Atom interferometry is a natural laboratory for precision tests of general relativity, but there are no known simple relationships between atom interferometer phase and geometric properties of spacetime. Here we show that a different atom interferometer observable, the phase shear, can be expressed directly as the integrated sectional curvature over a spacetime surface enclosed by the interferometer arms and final beamsplitter, which follows from a generalized Gauss-Bonnet theorem. This interface of quantum mechanics, relativity, and geometry is manifestly coordinate-free and representation-free and can provide a useful computational tool for predicting atom interferometer properties in arbitrary background spacetimes.