Research Article
BibTex RIS Cite

$K$-Ricci-Bourguignon Almost Solitons

Year 2024, Volume: 17 Issue: 1, 63 - 71, 23.04.2024

U.c. De Krishnendu De

https://doi.org/10.36890/iejg.1434598

Abstract

We in this current article introduce and characterize a $K$-Ricci-Bourguignon almost solitons in perfect fluid spacetimes and generalized Robertson-Walker spacetimes. First, we demonstrate that if a perfect fluid spacetime admits a $K$-Ricci-Bourguignon almost soliton, then the integral curves produced by the velocity vector field are geodesics and the acceleration vector vanishes. Then we establish that if perfect fluid spacetimes permit a gradient $K$-Ricci-Bourguignon soliton with Killing velocity vector field, then either state equation of the perfect fluid spacetime is presented by $p=\frac{3-n}{n-1}\sigma$ , or the gradient $K$-Ricci-Bourguignon soliton is shrinking or expanding under some condition. Moreover, we illustrate that the spacetime represents a perfect fluid spacetime and the divergence of the Weyl tensor vanishes if a generalized Robertson-Walker spacetime admits a $K$-Ricci-Bourguignon almost soliton.

Keywords

Perfect fluid spacetimes GRW spacetimes $K$ Ricci-Bourguignon solitons

References

  • [1] Alías, L., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson- Walker spacetimes. Gen. Relativ. Gravit. 27, 71-84 (1995).
  • [2] Aubin, T.: Métriques riemanniennes et courbure. J. Differential Geometry. 4, 383–424 (1970).
  • [3] Barton, G.: Introduction to the Relativity Principle. John Wiley & Sons Inc. (1999).
  • [4] Blaga, A.M.: Solitons and geometrical structures in a perfect fluid spacetime. Rocky Mountain J. Math. 50, 41-43 (2020).
  • [5] Bourguignon, J.P.: Ricci curvature and Einstein metrics. Global differential geometry and global analysis. 42–63 (1981).
  • [6] Brozos-Vazquez, M., Garcia-Rio, E., Vazquez-Lorenzo, R.: Some remarks on locally conformally flat static space–times. Journal of Mathematical Physics. 46, 022501 (2005).
  • [7] Catino, G., Cremaschi, L., Djadli, Z., Mantezza, C., Mazzieri, L.: The Ricci- Bourguignon flow. Pac. J. Math. 287, 337370 ( 2017 ).
  • [8] Chavanis, P.H.: Cosmology with a stiff matter era. Phys. Rev. D. 92, 103004 (2015).
  • [9] Chen, B.Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications. World Scientific. (2011).
  • [10] Chen, B.Y.: A simple characterization of generalized Robertson-Walker spacetimes. Gen. Relativ. Gravit. 46, 1833 (2014).
  • [11] Chen, B.Y.: Some results on concircular vector felds and their applications to Ricci solitons. Bull. Korean Math. Soc. 52, 1535–1547 (2015).
  • [12] Chen, B.Y., Deshmukh, S.: Ricci solitons and concurrent vector fields. Balkan J. Geom. Appl. 20, 14-25 (2015).
  • [13] De, K., De, U.C.: Investigation on gradient solitons in perfect fluid spacetimes. Reports on Math. Phys. 91, 277-289 (2023).
  • [14] De, K., De, U.C.: Ricci-Yamabe solitons in f(R)-gravity. International Electronic Journal of Geometry. 16 (1), 334-342 (2023).
  • [15] De, K., De, U.C., Gezer, A.: Perfect fluid spacetimes and k-almost Yamabe solitons. Turk J Math. 47, 1236-1246 (2023) .
  • [16] De, K., Khan, M.N., De, U.C.: Characterizations of GRW spacetimes concerning gradient solitons. heliyon (2024). http://dx.doi.org/10.1016/j.heliyon.2024.e25702
  • [17] De, K., De, U.C., Syied, A.A., Turki, N.B., Alsaeed, S.: Perfect Fluid Spacetimes and Gradient Solitons. Journal of Nonlinear Mathematical Physics. 29, 843-858 (2022).
  • [18] De, U.C., Mantica, C.A., Suh, Y.J.: Perfect Fluid Spacetimes and Gradient Solitons. Filomat. 36, 829-842 (2022).
  • [19] Duggal, K.L., Sharma, R.: Symmetries of spacetimes and Riemannian manifolds. 487, Kluwer Academic Press, Boston, London (1999).
  • [20] Gebarowski, A.: Doubly warped products with harmonic Weyl conformal curvature tensor. Colloq. Math. 67, 73-89 (1994).
  • [21] Hamilton, R.S.: The Ricci flow on surfaces, Mathematics and general relativity. Contemp. Math. 71, 237-262 (1998).
  • [22] Hervik, S., Ortaggio, M., Wylleman, L.: Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension. Class. Quantum Grav. 30, 165014 (2013).
  • [23] Lovelock, D., Rund, H.: Tensors, differential forms, and variational principles, Courier Corporation. (1989).
  • [24] Mantica, C.A., Molinari, L.G.: Generalized Robertson-Walker spacetimes-A survey. Int. J. Geom. Methods Mod. Phys. 14, 1730001 (2017).
  • [25] Mantica, C.A., Molinari, L.G.: On the Weyl and the Ricci tensors of generalized Robertson–Walker spacetimes. J. Math. Phys. 57 (10), 102502 (2016).
  • [26] Sánchez, M.: On the geometry of generalized Robertson–Walker spacetimes: Curvature and killing fields. Gen. Relativ. Gravit. 31, 1-15 (1999).
  • [27] Stephani, H., Kramer, D., Mac-Callum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press. Cambridge, (2009).

