# Isometry-invariant factorization

## Definition

### Symbol-free definition

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:

and