Milan Zlatanović

We consider the reciprocal cost function $J\left(x\right)={\displaystyle \frac{1}{2}}(x+x^{-1})-1$ and its n-dimensional extension $J(x_1,\dots ,x_n)={\displaystyle \frac{1}{2}}(R+R^{-1})-1,R={\displaystyle \prod _{i=1}^n}{x}_i^{{\alpha }_i},\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{R}}^n\setminus \left\{0\right\}.$ In logarithmic coordinates $t_i=\log x_i$, the Hessian of J has a rank of one at every point. The associated Hessian geometry is degenerate and does not define a Riemannian metric. To obtain a nondegenerate geometric structure, we introduce a family of Hessian metrics $h_{\lambda }$. Combining the rank-one tensor with the Hessian metric $h_{\lambda }$, we construct a $(1,1)$-tensor field $A_{\lambda }$. Its trace normalization defines a projector $P_{\lambda }$, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector $P_{\lambda }$ and the induced structures, including eigendistributions, parallelism, integrability, and curvature. The construction is given in an arbitrary dimension, and explicit formulas are obtained in the two-dimensional case. In particular, we show that the projector $P_{\lambda }$ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection ${\nabla }^{\lambda }$ of the Hessian metric $h_{\lambda }$.

In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural n-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination $S=\alpha ·t$, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an $(n-1)$-dimensional null distribution. On the other hand, when the same function is expressed in the original x-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in x-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura–Saito and Bregman divergences, and give a Fisher–Rao realization of the logarithmic Hessian metric.

We study functions satisfying the composition law $F\left(xy\right)+F(x/y)=P\left(F\right(x),F(y\left)\right)$ with a symmetric polynomial combiner P. We prove that symmetry together with a quadratic degree bound on P forces a composition law of d’Alembert type. We establish a degree mismatch exclusion criterion showing that symmetric polynomial combiners with $degP(u,v)\ge 3$ do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 1). For continuous nonconstant functions $F:{\mathbb{R}}_{>0}\to \mathbb{R}$ with $F\left(1\right)=0$ satisfying the composition law with a symmetric polynomial P of degree at most two, the combiner is necessarily of the form $P(u,v)=2u+2v+cuv$, $c\in \mathbb{R}$ (Theorem 3). The equation reduces in logarithmic coordinates to the classical d’Alembert functional equation. For $c\ne 0$, one obtains hyperbolic or trigonometric branches, while $c=0$ yields the squared-logarithm family. Under the cost-function assumptions $F\ge 0$ and convexity, only the hyperbolic branch with $c>0$ remains. A unit log-curvature calibration selects the canonical value $c=2$, which yields the canonical reciprocal cost $F\left(x\right)=\frac{1}{2}(x+x^{-1})-1$. For $c\ne 0$, the result extends to ${\mathbb{R}}_{>0}^n$: every solution depends only on a single linear combination of coordinate logarithms; for $c=0$, the solution is a general quadratic form ${\sum }_{i,j}{a}_{ij}ln{x}_{i}ln{x}_j$. In either case, nontrivial coordinate-wise separable costs are excluded.

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.

A.~Einstein considered a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric and $F\ne0$ is skew-symmetric, and a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. M. Prvanovi\'c (1995) obtained the explicit form of the Einstein connection of an almost Hermitian manifold. In this paper, first, we present the result above in coordinate-free form, and then extend it to almost contact metric mani\-folds satisfying the so-called $f^2$-torsion condition.We then derive the Einstein connection of nonsymmetric pseudo-Riemannian, in particular, weak almost Hermitian manifolds $(M,f,g)$, satisfying the $f^2$-torsion condition, where $F(X,Y)=g(X,fY)$,give explicit formulas for the torsion in terms of $\nabla^g F$, $dF$ and a new (1,1)-tensor $\widetilde Q:=-f^2-{\rm Id}$, and show that in the almost Hermitian case, our main result reduces to the coordinate-free form of Prvanovi\'c's solution.Finally, we describe special Einstein connections, i.e., the difference tensor has the property~$K_XY=-K_YX$, and indicate the Gray-Hervella classes. Illustrative examples are given, including the construction of a weighted product.

We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map $\overline{R}:{\mathcal{C}}_R\to \mathcal{E}$ is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C^*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on ${\mathbb{R}}^n$, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency.

Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where $F$ can be defined as an almost complex or almost contact metric (a.c.m.) structure. In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the second author and R.~Wolak), which generalize almost complex and a.c.m. structures. We consider linear connections with totally skew-symmetric torsion that satisfy both the Einstein metricity condition and the $A$-torsion condition, where $A$ is a skew-symmetric (1,1)-tensor adjoint to~$F$. In the almost Hermitian case, we prove that the manifold with such a connection is weak nearly K\" ahler, the torsion is completely determined by the exterior derivative of the fundamental 2-form and the Nijenhuis tensor, and the structure tensors are parallel, while in the weak a.c.m. case, the contact distribution is involutive, the Reeb vector field is Levi-Civita parallel, and the structure tensors are also parallel with respect to both connections. For rank$(F)=\dim M$, we apply weak almost Hermitian structures to fundamental results (by the first author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold equipped with an Einstein's connection is a weighted product of several nearly Kähler manifolds. For~rank$(F)<\dim M$ we apply weak almost Hermitian and weak a.c.m. structures and obtain splitting results for generalized Riemannian manifolds equipped with Einstein's connections.

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the K\"ahler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.