Year 2024, Volume: 17 Issue: 1, 63 - 71, 23.04.2024

U.c. De Krishnendu De

https://doi.org/10.36890/iejg.1434598

Abstract

References

  • [1] Alías, L., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson- Walker spacetimes. Gen. Relativ. Gravit. 27, 71-84 (1995).
  • [2] Aubin, T.: Métriques riemanniennes et courbure. J. Differential Geometry. 4, 383–424 (1970).
  • [3] Barton, G.: Introduction to the Relativity Principle. John Wiley & Sons Inc. (1999).
  • [4] Blaga, A.M.: Solitons and geometrical structures in a perfect fluid spacetime. Rocky Mountain J. Math. 50, 41-43 (2020).
  • [5] Bourguignon, J.P.: Ricci curvature and Einstein metrics. Global differential geometry and global analysis. 42–63 (1981).
  • [6] Brozos-Vazquez, M., Garcia-Rio, E., Vazquez-Lorenzo, R.: Some remarks on locally conformally flat static space–times. Journal of Mathematical Physics. 46, 022501 (2005).
  • [7] Catino, G., Cremaschi, L., Djadli, Z., Mantezza, C., Mazzieri, L.: The Ricci- Bourguignon flow. Pac. J. Math. 287, 337370 ( 2017 ).
  • [8] Chavanis, P.H.: Cosmology with a stiff matter era. Phys. Rev. D. 92, 103004 (2015).
  • [9] Chen, B.Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications. World Scientific. (2011).
  • [10] Chen, B.Y.: A simple characterization of generalized Robertson-Walker spacetimes. Gen. Relativ. Gravit. 46, 1833 (2014).
  • [11] Chen, B.Y.: Some results on concircular vector felds and their applications to Ricci solitons. Bull. Korean Math. Soc. 52, 1535–1547 (2015).
  • [12] Chen, B.Y., Deshmukh, S.: Ricci solitons and concurrent vector fields. Balkan J. Geom. Appl. 20, 14-25 (2015).
  • [13] De, K., De, U.C.: Investigation on gradient solitons in perfect fluid spacetimes. Reports on Math. Phys. 91, 277-289 (2023).
  • [14] De, K., De, U.C.: Ricci-Yamabe solitons in f(R)-gravity. International Electronic Journal of Geometry. 16 (1), 334-342 (2023).
  • [15] De, K., De, U.C., Gezer, A.: Perfect fluid spacetimes and k-almost Yamabe solitons. Turk J Math. 47, 1236-1246 (2023) .
  • [16] De, K., Khan, M.N., De, U.C.: Characterizations of GRW spacetimes concerning gradient solitons. heliyon (2024). http://dx.doi.org/10.1016/j.heliyon.2024.e25702
  • [17] De, K., De, U.C., Syied, A.A., Turki, N.B., Alsaeed, S.: Perfect Fluid Spacetimes and Gradient Solitons. Journal of Nonlinear Mathematical Physics. 29, 843-858 (2022).
  • [18] De, U.C., Mantica, C.A., Suh, Y.J.: Perfect Fluid Spacetimes and Gradient Solitons. Filomat. 36, 829-842 (2022).
  • [19] Duggal, K.L., Sharma, R.: Symmetries of spacetimes and Riemannian manifolds. 487, Kluwer Academic Press, Boston, London (1999).
