-
Previous Article
Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems
- JGM Home
- This Issue
- Next Article
A family of multiply warped product semi-Riemannian Einstein metrics
| 1. | Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India |
In this paper, we characterize multiply warped product semi -Riemannian manifolds when the base is conformal to an $ n $-dimensional pseudo-Euclidean space. We prove some conditions on warped product semi- Riemannian manifolds to be an Einstein manifold which is invariant under the action of an $ (n-1) $-dimensional translation group. After that we apply this result for the case of Ricci-flat multiply warped product space when the fibers are Ricci-flat. We also discuss the existence of infinitely many Ricci-flat multiply warped product spaces under the same action with null like vector.
References:
| [1] |
J. K. Beem and P. Ehrlich, Global Lorentzian Geometry, 1$^{st}$ edition, Markel-Deccer, New York, 1981. |
| [2] |
J. K. Beem, P. Ehrlich and K. Easley, Global Lorentzian Geometry, 2$^{nd}$ edition, Markel-Deccer, New York, 1996. |
| [3] |
A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., 3, Berlin, Heidelberg, New York, Springer Verlag, 1987.
doi: 10.1007/978-3-540-74311-8. |
| [4] |
R. L. Bishop and B. O'Neill,
Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
doi: 10.1090/S0002-9947-1969-0251664-4. |
| [5] |
P. G. C. Bonfim and R. Pina, Quasi-Einstein manifolds with structure of warped product, arXiv: 1905.02006.
doi: 128.84.4.27/1905.02006. |
| [6] |
M. Brózos-Vázquez, E. García-Río and R. Vázquez-Lorenzo, Some remarks on locally conformally flat static space-times, Journal of Mathematical Physics, 46 (2005), 022501.
doi: 10.1063/1.1832755. |
| [7] |
M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo,
Complete locally conformally flat manifolds of negative curvature, Pac. J. Math., 226 (2006), 201-219.
doi: 10.2140/pjm.2006.226.201. |
| [8] |
M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo, Warped product metrics and locally conformally flat structures, Matemática Contemporânea, in: SBM, 28 (2000), 91–110. |
| [9] |
Q. Chen and C. He,
On Bach flat warped product Einstein manifolds, Pac. J. Math., 265 (2013), 313-326.
doi: 10.2140/pjm.2013.265.313. |
| [10] |
J. Choi, Multiply warped products with nonsmooth metrics, Journal of Mathematical Physics, 41 (2000), 8163.
doi: 10.1063/1.1287432. |
| [11] |
F. Dobarro and B. Unal,
$\ddot{U}nal$, Curvature of multiply warped products, Journal of Geometry and Physics, 55 (2005), 75-106.
doi: 10.1016/j.geomphys.2004.12.001. |
| [12] |
D. Dumitru, On multiply Einstein warped products, Scientific Annals of the Alexandru Loan Cuza University of Laşi (New Series). Mathematics, Tomul LXII, 2016, F.1. |
| [13] |
F. Gholami, F. Darabi and A. Haji-Badali, Multiply-warped product metrices and reduction of Einstein equations, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750021.
doi: 10.1142/S0219887817500219. |
| [14] |
C. He, P. Petersen and W. Wylie,
On the classification of warped product Einstein metrics, Comm. Anal. Geom., 20 (2012), 271-312.
doi: 10.4310/CAG.2012.v20.n2.a3. |
| [15] |
B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied
Mathematics, 103, Academic Press, Inc., New York, 1983. |
| [16] |
S. Pahan, B. Pal and A. Bhattacharyya,
On Ricci flat warped products with a quarter-symmetric connection, J. Geom., 107 (2016), 627-634.
doi: 10.1007/s00022-015-0301-3. |
| [17] |
S. Pahan, B. Pal and A. Bhattacharyya, On Einstein warped products with a quarter-symmetric connection, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750050.
doi: 10.1142/S0219887817500505. |
| [18] |
M. L. D. Sousa and R. Pina,
A family of warped product semi-Riemannian Einstein metrics, Differential Geometry and its Applications, 50 (2017), 105-115.
doi: 10.1016/j.difgeo.2016.11.004. |
| [19] |
B. Unal,
Multiply warped products, J. Geom. Phys., 34 (2000), 287-301.
doi: 10.1016/S0393-0440(99)00072-8. |
show all references
References:
| [1] |
J. K. Beem and P. Ehrlich, Global Lorentzian Geometry, 1$^{st}$ edition, Markel-Deccer, New York, 1981. |
| [2] |
J. K. Beem, P. Ehrlich and K. Easley, Global Lorentzian Geometry, 2$^{nd}$ edition, Markel-Deccer, New York, 1996. |
| [3] |
A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., 3, Berlin, Heidelberg, New York, Springer Verlag, 1987.
