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Abstract
The concept of differential invariant, along with the concept of invariant differentiation, is the key in modern geometry [1]-[10]. In the Erlangen program [3] Felix Klein proposed a unified approach to the description of various geometries. According to this program, one of the main problems of geometry is to construct invariants of geometric objects with respect to the action of the group defining this geometry. This approach is largely based on the ideas of Sophus Lee, who introduced continuous geometry groups of transformations, now known as Lie groups, into geometry. In particular, when considering classification problems and equivalence problems in differential geometry, differential invariants with respect to the action of Lie groups should be considered. In this case, the equivalence problem of geometric objects is reduced to finding a complete system of scalar differential invariants. The interpretation of the k- order differential invariant as a function on the space of k- jets of sections of the corresponding bundle made it possible to operate with them efficiently, and using invariant differentiation, new differential invariants can be obtained. Differential invariants with respect to a certain Lie group generate differential equations for which this group is a symmetry group. This allows one to apply the well-known integration methods to such equations, and, in particular, the Li- Bianchi theorem [4]. Depending on the type of geometry, the orders of the first nontrivial differential invariants can be different. For example, in the space R3 equipped with the Euclidean metric, the complete system of differential invariants of a curve is its curvature and torsion, which are second and third order invariants, respectively. Note that scalar differential invariants are the only type of invariants whose components do not change when changing coordinates. For this reason, scalar differential invariants are effectively used in solving equivalence problems. In this paper differential invariants of Lie group of one parametric transformations of the space of two independent and three dependent variables are studied. It is shown method of construction of invariant differential operator. Obtained results applied for finding differential invariants of surfaces.
Highlights
Let G be a Lie group of transformations of Riemannian manifold M of dimension n with Riemannian metric g.Definition 1
The equivalence problem of geometric objects is reduced to finding a complete system of scalar differential invariants
It is known that any Lie group is similar to the group of translations
Summary
Let G be a Lie group of transformations of Riemannian manifold M of dimension n with Riemannian metric g. 2) f has a contact of k th order with g at the point p if the map (df ) : T M → T B has a contact of order (k−1) with map (dg) at each point TpM This fact can be written as follows: f ∼k g at the point p (k -positive number) [2]. The function I ∈ C∞(Jk(M, B)) is called a differential invariant of order k of the group G if it is preserved under the action of the k -th prolongation G on Jk(M, B), that is, g(I) = I for any transformation g ∈ G(k). The function I is an invariant of order k of the transformation group G if and only if it is the first integral of the infinitesimal generator of the group G(k)
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