# 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 $M = N \times P$ be Riemannian manifolds (the direct product is up to isometry). Let $p_1$ be the projection map from $M$ to $N$ and $p_2$ be the projection map from $M$ to $P$. Then we say that the factorization is isometry-invariant if for any isometry $\sigma$ of $M$, there are isometry $\sigma_1$ and $\sigma_2$ of $N$ and $P$ respectively, such that:

$p_1 \circ \sigma = \sigma_1 \circ p_1$

and

$p_2 \circ \sigma = \sigma_2 \circ p_2$