arXiv subjects

Milan Zlatanović

Publications and source records attributed to Milan Zlatanović.

15 recordsLinked to original sources

arXiv2026

Einstein connection of a weak almost contact metric manifold

Advances in modern physics since Einstein have made the nonsymmetric metric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne0$ is a skew-symmetric tensor associated with electromagnetism, more attractive than ever. Einstein considered a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=G(T(Y,X),Z)$. In this paper, we explicitly present the Einstein connection of $G=g+F$ using a weak almost contact structure $(f,ξ,η)$ with $g(X,fY)=F(X,Y)$ with a natural condition (trivial in the almost contact case). We discuss special Einstein connections, and give an example in terms of the weighted product of almost Hermitian manifold and a real line.

math.DG↗

arXiv2026

From one generator to loop order on the three-cube

We study the monoid $\mathcal W$ of based closed walks on the $1$-skeleton of the three-dimensional cube $Q_3$. The fundamental group of this graph is the free group $F_5$. We prove that every additive one-step reading, given by a sum of edge weights in an abelian group, factors through the directed transition counts, and we exhibit two closed walks with equal transition counts and different reduced loop words. Hence commutative aggregation does not determine the reduced loop word. The reduced loop word gives a proper noncommutative recognition quotient. We also determine the shortest closed walk with trivial abelianization but nontrivial degree-two commutator information. Its minimum length in the cube edge metric is $14$. We also obtain a separation between finite and unbounded memory. A two-state reading separates an order pair, while for every $k\geq1$, there is an explicit factorial pair which no $k$-state reading separates. An unbounded stack recovers the reduced loop word on every walk. For integer-valued functions on the vertex set, potential readings vanish on closed walks, while occupation binding is not rectangular. More generally, every occupation-based constraint is rectangular on a class of histories if and only if the occupation vector is constant on that class. Finally, a declared quarter-turn quaternion clock gives a second proper order-sensitive congruence, incomparable with the reduced-word quotient. The obstruction to commutative recovery, and the minimum $14$, remain valid on every hypercube $Q_n$, $n\geq3$.

math.CO↗

arXiv2026

The $δ$-calculus: from distinction to arithmetic

Let $δ$ denote the primitive act of distinction, formally realized as the one-step extension $r\mapsto Sr$ of a finite record. We prove that the $δ$-orbit is initial among $δ$-algebras: every $δ$-algebra admits a unique structure-preserving map from the orbit. The map is injective when the successor operation is injective and the base point is not a successor. If the $δ$-algebra also satisfies induction for all predicates, the map is bijective and gives the unique isomorphism with the generated $δ$-orbit. The $δ$-calculus is an intuitionistic first-order proof system over the signature $\{0,S,+,\cdot\}$. Each derivation carries a ledger recording the use of the law of excluded middle, the limited principle of omniscience, Markov's principle, and induction on quantified formulas. Every closed formula derivable in the forced fragment is true in the standard model. Starting from $δ$, we construct the choice-free number tower $δ\leadsto \mathbb{N}_δ\hookrightarrow \mathbb{Z}_δ\hookrightarrow \mathbb{Q}_δ$. The metatheoretic systems $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ each admit an explicit injection into $\mathbb{N}_δ$. Under the law of excluded middle, every recognizer is either injective or has kernel congruence $\equiv_{i,p}$ for a unique pair $i\geq0$, $p\geq1$. In the noninjective case, the quotient is a finite monogenic monoid $M(i,p)$. A decidable congruence together with an explicit pair of distinct related elements implies the classification without additional nonconstructive principles. For a decidable congruence different from equality, Markov's principle is needed. For an arbitrary congruence, the dichotomy requires the law of excluded middle. The reverse implications show that the last two prices are exact. They are distinct from the syntactic ledger of derivations in the $δ$-calculus.

math.LO↗

arXiv2026

Reciprocal Cost on the Positive Rationals

We study nonnegative solutions of the reciprocal cost law on the positive rationals and determine which of them admit regular extensions to the positive reals. Using the substitution $H=1+F$, we reduce the reciprocal composition law to d'Alembert's functional equation. We prove that every nonnegative solution on $\mathbb{Q}_{>0}$ is determined by one real weight $α_p$ for each prime $p$, with only a global sign identification. Thus the rational solution space is infinite dimensional. We prove that every nonnegative rational solution has an algebraic extension to $\mathbb{R}_{>0}$, but a regular extension exists exactly when the prime weights satisfy $α_p=λ\log p$ for some $λ\in\mathbb{R}$. In this case the extension is unique and belongs to the one-parameter family $F_λ(x)=\cosh(λ\log x)-1$. Otherwise the rational solution is unbounded on every nonempty open subset of $\mathbb{Q}_{>0}$, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of $\mathbb{R}_{>0}$, where the alternative is governed by rational rank. Finally, we show that unit logarithmic curvature selects the canonical reciprocal cost $J(x)=(x+x^{-1})/2-1$, while, for a carrier $G$ whose logarithm is dense in $\mathbb{R}$, the single asymptotic condition $F(e^s)\sim s^2/2$ implies both regularity and calibration.

