  Geometric phase, which is acquired after a system undergoing cyclic evolution
in the Hilbert space, is believed to be noise-resilient because it depends only
on the global properties of the evolution path. Here, we report geometric
control of energy transfer between two parametrically coupled mechanical modes
in an optomechanical system. The parametric pump is controlled along a closed
loop to implement geometric St\"uckelberg interferometry of mechanical motion,
in which the dynamical phase is eliminated using the Hahn-echo technique. We
demonstrate that the interference based solely on the geometric phase is robust
against certain noises. And, more remarkably, we show that the all-geometric
approach can achieve a high energy-transfer rate that is comparable to those
using conventional dynamical protocols.
