We present a formulation of the generalized uncertainty principle based on a commutator $[{\widehat{x}}^i,{\widehat{p}}_j]$ between position and momentum operators defined in a covariant manner using normal coordinates. We show how any such commutator can acquire corrections if the momentum space is curved. The correction is completely determined by the extrinsic curvature of the surface $p^2=\text{constant}$ in the momentum space, and results in noncommutativity of normal position coordinates $[{\widehat{x}}^i,{\widehat{x}}^j]\ne 0$. We then provide a construction for the momentum space geometry as a suitable four dimensional extension of a geometry conformal to the three dimensional relativistic velocity space—the Lobachevsky space—whose curvature is determined by the dispersion relation $F(p^2)=-m^2$, with $F(x)=x$ yielding the standard Heisenberg algebra.

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• General relativity
• Quantum gravity