The Jacobi-Maupertuis principle converts natural mechanical systems at fixed

energy into geodesic flow of a certain metric. We will find some sectional curvatures of

the metric associated to the four body problem with a 1/r² potential (forces proportional

to the inverse distances cubed) and equal masses. Then we will deduce some dynamical

consequences for the collinear, parallelogram, or trapezoidal four body subproblems.