An expression of a Riemannian manifold as a direct product of two Riemannian manifolds (up to isometry) is said to be isometry-invariant if the projections on both direct factors are isometry-invariant fibrations.
Definition with symbols
Let be Riemannian manifolds (the direct product is up to isometry). Let be the projection map from to and be the projection map from to . Then we say that the factorization is isometry-invariant if for any isometry of , there are isometry and of and respectively, such that: