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Understanding the relationship between quantum geometry and topologicalinvariants is a central problem in the study of topological states. In thiswork, we establish the relationship between the quantum metric and the Eulercurvature in two-dimensional systems with space-time inversion $I_{ST}$symmetry satisfying $I^2_{ST}=+1$. As $I_{ST}$ symmetry imposes the reality ofthe wave function with vanishing Berry curvature, the well-known inequalitybetween the quantum metric and the Berry curvature is not meaningful in thisclass of systems. We find that the non-abelian quantum geometric tensor of tworeal bands exhibits an intriguing inequality between the off-diagonal Berrycurvature and the quantum metric, which in turn gives the inequality betweenthe quantum volume and the Euler invariant. Moreover, we show that thesaturation condition of the inequality is deeply related to the ideal conditionfor Euler bands, which provides a criterion for the stability of fractionaltopological phases in interacting Euler bands. Our findings demonstrate thepotential of the quantum geometry as a powerful tool for characterizingsymmetry-protected topological states and their interaction effect.

Quantum geometric bound and ideal condition for Euler band topology