Riemannian submersion

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In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics.

Let (M, g) and (N, h) be two Riemannian manifolds and

$f:M\to N$

a submersion.

Then f is a Riemannian submersion if and only if the isomorphism

$df : \mathrm{ker}(df)^{\perp} \rightarrow TN$

is an isometry.

[edit] Examples

An example of a Riemannian submersion arises when a Lie group $G$ acts isometrically, freely and properly on a Riemannian manifold $(M, g)$ . The projection $\pi: M \rightarrow N$ to the quotient space $N = M / G$ equipped with the quotient metric is a Riemannian submersion.

[edit] Properties

The sectional curvature of the target space of a Riemannian submersion can be calculated from the curvature of the total space by O'Neill's formula:

$K_N(X,Y)=K_M(\tilde X, \tilde Y)+\tfrac34|[\tilde X,\tilde Y]^\top|^2$

where $X, Y$ are orthonormal vector fields on $N$ , $\tilde X, \tilde Y$ their horizontal lifts to $M$ , $[ * , * ]$ is the Lie brackets and $Z^\top$ is the projection of the vector field $Z$ to the vertical distribution.