  The geometry of the parameter space is encoded by the quantum geometric
tensor, which captures fundamental information about quantum states and
contains both the quantum metric tensor and the curvature of the Berry
connection. We present a formulation of the Berry connection and the quantum
geometric tensor in the framework of the phase space or Wigner function
formalism. This formulation is obtained through the direct application of the
Weyl correspondence to the geometric structure under consideration. In
particular, we show that the quantum metric tensor can be computed using only
the Wigner functions, which opens an alternative way to experimentally measure
the components of this tensor. We also address the non-Abelian generalization
and obtain the phase space formulation of the Wilczek-Zee connection and the
non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum
metric tensor involves only the non-diagonal Wigner functions. Then, we verify
our approach with examples and apply it to a system of $N$ coupled harmonic
oscillators, showing that the associated Berry connection vanishes and
obtaining the analytic expression for the quantum metric tensor. Our results
indicate that the developed approach is well adapted to study the parameter
space associated with quantum many-body systems.
