  We propose a generalized quantum geometric tenor to understand topological
quantum phase transitions, which can be defined on the parameter space with the
adiabatic evolution of a quantum many-body system. The generalized quantum
geometric tenor contains two different local measurements, the non-Abelian
Riemannian metric and the non-Abelian Berry curvature, which are recognized as
two natural geometric characterizations for the change of the ground-state
properties when the parameter of the Hamiltonian varies. Our results show the
symmetry-breaking and topological quantum phase transitions can be understood
as the singular behavior of the local and topological properties of the quantum
geometric tenor in the thermodynamic limit.
