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It is known that corresponding to each isometry there exist a conservedquantity. It is also known that the Lagrangian of the line element of a spaceis conserved. Here we investigate the possibility of the existence of "new"conserved quantities, i.e. other than the Lagrangian and associated with theisometries, for spaces of different curvatures. It is found that there existnew conserved quantities only for the spaces of zero curvature or having asection of zero curvature.

On the Existence of New Conservation Laws for the Spaces of Different Curvatures