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The quantum metric of single-particle wave functions in topological flat bands plays a crucial role in determining the stability of fractional Chern insulating (FCI) states. Here, we unravel that the quantum metric causes the many-body Chern number of the FCI states to deviate sharply from the expected value associated with partial filling of the single-particle topological flat band. Furthermore, the variation of the quantum metric in momentum space induces band dispersion through interactions, affecting the stability of the FCI states. This causes a reentrant transition into the Fermi liquid from the FCI phase as the interaction strength increases.