pith. sign in

arxiv: 2510.00849 · v2 · pith:RMWQ633Nnew · submitted 2025-10-01 · 🧮 math.DG · gr-qc

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

Miroslav Maksimovi\'c , Milan Zlatanovi\'c , Marija Najdanovi\'c This is my paper

Pith reviewed 2026-05-21 20:58 UTC · model grok-4.3

classification 🧮 math.DG gr-qc
keywords semi-symmetric metric connectionperfect fluid space-timephantom barrierGRW space-timetorse-forming vectorEinstein field equationscosmological constantquasi-Einstein manifold
0
0 comments X

The pith

Perfect fluid space-times admitting a unit timelike torse-forming vector have an equation of state at the phantom barrier.

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

The paper studies concircularly semi-symmetric metric connections on pseudo-Riemannian manifolds and establishes that the associated Ricci tensors are symmetric, yielding conditions under which the manifold is quasi-Einstein. When the connection is applied to a Lorentzian manifold with a unit timelike generator, the manifold reduces to a generalized Robertson-Walker space-time. For perfect fluid space-times that satisfy the Einstein field equations with a cosmological constant and admit such a vector, the scalar curvature is proven constant, which reduces the space-time to an Einstein manifold. This geometric setup implies that the equation of state parameter sits at the phantom barrier.

Core claim

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.

What carries the argument

Concircularly semi-symmetric metric connection with unit timelike torse-forming generator, which converts the Lorentzian manifold into a GRW space-time and enforces constant scalar curvature.

If this is right

  • The manifold reduces to a generalized Robertson-Walker space-time.
  • Scalar curvature takes a specific constant value that forces reduction to an Einstein manifold.
  • Ricci tensors with respect to the connection become symmetric.
  • The space-time satisfies the conditions for being quasi-Einstein.

Where Pith is reading between the lines

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

  • Such space-times could serve as geometric models for late-time acceleration driven by a cosmological constant at the boundary between quintessence and phantom regimes.
  • The torse-forming vector might supply a mechanism that stabilizes the equation of state exactly at the phantom divide without additional fields.
  • Relaxing the perfect-fluid assumption while keeping the same connection could link the result to more general matter models in GRW cosmologies.

Load-bearing premise

The Lorentzian manifold admits a concircularly semi-symmetric metric connection whose generator is a unit timelike torse-forming vector.

What would settle it

Derive the equation of state parameter directly from the Einstein equations with cosmological constant under the given connection and vector, then test whether its value equals negative one.

read the original abstract

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.

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

1 major / 2 minor

Summary. The paper studies a concircularly semi-symmetric metric connection on pseudo-Riemannian manifolds. It shows that the associated Ricci tensor is symmetric and uses this to obtain conditions under which the manifold is quasi-Einstein. Specializing to Lorentzian manifolds with a unit timelike torse-forming generator, the connection becomes a semi-symmetric metric P-connection and the manifold reduces to a generalized Robertson-Walker space-time. The manuscript proves that the scalar curvature of a perfect fluid space-time equipped with this connection is constant (and computes its value), implying the space-time is Einstein. It then applies the setting to general relativity, showing that a perfect fluid satisfying the Einstein field equations with cosmological constant and admitting such a vector field must have equation-of-state parameter corresponding to the phantom barrier.

Significance. If the central derivations are completed, the work supplies a geometric mechanism that forces the equation-of-state parameter of a perfect fluid in Einstein gravity with cosmological constant to lie in the phantom regime. The explicit reduction to GRW space-times and the constant-curvature result for the modified connection are technically concrete and could be of interest to researchers studying modified connections in Lorentzian geometry and their possible cosmological implications.

major comments (1)
  1. [application to the theory of relativity] The constancy of scalar curvature is established only for the concircularly semi-symmetric metric connection (abstract and the Lorentzian-manifold application paragraph). The Einstein field equations and the perfect-fluid stress-energy tensor, however, are written with respect to the Levi-Civita connection. No explicit relation is derived between the two scalar curvatures under the unit timelike torse-forming assumption, so it is not immediate that the constancy result constrains the standard curvature scalars that enter the Einstein equations and thereby force the phantom equation of state.
minor comments (2)
  1. Notation for the concircularly semi-symmetric metric connection and its generator should be introduced with a single consistent symbol and recalled at the beginning of each major section.
  2. The manuscript would benefit from a brief comparison, even if only in the introduction, with existing literature on semi-symmetric metric connections and torse-forming vector fields in Lorentzian geometry.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comment below and will revise the manuscript to make the connection between the curvatures explicit.

read point-by-point responses

  1. Referee: [application to the theory of relativity] The constancy of scalar curvature is established only for the concircularly semi-symmetric metric connection (abstract and the Lorentzian-manifold application paragraph). The Einstein field equations and the perfect-fluid stress-energy tensor, however, are written with respect to the Levi-Civita connection. No explicit relation is derived between the two scalar curvatures under the unit timelike torse-forming assumption, so it is not immediate that the constancy result constrains the standard curvature scalars that enter the Einstein equations and thereby force the phantom equation of state.

