Connections with (skew-symmetric) torsion on a non-symmetric Riemannian manifold satisfying the Einstein metricity condition (non-symmetric gravitation theory (NGT) with torsion) are considered. It is shown that an almost Hermitian manifold is NGT with torsion if and only if it is a nearly Kähler manifold. In the case of an almost contact metric manifold the NGT with torsion spaces are characterized and a possibly new class of almost contact metric manifolds is extracted. Similar considerations lead to a definition of a particular class of almost para-Hermitian and almost paracontact metric manifolds. Conditions are given in terms of the corresponding Nijenhuis tensors and the exterior derivative of the skew-symmetric part of the non-symmetric Riemannian metric.

We study G-almost geodesic mappings of the second type \(\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2\), θ = 1,2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e-structures that generate mappings of type \(\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2\), θ = 1,2. For a mapping \(\mathop {{\pi _2}}\limits_\theta (e,F),\theta = 1,2\), θ = 1,2, we determine the basic equations which generate them.

We study a rigidity problem for functions $F:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{\ge 0}$ that penalize deviation of a positive ratio from equilibrium $x=1$. Assuming (i) a d’Alembert-type composition law on ${\mathbb{R}}_{>0}$, and (ii) a single quadratic calibration at the identity (in logarithmic coordinates), we prove that F is uniquely determined. The composition law implies the normalization $F\left(1\right)=0.$ The unique solution is called the canonical reciprocal cost, namely the difference between the arithmetic and geometric means of x and its reciprocal. Our proof uses the logarithmic coordinates $H\left(t\right)=F\left(e^t\right)+1$, where the composition law becomes d’Alembert’s functional equation on $\mathbb{R}$. The calibration provides the minimal regularity needed to invoke the classical classification of continuous solutions and fixes the remaining scaling freedom, selecting the hyperbolic-cosine branch. We also establish the necessity of each assumption: without calibration the composition law admits a continuous one-parameter family; without the composition law the calibration does not determine the global form; and without regularity the composition law admits pathological non-measurable solutions. Finally, we establish a stability estimate for approximate solutions under bounded defect and characterize some properties of the canonical cost.

Considering a non-symmetric linear connection and its dual one appear 6 independent curvature tensors [15]. In the present paper we study properties of curvature tensors of semi-symmetric metric co...

We define and study the quarter-symmetric connection preserving the generalized metric G in the generalized Riemannian manifold. It is proved that skew-symmetric part F of the generalized metric G in the generalized Riemannian manifold with the quarter-symmetric generalized metric connection is closed and hence the even-dimensional manifold is a symplectic manifold. We also observed the properties of curvature tensors and connection transformations in which the Riemannian tensor of the Levi-Civita connection is invariant. Finally, we observed the quarter-symmetric connection with a special conditions.

We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condition is equivalent to the existence of a Sasaki–Einstein 5-manifold and vice versa, any Sasaki–Einstein 5-manifold generates a two parametric family of connections with skew torsion satisfying the Einstein metricity condition. Formulas for the curvature and the Ricci tensors of these connections are presented in terms of the Sasaki–Einstein SU(2) structures.