  The quantum geometric tensor, composed of the quantum metric tensor and Berry
curvature, fully encodes the parameter space geometry of a physical system. We
first provide a formulation of the quantum geometrical tensor in the path
integral formalism that can handle both the ground and excited states, making
it useful to characterize excited state quantum phase transitions (ESQPT). In
this setting, we also generalize the quantum geometric tensor to incorporate
variations of the system parameters and the phase-space coordinates. This gives
rise to an alternative approach to the quantum covariance matrix, from which we
can get information about the quantum entanglement of Gaussian states through
tools such as purity and von Neumann entropy. Second, we demonstrate the
equivalence between the formulation of the quantum geometric tensor in the path
integral formalism and other existing methods. Furthermore, we explore the
geometric properties of the generalized quantum metric tensor in depth by
calculating the Ricci tensor and scalar curvature for several quantum systems,
providing insight into this geometric information.
