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