Projection of Euclidean Space onto Euclidean Subspace is Riemannian Submersion

Theorem

Let $\struct {\R^{n + k}, g^E_{n + k}}$, $\struct {\R^n, g^E_n}$ be the real vector spaces with Euclidean metrics.

Let $\pi : \R^{n + k} \to \R^n$ be the projection onto the first $n$ coordinates.

Then $\pi$ is a Riemannian submersion.

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