Champ de vecteurs projectifs sur le fibré unitaire. (Projective vector field on unitary fibre bundle)
We give a characterization of a projective vector field on Finslerian unitary fibre bundle and we prove that, in the compact case, the existence of projective vectors is related to the sign of the sectional curvature. We define the notion of projective restricted infinitesimal transformations. A new projectively invariant tensor is introduced, we determine necessary and sufficient conditions of nullities of projective invariants. We study the case when the mean directional curvature (M.D.C.) is satisfied under certain conditions, especially when the curvature is constant (that is a generalization of Einstein's space). We demonstrate that, any Finsler manifold, of M.D.C. positive constant, simply connected, metrically complete, and admitting a proper projective vector field, leaving invariant the covector trace of torsion, is homeomorphic to an n-sphere. We shall consider the case of a manifold with Euclidean connection.
projective restricted infinitesimal transformations
scientific article; zbMATH DE number 3931934