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arxiv: 2604.08486 · v2 · submitted 2026-04-09 · 🧮 math.DG · math-ph· math.MP

Einstein connection of nonsymmetric pseudo-Riemannian manifold, II

Vladimir Rovenski , Milan Zlatanovi\'c , Miroslav Maksimovi\'c This is my paper

Pith reviewed 2026-05-10 17:17 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MP
keywords nonsymmetric metricEinstein connectionweak almost contact structuretorsionpseudo-Riemannian manifoldalmost Hermitian manifoldunified field theory
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The pith

A weak almost contact structure on a manifold with nonsymmetric metric G = g + F yields an explicit formula for the Einstein connection with torsion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives an explicit expression for the linear connection ∇ with torsion T that satisfies the Einstein condition (∇_X G)(Y, Z) = G(T(Y, X), Z) for the nonsymmetric tensor G = g + F. Here g is pseudo-Riemannian and F is skew-symmetric. The formula uses a weak almost contact structure (f, ξ, η) obeying g(X, fY) = F(X, Y) plus one natural compatibility condition. This revives Einstein's geometric approach to combining gravity and electromagnetism by supplying concrete coefficients rather than an existence proof alone. The authors also treat special cases that recover familiar connections and construct an example on a weighted product of an almost Hermitian manifold with the real line.

Core claim

If a pseudo-Riemannian manifold with nonsymmetric metric G = g + F admits a weak almost contact structure (f, ξ, η) satisfying g(X, fY) = F(X, Y) together with the stated natural condition, then the Einstein connection ∇ can be written down explicitly in terms of the Levi-Civita connection of g, the tensor F, and the structure tensors f, ξ, η. The resulting torsion T satisfies the defining relation of an Einstein connection.

What carries the argument

The weak almost contact structure (f, ξ, η) with g(X, fY) = F(X, Y) and the natural condition, which converts the skew-symmetric part F into an almost-complex-type operator and thereby determines the torsion terms in the connection.

If this is right

  • The explicit formula gives concrete Christoffel-like symbols that can be substituted into curvature or field equations.
  • When the structure reduces to a genuine almost contact structure the natural condition holds automatically and the connection simplifies.
  • The weighted product construction supplies an infinite family of examples on which the connection can be computed in coordinates.
  • Special Einstein connections arise by imposing further algebraic conditions on f or on the torsion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same structure might be used to define parallel transport or geodesic equations that mix gravitational and electromagnetic effects without additional postulates.
  • One could test the construction on explicit four-dimensional Lorentzian manifolds with a prescribed electromagnetic 2-form to see whether the resulting geodesics differ measurably from the Levi-Civita case.
  • The method may extend to higher-dimensional or degenerate metrics if a suitable generalization of the weak almost contact structure can be found.

Load-bearing premise

The manifold must admit a weak almost contact structure compatible with the skew-symmetric part via g(X, fY) = F(X, Y) and the natural condition.

What would settle it

A concrete pseudo-Riemannian manifold equipped with nonzero F for which no weak almost contact structure satisfying the compatibility condition exists, yet a torsion connection obeying (∇_X G)(Y, Z) = G(T(Y, X), Z) can be shown to exist or to fail by direct computation.

read the original abstract

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. A. 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 $(f,\xi,\eta)$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper claims to explicitly present the Einstein connection ∇ of the nonsymmetric pseudo-Riemannian metric G = g + F satisfying (∇_X G)(Y,Z) = G(T(Y,X),Z), expressed in terms of a weak almost contact structure (f, ξ, η) with the compatibility condition g(X, fY) = F(X, Y) together with a stated natural condition (trivial in the almost contact case). It discusses special Einstein connections and supplies one example via the weighted product of an almost Hermitian manifold with ℝ.

