Abstract
We formulate the quantum geometric tensor in a curved space background, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. The quantum metric tensor is obtained in two different ways: By using the definition of the infinitesimal distance between two states in the parameter space and via the fidelity-susceptibility approach. The usual Berry connection is modified into an object that transforms as a connection and density of weight one due to the modification of the inner product. Finally, we provide three examples in one dimension: two gauge-transformed harmonic oscillators and one gauge-transformed generalized harmonic oscillator.
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