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In this paper we study the preserving of Riemannian and Ricci tensors with respect to a diffeomorphism of spaces with affine connection. We consider geodesic and almost geodesic mappings of the first type. The basic equations of these maps form a closed system of Cauchy type in covariant derivatives. We determine the quantity of essential (substantial) parameters on which the general solution of this problem depends.

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This article studies the global conformal transformations f on a Finsler space (M, F ), which satisfy f * F = e c(x) F , where F := F (x, y) is a Finsler metric on M and x ∈ M , y ∈ T x M \ {0}. We obtain the relations between some important geometric quantities of F and their correspondences respectively, including Riemann curvatures, Ricci curvatures, Landsberg curvatures, mean Landsberg curvatures and S-curvatures. Then, we discuss the properties of those conformal transformations on (M, F ) which preserve Ricci curvature, Landsberg curvature, mean Landsberg curvature and S-curvature respectively.

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