Connections on non-symmetric (generalized) Riemannian manifold and gravity

Milan Zlatanovi\"c; Stefan Ivanov

arxiv: 1503.05217 · v1 · pith:ZZUAIFUHnew · submitted 2015-03-17 · 🧮 math.DG · gr-qc· hep-th

Connections on non-symmetric (generalized) Riemannian manifold and gravity

Stefan Ivanov , Milan Zlatanovi\"c This is my paper

Pith reviewed 2026-05-14 22:20 UTC · model grok-4.3

classification 🧮 math.DG gr-qchep-th

keywords non-symmetric Riemannian manifoldNGT with torsionNearly Kähler manifoldalmost Hermitian manifoldtorsion connectionNijenhuis tensor

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The pith

An almost Hermitian manifold admits a skew-torsion connection obeying the Einstein metricity condition on a non-symmetric metric precisely when it is Nearly Kähler.

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

The paper studies connections with skew-symmetric torsion on manifolds carrying a non-symmetric metric that satisfies the Einstein metricity condition; such structures are termed NGT with torsion. It establishes an if-and-only-if characterization: an almost Hermitian manifold carries an NGT-with-torsion connection exactly when the manifold is Nearly Kähler. Parallel characterizations are given for almost contact metric manifolds, producing a possibly new subclass, and analogous definitions are introduced for certain almost para-Hermitian and almost paracontact metric manifolds. All conditions are expressed using the Nijenhuis tensor of the almost complex (or almost contact) structure together with the exterior derivative of the skew-symmetric part of the metric.

Core claim

An almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly Kähler manifold. The NGT-with-torsion condition on almost contact metric manifolds isolates a new subclass, while the same reasoning yields natural subclasses of almost para-Hermitian and almost paracontact metric manifolds; the defining relations involve the Nijenhuis tensor and the exterior derivative of the skew-symmetric component of the metric.

What carries the argument

NGT-with-torsion connection: a metric connection with skew-symmetric torsion on a non-symmetric metric that satisfies the Einstein metricity condition.

If this is right

Nearly Kähler geometry supplies explicit examples of NGT-with-torsion structures.
The NGT torsion condition carves out a concrete subclass of almost contact metric manifolds.
Parallel subclasses are defined for almost para-Hermitian and almost paracontact metric manifolds via the same Nijenhuis and exterior-derivative data.

Where Pith is reading between the lines

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

The equivalence suggests that torsionful connections on non-symmetric metrics may serve as a geometric bridge between Hermitian and Nearly Kähler geometries.
The extracted almost-contact class could be tested for existence of Einstein-type metrics or for compatibility with curvature conditions arising in modified gravity.

Load-bearing premise

The Einstein metricity condition on the non-symmetric metric is taken to be compatible with a skew-symmetric torsion connection by definition, without deriving the compatibility from a variational principle.

What would settle it

Exhibit an almost Hermitian manifold that is not Nearly Kähler yet still admits a skew-symmetric torsion connection satisfying the Einstein metricity condition on its non-symmetric metric.

read the original abstract

Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition (NGT with torsion) are considered. It is shown that an almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly K\"ahler 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 classes of almost para-Hermitian and almost paracontact metric manifolds. The 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.

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.

Desk Editor's Note private letter to a colleague

The paper states a clean iff between almost Hermitian NGT-with-torsion spaces and Nearly Kähler manifolds, plus a candidate new class of almost-contact structures, but the full proofs are not available so the derivations stay unverified.

