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

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... ltonian 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.