  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 $N>3$.
