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Homogeneous Geometric Structures and Homogeneous Differential Equations.

机译：齐次几何结构和齐次微分方程。

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The authors invent the notion of a homogeneous geometric structure in the paper and show that these structures arise naturally both in geometric theory of differential equations and in differential geometry. The homogeneous geometric structures (under some additional conditions) may be endowed with a connection of special type which the authors refer to as a Cartan connection, because the construction of such connection is based on a Cartan distribution on a jet manifold. The homogeneity means in this case the triviality of the Weyl tensor of the Cartan connection and the later becomes a Bott connection. The paper contains 3 main sections. In the first section the authors recall in a suitable form the main properties of a metasymplectic structure on jet spaces, define Cartan connections and show that the restriction of the metasymplectic structure on the horizontal subspaces of the Cartan connection provides a curvature tensor. In the second section a definition of a homogeneous geometric structure and its generalizations is given. The authors also introduce a notion of a non-singular map with respect to a given geometrical structure. These non-singular mappings consist of a category of geometrical structures. In the third section Lie algebras of symmetries for differential equations are studied. The main results are Theorems 3.4 describing conditions for a symmetries algebra to be an elliptic or finite type.

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