  • [20] Gebarowski, A.: Doubly warped products with harmonic Weyl conformal curvature tensor. Colloq. Math. 67, 73-89 (1994).
  • [21] Hamilton, R.S.: The Ricci flow on surfaces, Mathematics and general relativity. Contemp. Math. 71, 237-262 (1998).
  • [22] Hervik, S., Ortaggio, M., Wylleman, L.: Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension. Class. Quantum Grav. 30, 165014 (2013).
  • [23] Lovelock, D., Rund, H.: Tensors, differential forms, and variational principles, Courier Corporation. (1989).
  • [24] Mantica, C.A., Molinari, L.G.: Generalized Robertson-Walker spacetimes-A survey. Int. J. Geom. Methods Mod. Phys. 14, 1730001 (2017).
  • [25] Mantica, C.A., Molinari, L.G.: On the Weyl and the Ricci tensors of generalized Robertson–Walker spacetimes. J. Math. Phys. 57 (10), 102502 (2016).
  • [26] Sánchez, M.: On the geometry of generalized Robertson–Walker spacetimes: Curvature and killing fields. Gen. Relativ. Gravit. 31, 1-15 (1999).
  • [27] Stephani, H., Kramer, D., Mac-Callum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press. Cambridge, (2009).
There are 27 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

U.c. De 0000-0002-8990-4609

Krishnendu De 0000-0001-6520-4520

Early Pub Date April 5, 2024
Publication Date April 23, 2024
Submission Date February 9, 2024
Acceptance Date March 10, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA De, U., & De, K. (2024). $K$-Ricci-Bourguignon Almost Solitons. International Electronic Journal of Geometry, 17(1), 63-71. https://doi.org/10.36890/iejg.1434598
AMA De U, De K. $K$-Ricci-Bourguignon Almost Solitons. Int. Electron. J. Geom. April 2024;17(1):63-71. doi:10.36890/iejg.1434598
Chicago De, U.c., and Krishnendu De. “$K$-Ricci-Bourguignon Almost Solitons”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 63-71. https://doi.org/10.36890/iejg.1434598.
EndNote De U, De K (April 1, 2024) $K$-Ricci-Bourguignon Almost Solitons. International Electronic Journal of Geometry 17 1 63–71.
IEEE U. De and K. De, “$K$-Ricci-Bourguignon Almost Solitons”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 63–71, 2024, doi: 10.36890/iejg.1434598.
ISNAD De, U.c. - De, Krishnendu. “$K$-Ricci-Bourguignon Almost Solitons”. International Electronic Journal of Geometry 17/1 (April 2024), 63-71. https://doi.org/10.36890/iejg.1434598.
JAMA De U, De K. $K$-Ricci-Bourguignon Almost Solitons. Int. Electron. J. Geom. 2024;17:63–71.
MLA De, U.c. and Krishnendu De. “$K$-Ricci-Bourguignon Almost Solitons”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 63-71, doi:10.36890/iejg.1434598.
Vancouver De U, De K. $K$-Ricci-Bourguignon Almost Solitons. Int. Electron. J. Geom. 2024;17(1):63-71.
 Download Cover Image