Gradient conformal vector fields and their applications in vacuum static spaces

Abstract

The purpose of this work is to study the conformal geometry of vacuum static spaces with a constant scalar curvature, which corresponds to an interesting generalization of the critical point of the equation. We prove that a vacuum static space admitting a non-trivial gradient conformal vector field is isometric to a Euclidean space or to a warped product $\mathrm{I}\times_\varphi\mathfrak{L}^{m-1}$ of an interval $\mathrm{I}$ and an Einstein manifold $\mathfrak{L}^{m-1}$, provided that scalar curvature of $\mathfrak{L}^{m-1}$ is constant.

Keywords

• vacuum static spaces
• closed conformal vector field
• gradient conformal vector field
• static space-time
• perfect fluid

Supporting Institution

The authors would like to express their sincere thanks to the referee for the valuable suggestions for the improvement of the paper.

Project Number

RGP2/297/46

References

1. [1] Ali, A., Alshehri, N., Mofarreh, F., and Li, Y., Geometry of gradient Einstein harmonic solitons in sequential warped products manifolds, Eur. Phys. J. Plus, 139 (4), 1–8, 2024.
2. [2] Ali, A., Mofarreh, F. and Lee, J.W., Classifications of gradient almost Ricci solitons within the framework of closed conformal vector fields, Turkish J. Math. 49 (6), 714– 726, 2025.
3. [3] Ali, A., Mofarreh, F. and Patra, D.S., Geometry of almost Ricci solitons on paracontact metric manifolds, Quaest. Math. 45 (8), 1167–1180, 2022.
4. [4] Alias, L. J., Caminha, A. and do Nascimento, F. Y., On complete Kaehlerian manifolds endowed with closed conformal vector fields, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 117(3), Paper No. 127, 2023.
5. [5] Amur, K. and Hedge, V. S., Conformality of Riemannian manifolds to spheres, J. Differential Geom. 9 (4), 571–576, 1974.
6. [6] Aubin, T., Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55, 269–296, 1976.
7. [7] Besse, A. L., Einstein Manifolds, Springer-Verlag, New York, 2008.
8. [8] Caminha, A., The geometry of closed conformal vector fields on Riemannian spaces, Bull. Braz. Math. Soc. (N.S.), 42, 277–300, 2011.

9. [9] Carroll, S. M., Spacetime and Geometry: An Introduction to General Relativity, Addison-Wesley, 2004.
10. [10] Case, J., Shu, Y.-J. and Wei, G., Rigidity of quasi-Einstein metrics, Differential Geom. Appl., 29 (1), 93–100, 2011.
11. [11] Castro, I., Montealegre, C. R., and Urbano, F., Closed conformal vector fields and Lagrangian submanifolds in complex space forms, Pacific J. Math. 199 (2), 269–302, 2001.
12. [12] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), 144 (1), 189–237, 1996.
13. [13] Da Silva Filho, J. F., Critical point equation and closed conformal vector fields, Math. Nachr. 293 (12), 2299–2305, 2020.
14. [14] Da Silva Filho, J. F., Quasi-Einstein manifolds admitting a closed conformal vector field, Differential Geom. Appl. 92, Paper No. 102083, 2024.
15. [15] Deshmukh, S., Characterizing spheres and Euclidean spaces by conformal vector fields, Ann. Mat. Pura Appl. 196 (6), 2135–2145, 2017.
16. [16] Deshmukh, S. and Al-Solamy, F. R., Conformal gradient vector fields on a compact Riemannian manifold, Colloquium Math. 112 (1), 157–161, 2008.
17. [17] Diógenes, R., Ribeiro Jr. E. and Silva Filho, J., Gradient Ricci solitons admitting a closed conformal vector field, J. Math. Anal. Appl. 455, 1975–1983, 2017.
18. [18] Hwang, S. and Yun, G., Vacuum static spaces and conformal vector fields, arXiv:2308.04927, 2023.
19. [19] Hwang, S. and Yun, G., Conformal vector fields and their applications to Einstein-type manifolds, Results Math. 79 (1), 45, 2024.
20. [20] Kim, D.-S. and Kim, Y. H., Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc. 131 (8), 2573–2576, 2003.
21. [21] Misner, C. W., Thorne, K. S. and Wheeler, J. A., Gravitation, W. H. Freeman, 1993.
22. [22] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
23. [23] Obata, M., Conformal transformations of Riemannian manifolds, J. Differential Geom., 4 (3), 311–333, 1970.
24. [24] Obata, M. and Yano, K., Conformal changes of Riemannian metrics, J. Differential Geom., 4 (1), 53–72, 1970.
25. [25] Qing, J. and Yuan, W., A note on static spaces and related problems, J. Geom. Phys. 74, 18–27, 2013.
26. [26] Petersen, P. and Wylie, W., On the classification of gradient Ricci solitons, Geom. Topol. 14 (4), 2277–2300, 2010.
27. [27] Patra, D.S., Ali, A. and Mofarreh, F., Characterizations of Ricci-Bourguignon almost solitons on pseudo-Riemannian manifolds, Mediterr. J. Math. 19 (4), 176, 2022.
28. [28] Poddar, R., Balasubramanian, S. and Sharma, R., Generalized m-quasi-Einstein manifolds admitting a closed conformal vector field, Ann. Mat. Pura Appl. 202 (6), 2687– 2699, 2023.
29. [29] Ros, A. and Urbano, F., Lagrangian submanifolds of $\mathbf{C}^{n}$ with conformal Maslov form and the Whitney sphere, J. Math. Soc. Japan, 50 (1), 203–226, 1998.
30. [30] Stephani, H., Exact solutions of Einstein’s field equations, Cambridge University Press, 2004.
31. [31] Suh, Y. J., Ali, A., Alshehri, N. and Mofarreh, F., On gradient Ricci-Bourguignon harmonic solitons in sequential warped products, Filomat, 39 (21), 7347–7357, 2025.
32. [32] Tashiro, Y., Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117, 251–275, 1965.
33. [33] Wald, R. M., General Relativity, University of Chicago Press, 2010.