Significance. If the derivation is sound, the explicit formula would provide a concrete construction for the Einstein connection in the subclass of nonsymmetric metrics that admit a compatible weak almost contact structure. This could be useful in geometric models of unified gravity-electromagnetism theories. The example strengthens applicability in specific settings, but the absence of a general existence result for the structure means the result does not cover arbitrary G as suggested by the abstract.

major comments (3)
  1. [Abstract] Abstract: The claim of an 'explicit presentation' of the Einstein connection for G = g + F is not supported in full generality, as the formula is derived only under the assumption that a weak almost contact structure (f, ξ, η) with g(X, fY) = F(X, Y) exists and satisfies the natural condition; no existence theorem is given for arbitrary nonsymmetric pseudo-Riemannian G.
  2. [Main derivation] Main derivation: The explicit formula for ∇ is obtained by assuming the manifold admits the required weak almost contact structure and natural condition, yet the paper provides neither a proof that such a structure exists independently of the connection nor verification that the natural condition holds automatically outside the almost contact case (only the weighted-product example is supplied).
  3. [Example] Example section: The single example (weighted product of almost Hermitian manifold with ℝ) illustrates the formula but does not address whether the construction extends to general F or verify the independence of the assumed structure from the resulting connection, leaving the scope of the central claim unclear.
minor comments (1)
  1. [Preliminaries] The distinction between 'weak almost contact structure' and standard almost contact structures could be clarified with explicit definitions or references in the preliminaries.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below, clarifying the scope of our results and indicating revisions to improve precision.

read point-by-point responses

  1. Referee: [Abstract] The claim of an 'explicit presentation' of the Einstein connection for G = g + F is not supported in full generality, as the formula is derived only under the assumption that a weak almost contact structure (f, ξ, η) with g(X, fY) = F(X, Y) exists and satisfies the natural condition; no existence theorem is given for arbitrary nonsymmetric pseudo-Riemannian G.

    Authors: We agree that the abstract would benefit from greater precision. The manuscript derives the explicit formula under the stated assumptions on the existence of the weak almost contact structure and the natural condition; it does not claim or prove existence for arbitrary G = g + F. The phrasing 'using a weak almost contact structure' is intended to indicate the conditional setting, but we will revise the abstract to explicitly note that the construction applies to those nonsymmetric metrics that admit such a compatible structure. revision: yes

  2. Referee: [Main derivation] The explicit formula for ∇ is obtained by assuming the manifold admits the required weak almost contact structure and natural condition, yet the paper provides neither a proof that such a structure exists independently of the connection nor verification that the natural condition holds automatically outside the almost contact case (only the weighted-product example is supplied).

    Authors: The derivation is conditional on the manifold being equipped with the weak almost contact structure satisfying the compatibility g(X, fY) = F(X, Y) together with the natural condition; the structure is introduced as data on the manifold prior to constructing ∇. The natural condition is part of the hypothesis and is trivial when the structure reduces to an almost contact structure. We verify it holds in the supplied example but do not assert it holds automatically for all F outside that case. We will add a clarifying sentence in the introduction stating that the structure is defined independently of ∇. revision: partial

  3. Referee: [Example] The single example (weighted product of almost Hermitian manifold with ℝ) illustrates the formula but does not address whether the construction extends to general F or verify the independence of the assumed structure from the resulting connection, leaving the scope of the central claim unclear.

    Authors: The example is meant to illustrate the formula in a concrete case where the weak almost contact structure arises naturally from the almost Hermitian data on the base manifold and the product construction. It confirms the conditions, including the natural condition, hold in this setting. The result does not claim to cover arbitrary F without the existence of the structure, as proving general existence lies outside the paper's scope. We will expand the example section with a short paragraph confirming the independence of the structure from the derived connection. revision: partial

Circularity Check

0 steps flagged

No circularity in the derivation chain

full rationale

The paper presents an explicit formula for the Einstein connection of the nonsymmetric metric G = g + F, constructed using the assumed existence of a compatible weak almost contact structure (f, ξ, η) with g(X, fY) = F(X, Y) and a natural condition. This is a conditional constructive result rather than a derivation that reduces by construction to its inputs. No self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling are present in the abstract or described chain. The result applies precisely when the structure exists (with one illustrative example given), making the derivation self-contained under its stated assumptions without tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests entirely on the existence of the weak almost contact structure and natural condition; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The manifold admits a weak almost contact structure (f, ξ, η) such that g(X, fY) = F(X, Y) and satisfies the natural condition.
    This assumption is required to enable the explicit construction stated in the abstract.

pith-pipeline@v0.9.0 · 5454 in / 1179 out tokens · 55815 ms · 2026-05-10T17:17:35.344029+00:00 · methodology

discussion (0)

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Reference graph

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