    Authors: We agree that the relation between the scalar curvature of the concircularly semi-symmetric metric connection and the Levi-Civita scalar curvature must be made explicit for the application to the Einstein field equations to be fully rigorous. Under the unit timelike torse-forming vector field, the two connections differ by a term proportional to the vector field and its covariant derivatives. We will add a new paragraph deriving that the difference in scalar curvatures is a constant determined by the norm of the vector field and the dimension; this constant is absorbed into the cosmological term, so constancy of the modified scalar curvature implies constancy of the Levi-Civita scalar curvature. Consequently the space-time is Einstein with respect to the Levi-Civita connection, and the perfect-fluid equation of state is forced to the phantom barrier as claimed. This clarification will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from geometric assumptions to EoS constraint via independent curvature calculations

full rationale

The paper defines a concircularly semi-symmetric metric connection on a Lorentzian manifold with unit timelike torse-forming generator, derives symmetry of the associated Ricci tensor, proves constancy of the scalar curvature for perfect fluid space-times (implying reduction to Einstein manifold), and applies the setup to the standard Einstein field equations with cosmological constant. The final claim that the equation-of-state parameter represents a phantom barrier follows from these steps and the perfect-fluid stress-energy tensor rather than reducing tautologically to the input definitions or via self-citation chains. No fitted parameters renamed as predictions or ansatzes smuggled through citations are present; the chain is self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard definitions of semi-symmetric metric connections and GRW space-times from the differential-geometry literature together with the assumption that a unit timelike torse-forming vector exists; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption A Lorentzian manifold admits a concircularly semi-symmetric metric connection with unit timelike generator that turns it into a GRW space-time.
    This assumption is invoked to obtain the semi-symmetric metric P-connection and the subsequent curvature results.

pith-pipeline@v0.9.0 · 5716 in / 1384 out tokens · 42595 ms · 2026-05-21T20:58:19.759936+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    L. J. Alias, A. Romero, M. Sanchez,Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker space-times, Gen. Relativity Gravitation, 27(1), 71–84, (1995)

    work page 1995

  2. [2]

    Blaga,Solitons and geometrical structures in a perfect fluid spacetime, Rocky Mountain J

    A. Blaga,Solitons and geometrical structures in a perfect fluid spacetime, Rocky Mountain J. Math., 50(1), 41-53, (2020)

    work page 2020

  3. [3]

    Blaga, C

    A. Blaga, C. ¨Ozg¨ ur,On torse-forming-like vector fields, Mediterr. J. Math., 21, 214, (2024)

    work page 2024

  4. [4]

    M. C. Chaki, R. K. Maity,On quasi Einstein manifolds, Publ. Math. Debrecen, 57 (3-4), 297-306, (2000)

    work page 2000

  5. [5]

    S. K. Chaubey, U. C. De, M. D. Siddiqi,Characterization of Lorentzian manifolds with a semi-symmetric linear connec- tion, J. Geom. Phys. 166, 104269, (2021)

    work page 2021

  6. [6]

    S. K. Chaubey, J. W. Lee, S. K. Yadav,Riemannian manifolds with a semi-symmetric metric P-connection, J. Korean Math. Soc., 56 (4), 1113-1129, (2019)

    work page 2019

  7. [7]

    S. K. Chaubey, Y. J. Suh,Riemannian concircular structure manifolds, Filomat, 36 (19), 6699-6711, (2022)

    work page 2022

  8. [8]

    S. K. Chaubey, Y. J. Suh,Characterizations of Lorentzian manifolds, J. Math. Phys., 63, 062501, (2022)

    work page 2022

  9. [9]

    S. K. Chaubey, Y. J. Suh, U. C. De,Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric metric connection, Anal. Math. Phys. 10, 61, (2020)

    work page 2020

  10. [10]

    B. Y. Chen,A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit. 46(12), 1833, (2014)

    work page 2014

  11. [11]

    Csillag, T

    L. Csillag, T. Harko,Semi-symmetric metric gravity: From the Friedmann-Schouten geometry with torsion to dynamical dark energy models, Phys. Dark Univ., 46, 101596, (2024)

    work page 2024

  12. [12]

    K. De, U. C. De,Perfect fluid spacetimes obeying certain restrictions on the energy-momentum tensor, Filomat, 37(11), 3483-3492, (2023)

    work page 2023

  13. [13]

    K. De, U. C. De, Lj. Velimirovi´ c,Some curvature properties of perfect fluid spacetimes, Quaestiones Mathematicae, 47(4), 751-764, (2024)

    work page 2024

  14. [14]

