Alternatives to the Egorov's and Kobayashi's theorems on complete affine connections

The article concludes the research of completeness of the symmetry and geodesic flow of affine connections. The study aims to find and describe linear spaces over complete fields on the tangent bundle. The article verifies 3 new results. The developed theorems are alternatives to the Kobayashi's theorem on complete affine connections. The study examines a metric situation (theorem 7) and an affine situation (theorem 8). The article verifies an alternative to the Egorov's lacunarity in complete manifold dimensions. The paper verifies the evaluation of dimension borders of the first-order complete field which allows completeness of the connection itself (theorem 10). The article suggests a scheme of obtaining an alternative evaluation of higher order complete field dimension borders. The study sets up the ways of further research. The results could be used to classify the affine connections that accept extensive symmetry algebras of the geodesic flow.