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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 whose real part is the quantum metric tensor. Here, we experimentally realize a complete Bloch-state tomography to directly measure the 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. Furthermore, we experimentally test a predicted inequality between the Berry curvature and the quantum metric tensor, which reveals a deep connection between the topology and geometry.