  The momentum space of topological insulators and topological superconductors
is equipped with a quantum metric defined from the overlap of neighboring
valence band states or quasihole states. We investigate the quantum geometrical
properties of these materials within the framework of Dirac models and
differential geometry. The Ricci scalar is found to be a constant throughout
the whole Brillouin zone, and the vacuum Einstein equation is satisfied in 3D
with a finite cosmological constant. For linear Dirac models, several
geometrical properties are found to be independent of the band gap, including
the straight line geodesic, constant volume of the curved momentum space,
exponential decay form of the nonlocal topological marker, and unity Euler
characteristic in 2D, indicating the peculiar yet universal quantum geometrical
properties of these models.
