The quantum mechanical wave function of electrons in a particular class of materials, called topological materials, displays intriguing quantum geometrical and topological properties. Mathematically, the topological order in these materials can be ubiquitously described by a topological invariant similar to the Gauss map. In addition, a quantum metric can be introduced from the overlap of neighboring quantum states in momentum space, whose determinant is related to the topological invariant through a metric-curvature correspondence. Through this quantum metric, we reveal a number of intriguing differential geometrical properties of these materials, including a maximally symmetric space, constant Ricci scalar, vacuum Einstein equation with a finite cosmological constant, and straight line geodesic.

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