-
Previous Article
Onset of Benard-Marangoni instabilities in a double diffusive binary fluid layer with temperature-dependent viscosity
- NACO Home
- This Issue
-
Next Article
Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate
Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation
Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, P.O.Box 1993893973, Iran |
Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra $ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.
References:
[1] |
A. Ali Al-Eid, Conformal Deformation of a Riemannian Metric, M.Sc. Thesis, 1423, 2001. Google Scholar |
[2] |
W. Batat,
Curvature properties and Ricci solitons of Lorentzian pr-wave manifolds, J. Geom. Phys., 75 (2014), 7-16.
doi: 10.1016/j.geomphys.2013.08.014. |
[3] |
A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Springer-Verlag, Netherlands, 2006. |
[4] |
A. Bejancu and H. R. Farran, Geometry of Pseudo-Finslerian Submanifolds, Springer, Netherlands, 2000.
doi: 10.1007/978-94-015-9417-2. |
[5] |
H. W. Brinkmann,
Einstein spaces which are mapped conformally on each other, Math. Ann., 94 (1925), 119-145.
doi: 10.1007/BF01208647. |
[6] |
G. de Rham,
Sur la r$\acute{ e }$ducibilit$\acute{ e }$ d'un espace de Riemann, Math. Helv., 26 (1952), 328-344.
doi: 10.1007/BF02564308. |
[7] |
A. Gray,
Einstein-Like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259-280.
doi: 10.1007/BF00151525. |
[8] |
W. K$\ddot{ u }$hnel,
Conformal transformation between Einstein spaces, Aspects of Math., E12 (1988), 105-146.
|
[9] |
W. K$\ddot{ u }$hnel and H.B. Rademacher,
Conformal diffeomorphisms preserving the Ricci tensor, Proc. of Amer. Math. Soc., 123 (1995), 2841-2848.
doi: 10.2307/2160584. |
[10] |
T. Leistner,
Screen bundles of Lorentzian manifolds and some generalizations of pp-waves, J. Geom. Phys., 56 (2006), 2117-2134.
doi: 10.1016/j.geomphys.2005.11.010. |
[11] |
T. Leistner, Holonomy and Parallel Spinors in Lorentzian Geometry, Logos Verlag, 2004. Google Scholar |
[12] |
M. M. Rezaii and A. Alipour-Fakhri,
On projectively related warped product finsler manifolds, Int. J. Geom. Methods Mod. Phys., 8 (2011), 953-967.
doi: 10.1142/S0219887811005464. |
[13] |
R. Schimming,
Riemannsche R$\ddot{ a }$ume mit ebenfrontiger und mit ebener symmetrie, Math. Nachr., 59 (1974), 128-162.
doi: 10.1002/mana.19740590111. |
[14] |
H. Wu,
On the de Rham decomposition theorem, Illinois J. Math., 8 (1964), 291-311.
|
show all references
References:
[1] |
A. Ali Al-Eid, Conformal Deformation of a Riemannian Metric, M.Sc. Thesis, 1423, 2001. Google Scholar |
[2] |
W. Batat,
Curvature properties and Ricci solitons of Lorentzian pr-wave manifolds, J. Geom. Phys., 75 (2014), 7-16.
doi: 10.1016/j.geomphys.2013.08.014. |
[3] |
A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Springer-Verlag, Netherlands, 2006. |
[4] |
A. Bejancu and H. R. Farran, Geometry of Pseudo-Finslerian Submanifolds, Springer, Netherlands, 2000.
doi: 10.1007/978-94-015-9417-2. |
[5] |
H. W. Brinkmann,
Einstein spaces which are mapped conformally on each other, Math. Ann., 94 (1925), 119-145.
doi: 10.1007/BF01208647. |
[6] |
G. de Rham,
Sur la r$\acute{ e }$ducibilit$\acute{ e }$ d'un espace de Riemann, Math. Helv., 26 (1952), 328-344.
doi: 10.1007/BF02564308. |
[7] |
A. Gray,
Einstein-Like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259-280.
doi: 10.1007/BF00151525. |
[8] |
W. K$\ddot{ u }$hnel,
Conformal transformation between Einstein spaces, Aspects of Math., E12 (1988), 105-146.
|
[9] |
W. K$\ddot{ u }$hnel and H.B. Rademacher,
Conformal diffeomorphisms preserving the Ricci tensor, Proc. of Amer. Math. Soc., 123 (1995), 2841-2848.
doi: 10.2307/2160584. |
[10] |
T. Leistner,
Screen bundles of Lorentzian manifolds and some generalizations of pp-waves, J. Geom. Phys., 56 (2006), 2117-2134.
doi: 10.1016/j.geomphys.2005.11.010. |
[11] |
T. Leistner, Holonomy and Parallel Spinors in Lorentzian Geometry, Logos Verlag, 2004. Google Scholar |
[12] |
M. M. Rezaii and A. Alipour-Fakhri,
On projectively related warped product finsler manifolds, Int. J. Geom. Methods Mod. Phys., 8 (2011), 953-967.
doi: 10.1142/S0219887811005464. |
[13] |
R. Schimming,
Riemannsche R$\ddot{ a }$ume mit ebenfrontiger und mit ebener symmetrie, Math. Nachr., 59 (1974), 128-162.
doi: 10.1002/mana.19740590111. |
[14] |
H. Wu,
On the de Rham decomposition theorem, Illinois J. Math., 8 (1964), 291-311.
|
[1] |
Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017 |
[2] |
Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001 |
[3] |
Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123 |
[4] |
Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021004 |
[5] |
Editorial Office. Retraction: Xiaohong Zhu, Zili Yang and Tabharit Zoubir, Research on the matching algorithm for heterologous image after deformation in the same scene. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1281-1281. doi: 10.3934/dcdss.2019088 |
[6] |
Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020464 |
[7] |
Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269 |
[8] |
Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 |
[9] |
Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320 |
[10] |
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 |
[11] |
Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer. Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1111-1122. doi: 10.3934/dcdss.2020383 |
[12] |
Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 |
[13] |
Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020408 |
[14] |
Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311 |
[15] |
Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020031 |
[16] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[17] |
Ömer Arslan, Selçuk Kürşat İşleyen. A model and two heuristic methods for The Multi-Product Inventory-Location-Routing Problem with heterogeneous fleet. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021002 |
[18] |
Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004 |
[19] |
Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011 |
[20] |
Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020377 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]