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

$p_{2}\circ \sigma =\sigma _{2}\circ p_{2}$