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We investigate relations between topology and the quantum metric oftwo-dimensional Chern insulators. The quantum metric is the Riemannian metricdefined on a parameter space induced from quantum states. Similar to the Berrycurvature, the quantum metric provides a geometrical structure associated toquantum states. We consider the volume of the parameter space measured with thequantum metric, which we call the quantum volume of the parameter space. Weestablish an inequality between the quantum volume of the Brillouin zone andthat of the twist-angle space. Exploiting this inequality and the inequalitybetween the Chern number and the quantum volume, we investigate how the quantumvolume can be used as a good measure to infer the Chern number. Theinequalities are found to be saturated for fermions filling Landau levels.Through various concrete models, we elucidate conditions when the quantumvolume gives a good estimate of the topology of the system.

Relations between topology and the quantum metric for Chern insulators