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.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics