This article presents calculations related to the covariance postulate, the equivalence principle, coordinate systems, and the conservation laws within the general theory of relativity. The free motion of a particle is analysed in the framework of space-time geometry and affine symmetry principles. The laws of free motion were evaluated using differential equations governing the relationships between the metric tensor of Euclidean space, hypersurfaces, and affine connections. The transformation of the equation of flat affine connection into the Newtonian form of the law of motion in a gravitational field is presented, along with the transformation of the equation of motion under Galilean groups. The relationships between symmetries, equations, and conservation laws of a physical system are systematized according to Noether’s theorem. Symmetry is described using Lie groups and Lie algebras, while weak conservation laws are converted into strong ones. The conservation laws in the theory of gravitation are defined by the gravitational action, and the field equation is expressed through the symmetric energy-momentum density tensor. Fundamental connections between the general and special theories of relativity, both accounting for and neglecting gravitation, are examined in the context of spatial metrics and conservation laws. The affine flat connection, absolute time, Euclidean space metric tensor, and Newtonian potential are analysed comprehensively using Cartesian coordinates on a hypersurface. All calculations are performed using the mathematical apparatus of differential geometry, tensor calculus, variational methods, and integral calculus.