Definition:Riemannian Submersion

Definition

Let $\struct {\tilde M, \tilde g}$, $\struct {M, g}$ be Riemannian manifolds.

Let $\pi : \tilde M \to M$ be a smooth submersion.

Let $x \in \tilde M$, $p \in M$ be points in $\tilde M$ and $M$.

Let $H_x$ be a horizontal tangent space at $x$.

Let $T_p M$ be a tangent space of $M$ at $p \in M$.

Let $g_p$ be a Riemannian metric at $p$.

Suppose for every $x$ the differential $\d \pi_x$ restricts to a linear isometry from $H_x$ to $T_{\map \pi x} M$:

$\forall v, w \in H_x : \map {\tilde g_x} {v, w} = \map {g_{\map \pi x} } { \map {\d \pi_x} v, \map {\d \pi_x} w}$

Then $\pi$ is said to be a Riemannian submersion.

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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