Abstract
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(n+1)⊕ d2 or so(n,1)⊕ d2 (where d 2 is the two-dimensional dilation algebra), while for those admitting so(n)⊕s ℝn (where ⊕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 non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by h ⊕ d 2, provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes h ⊕ sl(m+2).
| Original language | English |
|---|---|
| Pages (from-to) | 65-74 |
| Number of pages | 10 |
| Journal | Nonlinear Dynamics |
| Volume | 45 |
| Issue number | 1-2 |
| DOIs | |
| State | Published - Jul 2006 |
Bibliographical note
Funding Information:TF and AQ are grateful to the Higher Education Commission of Pakistan for providing an enabling grant for this work. FM thanks the Department of Mathematics of Quaid-i-Azam University for its warm hospitality during the course of this work. We are also grateful to AV Aminova for making available to us some of the results that appeared in the cited papers, which we would not otherwise have been able to see.
Keywords
- Geodesic equations
- Isometries
- Metric
- Symmetries
ASJC Scopus subject areas
- Control and Systems Engineering
- Aerospace Engineering
- Ocean Engineering
- Mechanical Engineering
- Electrical and Electronic Engineering
- Applied Mathematics