We reveal a fundamental geometric structure of momentum space arising from the nonadiabatic evolution of Bloch electrons. By extending semiclassical wave-packet theory to incorporate nonadiabatic effects, we introduce a momentum-space metric tensor—the nonadiabatic metric. This metric gives rise to two velocity corrections, dubbed geometric and geodesic velocities, providing a unified and intuitive framework for understanding nonlinear and nonadiabatic transport phenomena beyond Berry phase effects. The geometric velocity is related to the nonadiabatic metric itself, whereas the geodesic velocity is a Christoffel symbol of the nonadiabatic metric. As the nonadiabatic metric is related to the energy-gap renormalized quantum metric, it unifies the broad quantum-metric effects in electronic responses. When the nonadiabatic metric is constant, it reduces to an effective mass, modifying flat-band electron dynamics in confining potentials. In a flat Chern band with harmonic attractive interactions, the two-body wave functions mirror the Landau-level wave functions on a torus. Furthermore, we show that the nonadiabatic metric endows momentum space with a curved geometry, recasting wave-packet dynamics as forced geodesic motion.

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• Electrical conductivity
• Topological materials
• Variational wave functional methods