• 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
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 & 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. 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.  Google Scholar

[3]

A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Springer-Verlag, Netherlands, 2006.  Google Scholar

[4]

A. Bejancu and H. R. Farran, Geometry of Pseudo-Finslerian Submanifolds, Springer, Netherlands, 2000. doi: 10.1007/978-94-015-9417-2.  Google Scholar

[5]

H. W. Brinkmann, Einstein spaces which are mapped conformally on each other, Math. Ann., 94 (1925), 119-145.  doi: 10.1007/BF01208647.  Google Scholar

[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.  Google Scholar

[7]

A. Gray, Einstein-Like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259-280.  doi: 10.1007/BF00151525.  Google Scholar

[8]

W. K$\ddot{ u }$hnel, Conformal transformation between Einstein spaces, Aspects of Math., E12 (1988), 105-146.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Wu, On the de Rham decomposition theorem, Illinois J. Math., 8 (1964), 291-311.   Google Scholar

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.  Google Scholar

[3]

A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Springer-Verlag, Netherlands, 2006.  Google Scholar

[4]

A. Bejancu and H. R. Farran, Geometry of Pseudo-Finslerian Submanifolds, Springer, Netherlands, 2000. doi: 10.1007/978-94-015-9417-2.  Google Scholar

[5]

H. W. Brinkmann, Einstein spaces which are mapped conformally on each other, Math. Ann., 94 (1925), 119-145.  doi: 10.1007/BF01208647.  Google Scholar

[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.  Google Scholar

[7]

A. Gray, Einstein-Like manifolds which are not Einstein, Geom. Dedicata, 7 (1978), 259-280.  doi: 10.1007/BF00151525.  Google Scholar

[8]

W. K$\ddot{ u }$hnel, Conformal transformation between Einstein spaces, Aspects of Math., E12 (1988), 105-146.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. Wu, On the de Rham decomposition theorem, Illinois J. Math., 8 (1964), 291-311.   Google Scholar

[1]

Federico Cacciafesta, Anne-Sophie De Suzzoni. Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4359-4398. doi: 10.3934/dcds.2019177

[2]

Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

[3]

Patrick Foulon, Vladimir S. Matveev. Zermelo deformation of finsler metrics by killing vector fields. Electronic Research Announcements, 2018, 25: 1-7. doi: 10.3934/era.2018.25.001

[4]

Igor Rivin and Jean-Marc Schlenker. The Schlafli formula in Einstein manifolds with boundary. Electronic Research Announcements, 1999, 5: 18-23.

[5]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[6]

Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139

[7]

H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549

[8]

Yacine Aït Amrane, Rafik Nasri, Ahmed Zeglaoui. Warped Poisson brackets on warped products. Journal of Geometric Mechanics, 2014, 6 (3) : 279-296. doi: 10.3934/jgm.2014.6.279

[9]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335

[10]

Ali Hyder, Luca Martinazzi. Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 283-299. doi: 10.3934/dcds.2015.35.283

[11]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303

[12]

Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313

[13]

Jinghong Liu, Yinsuo Jia. Gradient superconvergence post-processing of the tensor-product quadratic pentahedral finite element. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 495-504. doi: 10.3934/dcdsb.2015.20.495

[14]

Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003

[15]

Sebastian Acosta. A control approach to recover the wave speed (conformal factor) from one measurement. Inverse Problems & Imaging, 2015, 9 (2) : 301-315. doi: 10.3934/ipi.2015.9.301

[16]

Julien Chambarel, Christian Kharif, Olivier Kimmoun. Focusing wave group in shallow water in the presence of wind. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 773-782. doi: 10.3934/dcdsb.2010.13.773

[17]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

[18]

Andrei Agrachev, Ugo Boscain, Mario Sigalotti. A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 801-822. doi: 10.3934/dcds.2008.20.801

[19]

Ludovic Rifford. Ricci curvatures in Carnot groups. Mathematical Control & Related Fields, 2013, 3 (4) : 467-487. doi: 10.3934/mcrf.2013.3.467

[20]

Tracy L. Payne. The Ricci flow for nilmanifolds. Journal of Modern Dynamics, 2010, 4 (1) : 65-90. doi: 10.3934/jmd.2010.4.65

 Impact Factor: 

Metrics

  • PDF downloads (115)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]