In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, $\sigma$), $\sigma =1,2,3$ tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule.

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion “degree of neighborhood” is used for the geometric description of polar subspaces of quadrics. The polarity induced by a quadric is also studied.

We investigate conformal Fedosov structures on (pseudo-) Riemannian manifolds of Roter type. Using the integrability condition for the conformal Fedosov equation $$\nabla \omega = \theta \otimes \omega $$ ∇ ω = θ ⊗ ω , we show that the Lee form is closed and can be gauged away, reducing the problem to the classical Fedosov case with a parallel symplectic form. By exploiting the Roter-type decomposition of the curvature tensor, we establish that any such manifold must have constant sectional curvature. In particular, conformal Fedosov structures do not exist on non-trivial Roter type manifolds. This result extends earlier non-existence theorems for symmetric, recurrent, and quasi-constant curvature manifolds to the full Roter type class, revealing the strong rigidity of conformally Fedosov geometry.