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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 $. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.
References:
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A. Ali Al-Eid, Conformal Deformation of a Riemannian Metric, M.Sc. Thesis, 1423, 2001. |
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W. Batat,
Curvature properties and Ricci solitons of Lorentzian pr-wave manifolds, J. Geom. Phys., 75 (2014), 7-16.
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A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Springer-Verlag, Netherlands, 2006. |
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A. Bejancu and H. R. Farran, Geometry of Pseudo-Finslerian Submanifolds, Springer, Netherlands, 2000.
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H. W. Brinkmann,
Einstein spaces which are mapped conformally on each other, Math. Ann., 94 (1925), 119-145.
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G. de Rham,
Sur la r$\acute{ e }$ducibilit$\acute{ e }$ d'un espace de Riemann, Math. Helv., 26 (1952), 328-344.
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A. Gray,
Einstein-Like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259-280.
doi: 10.1007/BF00151525. |
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W. K$\ddot{ u }$hnel,
Conformal transformation between Einstein spaces, Aspects of Math., E12 (1988), 105-146.
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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. |
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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. |
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T. Leistner, Holonomy and Parallel Spinors in Lorentzian Geometry, Logos Verlag, 2004. |
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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. |
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R. Schimming,
Riemannsche R$\ddot{ a }$ume mit ebenfrontiger und mit ebener symmetrie, Math. Nachr., 59 (1974), 128-162.
doi: 10.1002/mana.19740590111. |
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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. |
| [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. |
| [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.
|
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