No hyperbolic pants for the 4-body problem

الأصل · EN

The N-body problem with a 1/r² potential has, in addition to translation and rotational symmetry, an effective scale symmetry which allows its zero energy flow to be reduced to a geodesic flow on complex projective N-2-space, minus a hyperplane arrangement. When N=3 we get a geodesic flow on the two-sphere minus three points. If, in addition we assume that the three masses are equal, then it was proved in [1] that the corresponding metric is hyperbolic: its Gaussian curvature is negative except at two points. Does the negative curvature property persist for N=4, that is, in the equal mass 1/r² 4-body problem? Here we prove `no' by computing that the corresponding Riemannian metric in this N=4 case has positive sectional curvature at some two-planes. This `no' answer dashes hopes of naively extending hyperbolicity from N=3 to N>3.