The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces
 Tooba Feroze,
 F. M. Mahomed,
 Asghar Qadir
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so( \({\rm so}(n+1)\oplus d_{2}\) or \({\rm so}(n,1)\oplus d_{2}\) (where d _{2} is the twodimensional dilation algebra), while for those admitting \({\rm so}(n)\oplus_{\rm s}\mathbb{R}^{n}\) (where \(\oplus_{\rm s}\) represents semidirect sum) the algebra is sl(n+2). A corresponding result holds on replacing so(n) by so(p,q) with p+q = n. It is conjectured that if the isometry algebra of any underlying space of nonzero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by \(h\oplus d_{2}\) , provided that there is no crosssection of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes \(h\oplus {\rm sl}(m+2)\) ).
 Hawking, S. W., Ellis, G. F. R. (1973) The Large Scale Structure of SpaceTime. Cambridge University Press, Cambridge
 Ibragimov, N.H. (1999) Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley, Chichester
 Olver, P. J. (1993) Applications of Lie Groups to Differential Equations. SpringerVerlag, New York
 Stephani, H. (1989) Differential Equations: Their Solutions Using Symmetry. Cambridge University Press, New York
 Caviglia, G. (1983) Dynamical symmetries: An approach to Jacobi fields and to constants of geodesic motion. Journal of Mathematical Physics 24: pp. 20652069 CrossRef
 Hojman, S., Nunez, L., Patino, A., Rago, H. (1986) Symmetries and conserved quantities in geodesic motion. Journal of Mathematical Physics 27: pp. 281286 CrossRef
 Katzin, G. H., Levine, J. (1981) Geodesic first integrals with explicit pathparameter dependence in Riemannian spacetimes. Journal of Mathematical Physics 22: pp. 18781891 CrossRef
 Aminova, A. V. (1995) Projective transformations and symmetries of differential equations. Sbornik Mathematics 186: pp. 17111726 CrossRef
 Aminova, A. V., ‘Automorphisms of geometric structures as symmetries of differential equations (in Russian),’ Izv.Vyssh.Uchebn.Zaved.Mat. 2, 1994, 3–10; Aminova, A. V. and Aminav, N., ‘Projective geometry of systems of differential equations: General conceptions,’ Tensor, N.S. 62(1), 2000, 65–86.
 Aminova, A. V. (1978) Concircular vector fields and group symmetries in worlds of constant curvature (in Russian). Gravit.i Teoriya Otnotisel'n 14: pp. 416
 Feroze, T., ‘Some aspects of symmetries of differential equations and their connection with the underlying geometry,’ Ph.D. Thesis, QuaidiAzam University, Pakistan, 2005; Feroze, T. and Qadir, A., ‘Symmetries of second order vector differential equations,’ Preprint, Department of Mathematics, QuaidiAzam University, Pakistan, 2004.
 Title
 The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces
 Journal

Nonlinear Dynamics
Volume 45, Issue 12 , pp 6574
 Cover Date
 20060701
 DOI
 10.1007/s110710060729y
 Print ISSN
 0924090X
 Online ISSN
 1573269X
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 geodesic equations
 isometries
 metric
 symmetries
 Industry Sectors
 Authors

 Tooba Feroze ^{(1)} ^{(2)}
 F. M. Mahomed ^{(3)}
 Asghar Qadir ^{(2)} ^{(4)}
 Author Affiliations

 1. Department of Mathematics, QuaidiAzam University, Islamabad, Pakistan
 2. Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan
 3. Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, P.O. Wits 2050, South Africa
 4. Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dahran, Saudi Arabia