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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