  The quantum geometric tensor (QGT) characterizes the complete geometric
properties of quantum states, with the symmetric part being the quantum metric,
and the antisymmetric part being the Berry curvature. We propose a generic
Hamiltonian with global degenerate ground states, and give a general relation
between the corresponding non-Abelian quantum metric and unit Bloch vector.
This enables us to construct the relation between the non-Abelian quantum
metric and Berry or Euler curvature. To be concrete, we present and study two
topological semimetal models with global degenerate bands under CP and $C_2T$
symmetries, respectively. The topological invariants of these two degenerate
topological semimetals are the Chern number and Euler class, respectively,
which are calculated from the non-Abelian quantum metric with our constructed
relations. Based on the adiabatic perturbation theory, we further obtain the
relation between the non-Abelian quantum metric and the energy fluctuation.
Such a non-adiabatic effect can be used to extract the non-Abelian quantum
metric, which is numerically demonstrated for the two models of degenerate
topological semimetals. Finally, we discuss the quantum simulation of the model
Hamiltonians with cold atoms.
