  This paper focuses on characterizing the energy profile along pathways
connecting different regions of configuration space in the context of a
prototypical glass model, the pure spherical $p$-spin model with $p=3$. The
study investigates pairs of stationary points (local minima or rank-1 saddles),
analyzing the energy profile along geodesic paths and comparing them with
"perturbed" pathways correlated to the landscape curvature. The goal is to
assess the extent to which information from the local Hessian matrices around
stationary points can identify paths with lower energy barriers. Surprisingly,
unlike findings in other systems, the direction of softest local curvature is
not a reliable predictor of low-energy paths, except in the case in which the
direction of softest curvature corresponds to an isolated mode of the Hessian.
However, other information encoded in the local Hessian does allow the
identification of pathways associated with lower energy barriers. We conclude
commenting on implications for the system's activated dynamics.
