The Local Theory Of Timelike Submanifolds Of Codimension 2 In R₂～5

In this paper,our aim is to devoelop the local properties of indicators 2 timelike 3-submanifold in semi-Euclidean 5-space,and the normal of the submanifold only contains spacelike.The singularity of the line and surface in Euclidean n-space(n < 5),and the indicator is 1 in semi-Euclidean have been done([11][4][5][8][9][10]). [6] has developed the Lorentzian 3-submanifolds in semi-Euclidean 5-space.Above all, they used a very important tool—lightcone Gauss maps and lightcone height funtion. But in this paper ,we increased the genus of known Gauss maps.the detail method and main results are as follows:In this paper,we construct M's canal hypersurface CM,the timelike Gauss mapss and the timelike height function.At the last,prove the CM and M have the same singularities and the properties.have:Theorem Let M is timelike 3—submanifolds in R₂⁵. CM is M's canal hypersurface. The H, H is height function respectively of M and CM, the hessian matrix of H, H have:by the theorem,letM is timelike 3-submanifold in semi-Euclidean 5-space,CM is M's canal hypersurface, there is a mapsH : CM × S₂⁴ → R,so the following properties are equivalence(1) a fixed λ∈S₂⁴, P₀ ∈ CM is Gauss maps hλ's degenerate critical point.(2) P₀ is singularities of Gauss maps L,so that λ = L(p₀) ∈ S₂⁴.(3) when θ = arctan -c/c|p0 = arctan -e/e|P0,the K_t(cosθ,sinθ)(p0) = 0. we can get M and CM has the same singularities.