  The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, a finite-momentum
superconducting pairing state, has been extensively studied from the
perspective of mismatched Fermi surfaces of paired electrons. In this work, we
propose a distinctive mechanism to realize FFLO states by creating an imbalance
in the quantum geometry of paired electrons on an isolated flat band, which we
term "Quantum Geometry Discrepancy (QGD)". Based on a flat-band electronic
Hamiltonian with continuously tunable quantum metrics for each spin species, we
analytically investigate the QGD-induced FFLO instability near the
superconducting critical temperature through the band-projection method. To
obtain the phase diagram of the BCS-FFLO transition driven by QGD, we perform
numerical calculations using self-consistent mean-field theory, which aligns
well with the analytical results. Additionally, we discuss the stability of the
flat-band FFLO state when a finite band dispersion is turned on. We point out
that QGD serves as a new protocol for stabilizing the FFLO states in flat-band
superconductors.