    U. C. De, K. De, S. G¨ uler,Characterizations of a Lorentzian manifold with a semi-symmetric metric connection, Publicationes Mathematicae Debrecen, 104(3-4), 329-341, (2024)

    work page 2024

  15. [15]

    U. C. De, G. C. Ghosh,On weakly concircular Ricci symmetric manifolds, South East Asian J. Math. and Math. Sci., 3(2), (2005), 9–15

    work page 2005

  16. [16]

    U. C. De, C. A. Mantica, Y. J. Suh,Perfect fluid spacetimes and gradient solitons, Filomat, 36(3), 829-842, (2022). 12

    work page 2022

  17. [17]

    Deshmukh, I

    S. Deshmukh, I. Al-Dayel, D. M. Naik,On an anti-torqued vector field on Riemannian manifolds, Mathematics, 9(18), 2201, (2021)

    work page 2021

  18. [18]

    Deszcz, M

    R. Deszcz, M. Glogowska, M. Hotlos, Z. Sent¨ urk,On certain quasi-Einstein semisymmetric hypersurfaces, Annales Univ. Sci. Budapest, 41, 151-164, (1998)

    work page 1998

  19. [19]

    K. L. Duggal, R. Sharma,Symmetries of spacetimes and Riemannian manifolds, Mathematics and its Applications, Kluwer Academic Publishers, (1999)

    work page 1999

  20. [20]

    S. I. Goldberg,On pseudo-harmonic and pseudo-Killing vectors in metric manifolds with torsion, Annals of Mathematics, 64(2), 364-373, (1956)

    work page 1956

  21. [21]

    S. I. Goldberg, I. Vaisman,On compact locally conformal Kaehler manifolds with non-negative sectional curvature, Ann. Fac. Sci. Toulouse Math. Ser. V, 2(2), 117–123, (1980)

    work page 1980

  22. [22]

    I. A. Gordeeva, V. I. Pan’zhenskii, S. E. Stepanov,Riemann-Cartan manifolds, Journal of Mathematical Sciences, 169(3), 342-361, (2010)

    work page 2010

  23. [23]

    Gutierrez, B

    M. Gutierrez, B. Olea,Global decomposition of a Lorentzian manifold as a generalized Robertson-Walker space, Differ. Geom. Appl. 27, 146-156, (2009)

    work page 2009

  24. [24]

    Y. Han, A. De, P. Zhao,On a semi-quasi-Einstein manifold, J. Geom. Phys., 155, 103739, (2021)

    work page 2021

  25. [25]

    Kranas, C

    D. Kranas, C. G. Tsagas, J. D. Barrow, D. Iosifidis,Friedmann-like universes with torsion, Eur. Phys. J. C, 79, 341, (2019)

    work page 2019

  26. [26]

    Y. Li, H. A. Kumara, M. S. Siddesha, D. M. Naik,Charachterization of Ricci almost solitons on Lorentzian manifolds, Symmetry, 15, 1175, (2023)

    work page 2023

  27. [27]

    Y. Li, M. S. Siddesha, H. A. Kumara, M. M. Praveena,Charachterization of Bach and Cotton tensors on a class of Lorentzian manifolds, Mathematics, 12, 3130, (2024)

    work page 2024

  28. [28]

    Maksimovi´ c, M

    M. Maksimovi´ c, M. Petrovi´ c, N. Vesi´ c, M. Zlatanovi´ c,Concircularly semi-symmetric metric connection, Quaest. Math., 47(3), 557-576, (2024)

    work page 2024

  29. [29]

    Maksimovi´ c, M

    M. Maksimovi´ c, M. Zlatanovi´ c,Einstein type curvature tensors and Einstein type tensors of generalized Riemannian space in the Eisenhart sense, Mediterr. J. Math., 19, 217, (2022)

    work page 2022

  30. [30]

    Maksimovi´ c, M

    M. Maksimovi´ c, M. Zlatanovi´ c, M. Vuˇ curovi´ c,Lorentzian manifolds equipped with a concircularly semi-symmetric metric connection, J. Geo. Phys., 217, 105633, (2025)

    work page 2025

  31. [31]

    C. A. Mantica, L. G. Molinari,Generalized Robertson-Walker spacetimes: A survey, Int. J. Geom. Methods Mod. Phys. 14 (3), 1730001, (2017)

    work page 2017

  32. [32]

    C. A. Mantica, L. G. Molinari,A note on concircular structure space-times, Commun. Korean Math. Soc. 34(2), 633-635, (2019)

    work page 2019

  33. [33]

