GEOMETRIC AND ALGEBRAIC PROPERTIES OF FIRST-ORDER DIFFERENTIAL EQUATIONS ON SMOOTH FINITE-DIMENSIONAL REAL MANIFOLDS

Abstract

We consider the geometric and algebraic properties of the first-order differential equation on smooth finite-dimensional real manifolds. An affine connection without torsion is compared with a differential flow (autonomic or non-autonomic) on a manifold, with all the original trajectories being some geodesic lines of this affine connection. Using differential-algebraic characteristics of affine connectivity, we study some classes of first-order equations on smooth finite-dimensional real differentiable manifolds.

Keywords

• systems of ordinary differential equations
• smooth manifolds
• affine connections
• universal algebras
• quasi-groups