math.GM↗

arXiv2026

Golden and Metallic Structures on Hessian Manifolds

We consider the reciprocal cost function $ J(x)=\frac12(x+x^{-1})-1 $ and its $n$-dimensional extension $J(x_1,\ldots,x_n) = \frac12(R+R^{-1})-1, R=\prod\limits_{i=1}^n x_i^{α_i}, α=(α_1,\ldots,α_n)\in\mathbb{R}^n\setminus\{0\}.$ 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 $h_λ$. Combining the rank-one tensor with the Hessian metric $h_λ$, we construct a $(1,1)$-tensor field $A_λ$. Its trace normalization defines a projector $P_λ$, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector $P_λ$ 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 $P_λ$ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection $\nabla^λ$ of the Hessian metric $h_λ$.

math.DG↗

arXiv2026

Multidimensional cost geometry

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 $S=α\cdot t$, 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.

math.DG↗

arXiv2026

A semi-symmetric metric connection, perfect fluid space-time and phantom barrier

We consider a concircularly semi-symmetric metric connection and its application. The Ricci tensors with respect to the concircularly semi-symmetric metric connection are symmetric, and they are used to define Einstein type manifolds. In this way, conditions under which a pseudo-Riemannian manifold is quasi-Einstein are obtained. On a Lorentzian manifold, a concircularly semi-symmetric metric connection with a unit timelike generator becomes a semi-symmetric metric $P$-connection, and a Lorentzian manifold becomes a GRW space-time. { It is proven in which case the scalar curvature of a perfect fluid space-time with that connection is constant and what its value is, which implies a reduction to Einstein manifold. } Furthermore, an application to the theory of relativity is presented, and the value of the equation of state is examined. It is ultimately shown that the equation of state in a perfect fluid space-time that satisfies Einstein field equation with cosmological constant and admits a unit timelike torse-forming vector represents a phantom barrier.

math.DG↗

arXiv2026

The d'Alembert Inevitability Theorem

We study functions satisfying the composition law $F(xy)+F(x/y)=P(F(x),F(y))$ 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 $\mbox{deg} P(u,v) \ge 3$ do not admit nonconstant continuous solutions, provided the leading term does not cancel (Theorem 3.1.). For continuous nonconstant functions $F:\mathbb{R}_{>0}\to\mathbb{R}$ with $F(1)=0$ satisfying the composition law with a symmetric polynomial $P$ of degree at most two, the combiner is necessarily of the form $P(u,v)=2u+2v+c\,uv$, $c\in\mathbb{R}$ (Theorem 3.3.). The equation reduces in logarithmic coordinates to the classical d'Alembert functional equation. For $c\neq 0$, 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 $F(x)=\tfrac12(x+x^{-1})-1$. For $c\neq0$, 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.

math.CA↗

arXiv2026

Einstein connection of nonsymmetric pseudo-Riemannian manifold

A.Einstein considered a linear connection $\nabla$ with torsion $T$ on a smooth manifold equipped with a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne0$ is a skew-symmetric tensor associated with electromagnetism, such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $F$, satisfying the $f^2$-torsion condition $T(f^2X,Y)=T(X,f^2Y)=f^2 T(X,Y)$, where $g(X,fY)=F(X,Y)$, and show that in the almost Hermitian case, it reduces to the M.Prvanović's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $f^2$-torsion condition, discuss special Einstein connections, and give example in terms of weighted product of almost Hermitian manifolds.

math.DG↗

arXiv2026

Uniqueness of the Canonical Reciprocal Cost

We study a rigidity problem for functions $F:\R_{>0}\to\R_{\ge 0}$ that penalize deviation of a positive ratio from equilibrium $x=1$. Assuming (i) a d'Alembert-type composition law on $\R_{>0}$, 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 $F(1)=0.$ 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 $H(t)=F(e^t)+1$, where the composition law becomes d'Alembert's functional equation on $\R$. 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 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.

math.CA↗

arXiv2025

Weak metric structures on generalized Riemannian manifolds

In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the first author and R. Wolak), which generalize almost complex and almost contact metric $f$-contact structures. We consider generalized metric connections (i.e., linear connections preserving $G$) with totally skew-symmetric torsion (0,3)-tensor. For rank$(F)=\dim M$ and non-conformal tensor $A^2$, where $A$ is a skew-symmetric (1,1)-tensor adjoint to $F$, we apply weak almost Hermitian structures to fundamental results (by the second author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold is a weighted product of several nearly Kähler manifolds corresponding to eigen-distributions of $A^2$. For rank$(F)<\dim M$ we apply weak $f$-structures and obtain splitting results for generalized Riemannian manifolds.

math.DG↗

arXiv2025

Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory

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.

math.DG↗

arXiv2023

Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

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.

math.DG↗

arXiv2019

Non-symmetric Riemannian gravity and Sasaki-Einstein 5-manifolds

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.

math.DG↗

arXiv2013

Geodesic mapping onto Kählerian space of the first kind

In the present paper a generalized Kählerian space $\mathbb{G}\underset 1 {\mathbb{K}}{}_N$ of the first kind is considered, as a generalized Riemannian space $\mathbb{GR}_N$ with almost complex structure $F^h_i$, that is covariantly constant with respect to the first kind of covariant derivative. Using the non-symmetric metric tensor we find necessary and sufficient conditions for a geodesic mapping $f:\mathbb{GR}_N\to \mathbb{G}\underset 1 {\mathbb{\bar{K}}}{}_N$ with respect to the four kinds of covariant derivatives.

math.DG↗