doi: 10.1007/978-3-540-74311-8. |
| [4] |
R. L. Bishop and B. O'Neill,
Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.
doi: 10.1090/S0002-9947-1969-0251664-4. |
| [5] |
P. G. C. Bonfim and R. Pina, Quasi-Einstein manifolds with structure of warped product, arXiv: 1905.02006.
doi: 128.84.4.27/1905.02006. |
| [6] |
M. Brózos-Vázquez, E. García-Río and R. Vázquez-Lorenzo, Some remarks on locally conformally flat static space-times, Journal of Mathematical Physics, 46 (2005), 022501.
doi: 10.1063/1.1832755. |
| [7] |
M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo,
Complete locally conformally flat manifolds of negative curvature, Pac. J. Math., 226 (2006), 201-219.
doi: 10.2140/pjm.2006.226.201. |
| [8] |
M. Brozos-Vázquez, E. Garcia-Rio and R. Vázquez-Lorenzo, Warped product metrics and locally conformally flat structures, Matemática Contemporânea, in: SBM, 28 (2000), 91–110. |
| [9] |
Q. Chen and C. He,
On Bach flat warped product Einstein manifolds, Pac. J. Math., 265 (2013), 313-326.
doi: 10.2140/pjm.2013.265.313. |
| [10] |
J. Choi, Multiply warped products with nonsmooth metrics, Journal of Mathematical Physics, 41 (2000), 8163.
doi: 10.1063/1.1287432. |
| [11] |
F. Dobarro and B. Unal,
$\ddot{U}nal$, Curvature of multiply warped products, Journal of Geometry and Physics, 55 (2005), 75-106.
doi: 10.1016/j.geomphys.2004.12.001. |
| [12] |
D. Dumitru, On multiply Einstein warped products, Scientific Annals of the Alexandru Loan Cuza University of Laşi (New Series). Mathematics, Tomul LXII, 2016, F.1. |
| [13] |
F. Gholami, F. Darabi and A. Haji-Badali, Multiply-warped product metrices and reduction of Einstein equations, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750021.
doi: 10.1142/S0219887817500219. |
| [14] |
C. He, P. Petersen and W. Wylie,
On the classification of warped product Einstein metrics, Comm. Anal. Geom., 20 (2012), 271-312.
doi: 10.4310/CAG.2012.v20.n2.a3. |
| [15] |
B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied
Mathematics, 103, Academic Press, Inc., New York, 1983. |
| [16] |
S. Pahan, B. Pal and A. Bhattacharyya,
On Ricci flat warped products with a quarter-symmetric connection, J. Geom., 107 (2016), 627-634.
doi: 10.1007/s00022-015-0301-3. |
| [17] |
S. Pahan, B. Pal and A. Bhattacharyya, On Einstein warped products with a quarter-symmetric connection, International Journal of Geometric Methods in Modern Physics, 14 (2017), 1750050.
doi: 10.1142/S0219887817500505. |
| [18] |
M. L. D. Sousa and R. Pina,
A family of warped product semi-Riemannian Einstein metrics, Differential Geometry and its Applications, 50 (2017), 105-115.
doi: 10.1016/j.difgeo.2016.11.004. |
| [19] |
B. Unal,
Multiply warped products, J. Geom. Phys., 34 (2000), 287-301.
doi: 10.1016/S0393-0440(99)00072-8. |
| [1] |
Federico Cacciafesta, Anne-Sophie De Suzzoni. Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4359-4398. doi: 10.3934/dcds.2019177 |
| [2] |
Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97. |
| [3] |
Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20. |
| [4] |
Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks and Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021 |
| [5] |
Brian Smith and Gilbert Weinstein. On the connectedness of the space of initial data for the Einstein equations. Electronic Research Announcements, 2000, 6: 52-63. |
| [6] |
Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015 |
| [7] |
Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098 |
| [8] |
Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799 |
| [9] |
Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173 |
| [10] |
Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905 |
| [11] |
Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56. |
| [12] |
Jordi Gaset, Narciso Román-Roy. New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity. Journal of Geometric Mechanics, 2019, 11 (3) : 361-396. doi: 10.3934/jgm.2019019 |
| [13] |
Jongmin Han, Juhee Sohn. On the self-dual Einstein-Maxwell-Higgs equation on compact surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 819-839. doi: 10.3934/dcds.2019034 |
| [14] |
Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651 |
| [15] |
Uta Renata Freiberg. Einstein relation on fractal objects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 509-525. doi: 10.3934/dcdsb.2012.17.509 |
| [16] |
Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems and Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1 |
| [17] |
Simone Calogero, Stephen Pankavich. On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field. Kinetic and Related Models, 2018, 11 (5) : 1063-1083. doi: 10.3934/krm.2018041 |
| [18] |
César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067 |
| [19] |
Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 |
| [20] |
Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223 |
2020 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]