The authors of this paper consider the \(N\)-body problem with strong potential (proportional to \(1/r^2\)). In this situation there is an effective scale symmetry in addition to the usual translational and rotational symmetries. The scale symmetry enables a reduction of the zero-energy flow to a geodesic flow on an \((N-2)\)-dimensional complex projective plane \(\mathbb{C}\mathbb{P}^{N-2}\) with the projective image of the “fat diagonals” \(\Delta\) removed. Here they focus on the \(N=4\) case.
Previously Montgomery has shown that for \(N=3\) there is a geodesic flow on the two-sphere less three points. (The two-sphere less three points is the “pants” of the title.) In addition, in the case of equal masses, the Jacobi-Maupertuis metric is hyperbolic and its Gaussian curvature is negative except at two points where it is zero.
The authors’ general approach is to form a Riemannian quotient via the quotient map \(\pi:\mathbb{C}^N-\Delta\to Y_N+\mathbb{C}\mathbb{P}^{N-2}-\Delta\) so that the zero-energy, zero-angular momentum \(1/r^2\) potential \(N\)-body problem becomes equivalent to finding geodesics for the metric defined by the Riemannian submersion \(\pi\) on \(Y_N\).
The main result of the paper is the following: If \(N=4\), if the Jacobi-Maupertuis metric on \(Y_4\) is induced as described above, and if the four masses are equal and interacting under a \(1/r^2\) potential, then there are \(2\)-planes \(\sigma\) tangent to \(Y_4\) at which the Riemannian sectional curvature \(K(\sigma)\) is positive. Consequently, for \(N=4\) hyperbolicity fails.