Three-dimensional Nilpotent Manifold And Discrete Subgroups

In this paper, we first discuss discrete subgroups of connected and simply connected nilpotent Lie group of dimension three, which we denote is Nil, by using the generators. And we gain the results about discrete subgroups as follows.When there is only one generator α₀= (x₀,y₀,z₀) e Ml selected, the cyclic subgroup is always discrete whatever the components x₀,y₀,z₀ ofthe generator are except that the generator equals to the identity element. When there are two generators α₁= (x₁,y₁,z₁),α₂= (x₂,y₂,z₂)∈Nil, which are commutative, the abelian subgroup <α₁,α₂> is discrete if and only if the components of the two generators satisfying the following conditions:(1)x₁y₂=x₂y₁;(2)α₁(?)<α₂>,α₂(?)<α₁>;(2)x_i²+y_i²+z_i²≠0,x₁²+y₁²+x₁²+y₂²≠0When the two generators are not commutative, the subgroup < α₁,α₂> is discrete if and only if the components of the two generators satisfying the following conditions:(2)α₁(?)<α₂>,α₂(?)<α₁>.Then we get the mainly results about the classification of 3-nilpotent manifolds of this paper using those discrete subgroups. Corresponding to the numbers of the generators, there are two 3-nilpotent manifolds. Whenthere is one generator, all the corresponding orbit spaces are diffeomorphic to R²×S¹ .When there are two generators which satisfy the conditions of discreteness, all the corresponding orbit spaces are diffeomorphic to T²×R.