The kinematic formula in Riemannian homogeneous spaces

Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form ΩG. Let M and N be compact submanifolds of G/K, andI(M∩gN) an “integral invariant ” of the intersection M ∩ gN. Then the integral