  In this paper, we extend the quantum geometric tensor for parameter-dependent
curved spaces to higher dimensions, and introduce an equivalent definition that
generalizes the Zanardi, et al, formulation of the tensor. The
parameter-dependent metric modifies the behavior of both the quantum metric
tensor and Berry curvature in a purely geometric way. Our focus is on
understanding the distinctions in higher dimensions that emerge when using the
generalized tensor compared to the conventional one. Through a comparative
analysis, illustrated with examples in two dimensions, we highlight unique
quantum geometric properties for both the quantum metric tensor and the Berry
curvature. Additionally, we explore differences between analytical and
perturbative approaches in solving the problems.
