The Connection Between Isometries and Symmetries of Geodesic Equations of the Underlying Spaces

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) ⊕ d 2 or so(n, 1) ⊕ d 2 (where d 2 is the two-dimensional dilation algebra), while for those admitting so(n) ⊕ s R n (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).

Geometric Linearization of Ordinary Differential Equations

“…A conjecture that was at the base of the connection between geometry and symmetry analysis found by Feroze et al [15] was stated and proved as a theorem. The proof given demonstrates that there is at least one extra symmetry generator of the Lagrangian apart from the isometry algebra, and that the symmetry algebra of the geodesic equations contains the symmetry algebra of the Lagrangian.…”

“…In this paper a review has been provided of the geometric approach to symmetry analysis using the connection between symmetries of geometry and of differential equations provided by the system of geodesic equations, noted by Aminova and Aminov [13] and separately by Feroze et al [15]. This leads to a new approach to investigate the linearizability of ODEs.…”

“…In [15] maximal curvature spaces were considered and it was found that for flat spaces of dimension n the symmetry algebra is sl(n+2, R), for positive curvature so(n+1), or for negative curvature a corresponding non-compact version. There was no result for spaces that are not maximally symmetric.…”

“…(A recent paper [14], and references therein, detail relations between projective algebra (point symmetries of a particular class of second order ODEs) and algebra of isometries, providing a more complete background of geometry for the purpose.) Independently, the same idea was developed and carried further to construct the Lie algebra of the system of geodesic equations on maximally symmetric spaces [15]. There a conjecture was stated for all spaces.…”

Killing and Noether Symmetries of Plane Symmetric Spacetime

“…The importance of Noether symmetries can be seen from the fact that these symmetries may recover some lost conservation laws and symmetry generators of spacetimes [13]. Sharif and Saira [14] calculated the energy contents of colliding plane waves by using approximate symmetries.…”