In this paper, a generalization and a covariant form of the Beltrami-Mitchell equations are derived for the case of incompatible deformations. The representation of the internal material metric and the associated coefficients of the non-Euclidean connectivity are given. A relation between them and the Euclidean connectivity coefficients defined by the deformation properties of the material is obtained. An incompatibility condition in covariant form is presented. The paper introduces an additional intrinsic non-Euclidean material characterization in terms of the Ricci tensor. The resulting Beltrami-Mitchell equations are given in Cartesian rectangular and cylindrical coordinate systems. When considering the scheme of the non-Euclidean model, the Ricci tensor is emphasized as a geometric parameter responsible for inelastic deformation. The analytical form of an additional internal non-Euclidean material characteristic for the problem of the plane deformed state in an axisymmetric cylinder is given. The analytical solution of the Beltrami-Mitchell equations for the problem of the plane deformed state in an axisymmetric cylinder is obtained.