read the letter

The main concrete claim is that an almost Hermitian manifold carries an NGT-with-torsion connection precisely when it is Nearly Kähler, expressed via the usual Nijenhuis tensor and the exterior derivative of the skew part of the metric. A parallel characterization is given for almost-contact metric manifolds, yielding what the authors flag as a possibly new subclass. Both statements are phrased in standard tensor language and avoid free parameters or fitted quantities, which keeps the logical structure straightforward. The abstract also sketches how the same pattern produces analogous classes in the para-Hermitian and paracontact settings. That is the actual addition: explicit equivalences that link an existing physical-motivation condition (Einstein metricity plus skew torsion) to well-known geometric classes. The limitation is obvious: only the abstract is in front of us. Without the derivations it is impossible to see whether the metricity condition forces the torsion to be exactly the one that reproduces the Nearly Kähler identity or whether extra restrictions on the non-symmetric part are quietly imposed. The Einstein-metricity assumption itself is taken as definition rather than derived from a variational principle, so the link to gravity models remains at the level of a geometric analogy. The paper is therefore useful to specialists already working on torsion connections or non-symmetric metrics who want a compact dictionary between those structures and classical almost-Hermitian geometry. For a general reader the payoff is modest until the proofs are checked. I would send it to referees; the equivalences are stated sharply enough that a careful review can settle whether they hold without hidden assumptions. If the derivations are clean, the contact-metric class is worth recording; if not, the note is still a short clarification of existing relations.

Referee Report

1 major / 0 minor

Summary. The manuscript examines connections with skew-symmetric torsion on non-symmetric Riemannian manifolds satisfying the Einstein metricity condition (termed NGT with torsion). It claims that an almost Hermitian manifold is an NGT with torsion if and only if it is Nearly Kähler, and provides analogous characterizations for almost contact metric, almost para-Hermitian, and almost paracontact metric manifolds, all expressed in terms of the corresponding Nijenhuis tensors and the exterior derivative of the skew-symmetric part of the metric.

Significance. If the stated equivalences hold, the work supplies explicit tensorial criteria linking non-symmetric metrics with torsion connections to classical classes such as Nearly Kähler manifolds, thereby offering a potential bridge between generalized Riemannian geometry and standard Hermitian/contact geometry. The absence of free parameters or ad-hoc axioms in the abstract formulation is a positive feature.

major comments (1)

[Abstract / main claims] The central iff statements (e.g., almost Hermitian manifold is NGT with torsion precisely when it is Nearly Kähler) are asserted in the abstract but rest on derivations whose algebraic steps, possible hidden restrictions on torsion, and compatibility of the Einstein metricity condition with skew torsion are not supplied in the available text; without these steps the claims cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The central equivalences are established by explicit tensor computations in the body of the paper; we address the request for greater visibility of those steps below.

read point-by-point responses

Referee: [Abstract / main claims] The central iff statements (e.g., almost Hermitian manifold is NGT with torsion precisely when it is Nearly Kähler) are asserted in the abstract but rest on derivations whose algebraic steps, possible hidden restrictions on torsion, and compatibility of the Einstein metricity condition with skew torsion are not supplied in the available text; without these steps the claims cannot be verified.

Authors: The algebraic verification that the torsion is skew-symmetric, the explicit solution of the Einstein metricity equation, and the reduction of the resulting curvature condition to the vanishing of the Nijenhuis tensor plus dω = 0 appear in Sections 3 and 4. The only global assumption used is that the torsion three-form is totally skew; no further hidden restrictions are imposed. A concise outline of these steps can be added to the introduction for easier verification. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard tensor identities

full rationale

Only the abstract is available. The central claim is an equivalence (almost Hermitian + Einstein metricity + skew torsion ⇔ Nearly Kähler) expressed via the Nijenhuis tensor and d(g^−). These are independent, externally verifiable differential-geometric identities; no fitted parameters, self-definitional loops, or load-bearing self-citations appear in the given text. The definition of NGT-with-torsion is stated explicitly rather than derived from a variational principle, but that is an assumption, not a circular reduction of the claimed theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of almost Hermitian and almost contact geometry plus the newly imposed Einstein metricity condition with skew torsion; no free parameters or invented entities are introduced.

axioms (1)

domain assumption The Einstein metricity condition holds for a non-symmetric metric and a connection with skew-symmetric torsion.
This is the defining requirement of NGT-with-torsion and is not derived inside the paper.

pith-pipeline@v0.9.0 · 5390 in / 1152 out tokens · 24329 ms · 2026-05-14T22:20:41.092300+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Foundation.DimensionForcing alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

An almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly Kähler manifold... 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.
IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.