A new basis of the conformal algebra is proposed, which makes appear two conjugated Schrδdinger algebras. This basis allows to exhibit a chain which does not contain the Poincare algebra, between the £^$(4,2) algebra and the (two-dimensional) extended Galilean one. This non-relativistic structure of the conformal algebra is well adapted to discuss some extreme models of hadrons based on collinear massless particles.
A new geometry is constructed over Galilean manifolds expressing the compatibility requirement between the conformal equivalence notion of two Galilean structures and the projective equivalence notion of two affine connections. It is shown that it is the very geometry of the Newtonian cosmology (chronoprojective flatness is equivalent to isotropy of Newtonian cosmological models); moreover, it also explains various ‘‘accidental’’ symmetries in classical mechanics.
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