On the singularities of the curved n-body problem

Abstract. We study singularities of the n-body problem in spaces of con-stant curvature and generalize certain results due to Painlevé, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection—a property that offers a new method of approaching the problem in this particular case. 1