Geometry of normal differential eguations

In the present Paper, the term „differential equations“ means systems of differential equations with partial derivatives. A uniform concept of geometry of differential equations does not exist in modern differential geometry. Representatives of different trends in geometry perceive geometry of differential equation in different ways. The theory of compatibility of differential equations and existence of solutions developed by Spenser-Goldsmidt is one of the most „geometrical“ among them. Later it was systematized and extended at the School of Moscow Mathematicians managed by A.Vinogradov (see [1]) and works of mathematicians from other countries [2]. Geometry of differential equations is also found in subjects of Lithuanian geometricians, first of all in investigations carried out by prof. V.Bliznikas and his disciples. The principal object of the said investigations was construction of internal bindings in differential equations. The object of my Paper is an investigation on geometry of one class of systems of normal linear differential equations with partial derivatives. The said class of equations is presented by differential equations of the second order, provided in tangential stratifications of smooth manifolds. Pursuing the said object, tensor fields are attributed to the differential equations under investigation in the internal manner and internal geometrical bindings of three kinds are developed. The developed bindings enable us to use their covariant differentiation operators, curvature morphisms and curvature tensors, to search for covariantly constant sections of the stratifications under investigation, to find internal characteristics of the given differential equations expressed by tensor equations. In the Paper, the theory of compatibility of differential equations and existence of solutions developed by Spenser (that may be considered a classic one already) is applied constantly. In the mentioned theory, symbols of differential equations and -cohomologies of the said symbols are developed, the involutiveness of differential equation is established, the criteria of formal integratability are found, existence of solutions is established, Lie differential equations are explored. These actions are carried out applying Spenser’s differential operators as well as Spenser’s sets developed with the said operators. For convenience of a reader, the principal concepts of Spenser’s theory as well as some constructions and results are provided herein. Afinal, Г1,2 and tensor bindings of smooth manifolds are discussed upon as well and new structures of theories of these bindings are provided. In the principal part of the Paper, the theorems on geometrical features of the differential equations under discussion are proved. The content sapidity of the geometrical theory of these equations is confirmed by the obtained results.Švietimo akademijaVytauto Didžiojo universiteta