    Mikeˇ s, L

    J. Mikeˇ s, L. Rachunek,Torse-forming vector fields inT-semisymmetric Riemannian manifolds, In Steps in Diff. Geom., Proc. of the Colloquium on Diff. Geom., Univ. Debrecen, Hungary, 219–229, (2000)

    work page 2000

  34. [34]

    D. M. Naik,Ricci solitons on Riemannian manifolds admitting certain vector field, Ric. Mat. 73, 531-546, (2024)

    work page 2024

  35. [35]

    O’Neill,Semi-Riemannian geometry with applications to the relativity, Academic Press, New York-London, (1983)

    B. O’Neill,Semi-Riemannian geometry with applications to the relativity, Academic Press, New York-London, (1983)

    work page 1983

  36. [36]

    Pak,On the pseudo-Riemannian spaces, J

    E. Pak,On the pseudo-Riemannian spaces, J. Korean Math. Soc., 6, 23-31, (1969)

    work page 1969

  37. [37]

    R. J. Petti,Derivation of Einstein–Cartan theory from general relativity, International Journal of Geometric Methods in Modern Physics, 18(06), 2150083, (2021)

    work page 2021

  38. [38]

    Petrovi´ c, M

    M. Petrovi´ c, M. Stankovi´ c, P. Peˇ ska,On conformal and concircular diffeomorphisms of Eisenhart’s generalized Rieman- nian spaces, Mathematics, 7, 626, (2019)

    work page 2019

  39. [39]

    Petrovi´ c, N

    M. Petrovi´ c, N. Vesi´ c, M. Zlatanovi´ c,Curvature properties of metric and semi-symmetric linear connections, Quaest. Math. 45(10) (2022) 1603-1627

    work page 2022

  40. [40]

    M. D. Siddiqi,Ricciρ-soliton and geometrical structure in a dust fluid and viscous fluid spacetime, Bulg. J. Phys. 46, 163-173, (2019). 13

    work page 2019

  41. [41]

    M. D. Siddiqi, U. C. De,Relativistic perfect fluid spacetime and Ricci-Yamabe solitons, Letters Math. Phys., 112(1), (2022)

    work page 2022

  42. [42]

    N. S. Sinyukov,Geodesic mappings of Riemannian spaces, Nauka, Moscow, (1979). (in Russian)

    work page 1979

  43. [43]

    A. A. Shaikh,On Lorentzian almost paracontact manifolds with a structure of the concircular type, Kyungpook Math. J. 43(2), 305-314, (2003)

    work page 2003

  44. [44]

    Slosarska,On some invariants of Riemannian manifold admitting a concircularly semi-symmetric metric connection, Demonstr

    W. Slosarska,On some invariants of Riemannian manifold admitting a concircularly semi-symmetric metric connection, Demonstr. Math., 17 (1), 251-257, (1984)

    work page 1984

  45. [45]

    Stavre,On theS-concircular andS-conharmonic connections, Tensor N.S., 38, 103-107, (1982)

    P. Stavre,On theS-concircular andS-conharmonic connections, Tensor N.S., 38, 103-107, (1982)

    work page 1982

  46. [46]

    S. E. Stepanov, I. A. Gordeeva,Pseudo-Killing and pseudoharmonic vector fields on a Riemann–Cartan manifold, Math. Notes, 87(2), 238-247, (2010)

    work page 2010

  47. [47]

    Stephani, D

    H. Stephani, D. Kramer, M. Mac-Callum, C. Hoenselaers, E. Hertl,Exact Solutions of Einstein’s Field Equations, 2nd edn., Cambridge Monographs on Mathematical Physics, Cambridge University Press, (2009)

    work page 2009

  48. [48]

    Y. J. Suh, S. K. Chaubey, M. N. I. Khan,Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection, Rev. Math. Phys. 36(03), 2450001, (2024)

    work page 2024

  49. [49]

    Yano,Concircular geometry I

    K. Yano,Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Tokyo, 16 (6), 195-200, (1940)

    work page 1940

  50. [50]

    Yano,On the torse-forming direction in a Riemannian spaces, Proc

    K. Yano,On the torse-forming direction in a Riemannian spaces, Proc. Imp. Acad. Tokyo, 20 (6), 340-345, (1944)

    work page 1944

  51. [51]

    K. Yano, S. Bochner,Curvature and Betti numbers, Annals of Mathematics Studies, 32 , Princeton University Press, (1954)

    work page 1954

  52. [52]

    H. B. Yilmaz,On pseudo Z-symmetric Lorentzian manifolds admitting a type of semi-symmetric metric connection, Anal. Math. Phys. 13, 18, (2023). Miroslav D. Maksimovi´ c University of Priˇ stina in Kosovska Mitrovica, Faculty of Sciences and Mathematics, Department of Mathematics, Kosovska Mitrovica, Serbia, and Institute of Mathematics and Informatics, Bu...

    work page 2023