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(${\rm so}(n+1)\oplus d_{2}$ or ${\rm so}(n,1)\oplus d_{2}$ (where d ₂ is the two-dimensional 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 non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by $h\oplus d_{2}$, 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 $h\oplus {\rm sl}(m+2)$).