  Based on the covariant Wigner function approach we derive the quantum
Boltzmann equation for fermions with flavor mixing in general curved spacetime.
This work gives a rigorous theoretical framework to investigate the flavor
oscillation phenomena taking the gravitational effect into account. It is shown
that the Boltzmann equation of the lowest order of the expansion with respect
to $\hbar$ reproduces the previous result which was derived in the relativistic
limit on the Minkowski background spacetime. It is demonstrated that the
familiar formula for the vacuum neutrino oscillation can be obtained by solving
the Boltzmann equation. Higher order effects of the $\hbar$-expansion is also
briefly discussed.
