  Despite its abundance in nature, predicting the occurrence of ferromagnetism
in the ground state is possible only under very limited conditions such as in a
flat band system with repulsive interaction or in a band with a single hole
under infinitely large Coulomb repulsion, etc. Here, we propose a general
condition to achieve saturated ferromagnetism based on the quantum geometry of
electronic wave functions in itinerant electron systems. By analyzing
multi-band repulsive Hubbard models with an integer band filling, relevant to
either ferromagnetic insulators or semimetals, we propose a rigorous quantum
geometric upper bound on the spin stiffness. By employing this geometric bound,
we establish that saturated ferromagnetism is prohibited in the absence of
interband coupling, even when the local Hubbard repulsion is infinitely large.
As a corollary, this shows that saturated ferromagnetism is forbidden in any
half-filled Hubbard model. We also derive the condition that the upper bound of
the spin stiffness can be completely characterized by the Abelian quantum
metric. We believe that our findings reveal a profound connection between
quantum geometry and ferromagnetism, which can be extended to various
symmetry-broken ground states in itinerant electronic systems.
