  In Hilbert space, the geometry of the quantum state is identified by the
quantum geometric tensor (QGT), whose imaginary part is the Berry curvature and
real part is the quantum metric tensor. Here, we propose and experimentally
implement a complete Bloch state tomography to directly measure eigenfunction
of an optical Raman lattice for ultracold atoms. Through the measured
eigenfunction, the distribution of the complete QGT in the Brillouin zone is
reconstructed, with which the topological invariants are extracted by the Berry
curvature and the distances of quantum states in momentum space are measured by
the quantum metric tensor. Further, we experimentally test a predicted
inequality between the Berry curvature and quantum metric tensor, which reveals
a deep connection between topology and geometry.
