Finsleroid-finsler spaces of positive-definite and relativistic types

We formulate the notion of the Finsleroid-Finsler space, including the positive-definite as well as indefinite cases. The associated concepts of angle, scalar product, and the distance function are proposed. If the Finsleroid-Finsler space is of Landsberg type, then the Finsleroid charge is a constant. The Finsleroid-Finsler space proves to be a Berwald space if and only if the Finsleroid-axis 1-form is parallel with respect to the associated Riemannian metric and, simultaneously, the Finsleroid charge is a constant. The necessary and sufficient conditions for the Finsleroid-Finsler space to be of the Landsberg type are found, they are explicit and simple. The structure of the basic associated curvature tensors is elucidated. The pseudo-Finsleroid gravitational field equations are formulated.