No hyperbolic pants for the 4-body problem

TL;DR

This paper investigates the geometric properties of the equal mass 4-body problem with a 1/r^2 potential, showing that its associated metric does not maintain negative curvature, unlike the 3-body case, thus challenging extensions of hyperbolic behavior.

Contribution

It proves that the Riemannian metric for the equal mass 4-body problem does not have negative curvature everywhere, disproving the extension of hyperbolic properties from the 3-body case.

Findings

The 4-body problem metric has positive sectional curvature at some planes.

Negative curvature does not extend from 3-body to 4-body problem.

Naive hyperbolicity extension from N=3 to N>3 is not valid.

Abstract

The $N$-body problem with a $1/r^2$ 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 $1/r^2$ 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 $N>3$.