Some connections and variational principle to the Finslerian scalar-tensor theory of gravitation

A generalized scalar-tensor theory of the Finslerian gravitational field is constructed on the basis of the fibered bundle of a base manifold and consists of two G-numbers (Grassmannian noncommutative) y+, y−, playing the role of fibres. In the framework of our approach, we study: the connection structures, torsions, curvatures and the gravitational field equations derived from variational principles of the appropriate Lagrangian densities.