Geometry of linear and angular momenta of NN-particles in the Riemannian space forms and de Sitter space

The total linear and angular momenta are the conserved quantities for the motions of NN-body problem. We are concerned with the geometry of the tangential (or normal) lines for the orbit curves of the motions of NN-particles. We investigate when such NN-tangential (or normal) lines meet at a point in the ambient space, where we consider 22-dimensional Riemannian space form or de Sitter space as the ambient space. We have three applications. The first one is to give the unified interpretation for the existence of the various centers of the triangles, and the second is to obtain the spherical Desargues’ theorem. The third is to answer the question when NN-geodesics in Riemann surface meet at a point.