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The concept of centre of mass of two particles in 2D spaces of constantGaussian curvature is discussed by recalling the notion of "relativistic ruleof lever" introduced by Galperin [Comm. Math. Phys. 154 (1993), 63--84] andcomparing it with two other definitions of centre of mass that arise naturallyon the treatment of the 2-body problem in spaces of constant curvature: firstlyas the collision point of particles that are initially at rest, and secondly asthe centre of rotation of steady rotation solutions. It is shown that if theparticles have distinct masses then these definitions are equivalent only ifthe curvature vanishes and instead lead to three different notions of centre ofmass in the general case.

Some remarks about the centre of mass of two particles in spaces of constant curvature