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

TL;DR

This paper investigates the potential for new conserved quantities in spaces of different curvatures, finding such quantities only in zero curvature spaces or those with a zero curvature section.

Contribution

It demonstrates the existence of new conserved quantities beyond known isometry-related ones, specifically in zero curvature or partially zero curvature spaces.

Findings

New conserved quantities exist only in zero curvature spaces.

Spaces with a section of zero curvature also admit new conserved quantities.

Conservation laws are limited to specific curvature conditions.

Abstract

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