-Space

In this paper,we mainly introduce some properties of pseudo-spherical Dar-boux surfaces of curves on timelike hypersurface in four dimensional Minkowski space.The research has shown that the curves on timelike hypersurfaces in four dimensional Minkowski space may be three types:timelike,spacelike and lightlike.The cases of timelike and spacelike all exist two situations.{t(s)?n?(s),?"(s)? may be linearly independent or linearly dependent.We establish some moving frames for these cases respectively.For these situations,the spacelike height function and the timelike height function are established for these three curves.By establishing the height function,we obtain three geometric invariants ?1(s),?2(s)and ?3(s),then define four pseudo-spherical Darboux surfaces in the process.In addition,we also define the de Sitter slice and hyperbolic slice on timelike hypersurface M in special cases.It is shown that the equivalence conditions for the existence of singularities on these Darboux surfaces are closely related to those geometric invariants,and are closely related to the contact between the curve ? and the tangent plane,and the contact between the curve ? and the two slices.