Classification of curves on de Sitter plane

In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described. Анотація. В 1917 році де Сіттер використав модифіковане рівняння Ейнштейна і запропонував модель Всесвіту без фізичної матерії але з космологічною постійною. Геометрія Де Сіттера, як і геометрія Мінковського, є максимально симетричною. Однак геометрія де Сіттера краще підходить для опису гравітаційних полів. Зокрема, вважається, що розширення Всесвіту на самих ранніх стадіях може бути описане саме моделлю де Сіттера (інфляційна модель Всесвіту). Дана стаття присвячена проблемі класифікації регулярних кривих в просторі де Сіттера, а в якості моделі площини де Сіттера обрана верхня півплощина, на якій задана метрика. Для розв’язання поставленої задачі побудовано алгебру диференціальних інваріантів кривих відносно рухів UDC 514.75+517.95


INTRODUCTION
Like Minkowski geometry, de Sitter geometry is one of the formalizations of the theory of relativity. Model of de Sitter universe was proposed in [3,2]. Unlike Minkowski geometry, it is better suited for descriptions of a nontrivial gravitational field, in particular, the expansion effect of the universe [7].
At the same time, there is much in common between Minkowski and de Sitter geometries: both of them are maximally symmetrical. The group of motions of Minkowski space, that is, the group of transformations preserving the metric, is the 10-parameters Poincaré group, consisting of four translations (three spatial and one temporal) three purely spatial rotations and three space-time rotations. The last six transformations form a subgroup of the Poincaré group (the group of Lorentz transformations).
De Sitter's space also has a 10-parametric dimensional group of motions. A review of de Sitter geometry is given in [6].
This article is devoted to the problem of classification of curves in 2dimensional de Sitter space. We consider the upper half-plane as a model of this space. This half-plane will be called de Sitter plane.
The proper motions generate a 3-dimensional Lie group, which we denote by G S and call de Sitter group. This group is generated by translations by x, the transformation , y Þ ÝÑ 4y t 2 x 2´4 tx´t 2 y 2 + 4 and homotheties.
Classification of curves on de Sitter plane 3 The corresponding Lie algebra G S is generated by the vector fields Consider the problem of classifying curves on de Sitter plane that defined as function graphs y = f (x).

THE DIMENSION OF THE ALGEBRA OF DIFFERENTIAL INVARIANTS
Let J k (π) be the space of k-jets of curves bundle, and let x, y 0 , y 1 , . . . , y k be the local canonical coordinates in the space J k (π).
The generating functions [4] of the vector fields X, Y, H are

respectively.
A function J on the space of k-jets is called a differential invariant of order ď k of the Lie group G S , if φ˚(J) = J for each transformation φ P G S , [1].
Define the projection from J k (π) to J 0 (π) by: Let a be some point on the manifold M . Since the Lie group G S acts transitively on this manifold, N k a = π´1 k (a) is a smooth manifold with coordinates y 1 , . . . , y k . Let G a = tg P G S | ga = au Ă G S be the isotropy group of the point a. Then transformations from the prolonged Lie group G (k) a preserve the manifold N k a . Let G a Ă G S be the corresponding to G a Lie subalgebra. We call this subalgebra the stabilizer of the point a. Note that the point a is a singular point for each vector field from G a .
Let us describe the stabilizer G a of the point a(a x , a y ) P M . For this purpose take a vector field Z a P G a with singularity at the point a, so Since the point a is singular, we have where β is arbitrary parameter. Put β = 1. Then we get The generating function of the vector field Z a is So, dim G a = 1 and G a = RZ a . Let S Za be the evolutionary part of the vector fields Z. Recall, [4], that Notice that the vector fields S  . Since a y ą 0, the rank is zero if 1´y 2 1 = y 2 = 0. Otherwise it equals 1. Therefore, the set Σ S = t1´y 2 1 = 0, y 2 = 0u consists of singular points of the G (k) a -orbit. The general curve has no singular points, therefore the codimension of its orbit is equal to 1.

CURVATURES OF CURVES ON DE SITTER PLANE
According to Theorem 2.1, the first differential invariants arise in the second order. Let us find them. The prolongations of the vector fields X, Y, H into J 2 (R) are Solving the system of differential equations we will find a first differential invariant The function is called a de Sitter curvature of the curve y = f (x).

EQUIVALENCE OF CURVES
A curve s on de Sitter is called regular if |dJ 2 | s ‰ 0. This means that the restriction J 2 | s of J 2 to s can be taken as a new parameter on the curve. Therefore the restriction of the differential invariant J 3 to the curve s can be represented as a function of τ : Proof. Let us prove necessity. Suppose that the curves s ands are locally equivalent. This means that there exists a transformation Φ P G S that transforms the curve s intos. Therefore the prolongations s (2) ands (2) of these curves to the 2-jet space J 2 (R) lie on the same connected component. Moreover, since J 2 and J 3 are differential invariants, we have F "F . Prove that condition F "F is sufficient for local equivalence of the curves s ands. Suppose that the curves s ands are graphs of the functions y = f (x) and y =f (x) respectively. The ordinary third order differential equation defines the hypersurface E in the space J 3 (R). The functions y = f (x) and y =f (x) are solutions of this equation and the curves s (3) ands (3) lie on this hypersurface. Due to condition (2) of the theorem, the restrictions of the function 1´y 2 1 to these curves are everywhere not zero. Therefore equation (5.1) can be solved with respect to the highest derivative: The generating functions [4] The restrictions S X , S Y , S H to the hypersurface E are shuffling symmetries of equation (5.2). These restrictions are linearly independent if (y 2 1´1 )F ‰ 0. Due to condition (1) of the theorem we have F (τ ) ‰ 0 and due to condition (2) we also have that y 2 1´1 ‰ 0. Therefore the last inequality is true and the vector fields S X , S Y , S H are linearly independent at each point of a connected component. So, the prolonged Lie group G S acts transitively on each connected component.
Choose on each curve the points: a 0 P s (3) and b 0 Ps (3) . Since the shift by variable x belongs to G S , we can assume that x(a 0 ) = x(b 0 ) = a. Thus, using a suitable transformation of this Lie group, we can achieve that a 0 = b 0 . So, without loss of generality, we can assume that the functions y = f (x) and y =f (x) being solutions of differential equation (5.2), have the same initial data. By the uniqueness theorem for solutions of differential equations, the functions f andf coincide at least locally.
Thus, using a suitable motion, the curve s can be locally translated into the curves. □ Remark 5.2. The described method for finding differential invariants can be applied to problems of classifying curves in other geometries. In addition, it is applicable to the classification problems of certain classes of differential equations, see [5].