Projection of Warped Product Manifold onto Unwarped Factor Manifold is Riemannian Submersion

Theorem

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

Let $f : M \to \R_{\mathop > 0}$ be a strictly positive smooth function.

Let $M \times_f N$ be the warped product manifold endowed with the metric $g = g^M \oplus f^2 g^N$.

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

$\pi_M : M \times_f N \to M$

$\pi_N : M \times_f N \to N$

Then $\pi_M$ is a Riemannian submersion, but $\pi_N$ is not necessarily so.

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