  Quantum geometry has emerged as a central and ubiquitous concept in quantum
sciences, with direct consequences on quantum metrology and many-body quantum
physics. In this context, two fundamental geometric quantities are known to
play complementary roles: the Fubini-Study metric, which introduces a notion of
distance between quantum states defined over a parameter space, and the Berry
curvature associated with Berry-phase effects and topological band structures.
In fact, recent studies have revealed direct relations between these two
important quantities, suggesting that topological properties can, in special
cases, be deduced from the quantum metric. In this work, we establish general
and exact relations between the quantum metric and the topological invariants
of generic Dirac Hamiltonians. In particular, we demonstrate that topological
indices (Chern numbers or winding numbers) are bounded by the quantum volume
determined by the quantum metric. Our theoretical framework, which builds on
the Clifford algebra of Dirac matrices, is applicable to topological insulators
and semimetals of arbitrary spatial dimensions, with or without chiral
symmetry. This work clarifies the role of the Fubini-Study metric in
topological states of matter, suggesting unexplored topological responses and
metrological applications in a broad class of quantum-engineered systems.
