Projection of Product Manifold onto Factor Manifold is Riemannian Submersion

Theorem
Let $\struct {M, g^M}$ and $\struct {N, g^N}$ be Riemannian manifolds.

Let $M \times N$ be the product manifold endowed with the product metric $g = g^M \oplus g^N$.

Let $\pi_M$, $\pi_N$ be projections such that:


 * $\pi_M : M \times N \to M$


 * $\pi_N : M \times N \to N$

Then $\pi_M$ and $\pi_N$ are Riemannian submersions.