, Volume 45, Issue 1, pp 6574
First online:
The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces
 Tooba FerozeAffiliated withDepartment of Mathematics, QuaidiAzam UniversityCentre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of College of Electrical and Mechanical Engineering
 , F. M. MahomedAffiliated withCentre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand Email author
 , Asghar QadirAffiliated withCentre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of College of Electrical and Mechanical EngineeringDepartment of Mathematical Sciences, King Fahd University of Petroleum and Minerals
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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 \(h\oplus {\rm sl}(m+2)\)).
Keywords
geodesic equations isometries metric symmetries 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
 200607
 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