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December  2019, 9(4): 401-412. doi: 10.3934/naco.2019039

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

* Corresponding author

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

Received  December 2017 Revised  September 2018 Published  August 2019

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.

Citation: Fatemeh Ahangari. Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 401-412. doi: 10.3934/naco.2019039
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. 

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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