Geometry of foliations of Minkowski spaces

. As is well known, foliations of constant curvature and foliations generated by the orbits of Killing vector fields are important classes of foliations from a geometric point of view. The paper studies the geometry of some foliations of the Minkowski space, which arise in a natural way. It is shown that these foliations are foliations of constant curvature. The paper also studies the geometry of some singular foliations generated by the orbits of Killing vector fields.


Introduction
The Minkowski space can serve as a model of the space-time of the special theory of relativity. The Minkowski space is the basic space model of quantum physics that plays an important role in general relativity. In recent years, with the development of the theory of relativity, physicians and geometers extended the topics in classical ifferential geometry of Riemannian manifolds to that of Lorentzian manifolds. It is clearly demonstrated by the fact that many works in Euclidean space have found their counterparts in Minkowski space [1][2][3][4][5][6]. A. Ya. Narmanov and Zh. Aslonov obtained a complete classification of the singular Riemannian foliations of three-dimensional Euclidean space generated by the orbits of Killing vector fields [9]. The reachability set geometries of the Killing family of vector fields were studied by S.S.Saitova [10]. Minkowski space and its sub-space geometries are well studied. Its differential geometry [1][2][3][4][13][14][15] is given in these works 1 Basic concepts of pseudo-Euclidean spaces and foliation theory. Let 3 V denote the real vector space with its usual vector structure. Denote by 1 2 3 { , , } B e e e  the canonical basis of 3 R , that is 1 2 3 (1,0,0), (0,1,0), (0,0,1). e e e    We denote   , , x y z the coordinates of a vector with respect to B . We also consider in 3 R its affine structure, and we will say "horizontal" or "vertical" in its usual sense.
We say the scalar product 3 ( , , ) V  of the vectors { , , } Y x y z the number defined by the following rule: (1) and the exterior product by . We also use the terminology Minkowski space and Minkowski metric to refer the space and the metric, respectively. The Minkowski metric is a non-degenerate metric of index 1.
The norm of a vector || x is defined as the square root of the scalar square of the vector, and the distance between two points is defined as the norm of the vector connecting these points.
We present the definition of a foliation given in [8].
Let M be a smooth connected manifold of dimension n . Smoothness in this paper means the class Let us recall the definition of a foliation. R as unions of 2 -dimensional parallel planes along the 1 x  axis.. Definition 4. A partition F of a manifold M into leaves is called a smooth (from the class r C ) singular foliation (that is, a foliation with singularities) if the following conditions are satisfied: If the dimensions of the leaves of a foliation with singularities are the same, then it is a regular foliation in the sense of the definition given in [].
Let -X a vector field on M , M x  ,   x X t be an integral curve of the vector field X passing through the point is defined in some region   x I , which generally depends not only on the field X , but also on the starting point x .
In what follows, everywhere in formulas of the form   x X t we will assume that   Hence, there are values of the parameters ,, ts i.e. from any point 1 Let us show that from the point 1    XX generate a singular foliation whose singular leaf is a point, and whose regular leaves are a cone