One-loop effective action for the gauge field in curved space–time

We consider the one-loop contribution of gluon fluctuations to the gauge-field effective action in a constant homogeneous Abelian-like external (background) field in a space with nonzero curvature. We calculate the real and imaginary parts of the effective action and show that the chromomagnetic vacuum remains unstable in spite of the finite curvature of the space.