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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, meaning that it is an orthogonal projection on tangent spaces.

Let (*M*, *g*) and (*N*, *h*) be two Riemannian manifolds and a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution is a sub-bundle of the tangent bundle of which depends both on the projection and on the metric .

Then, *f* is called a Riemannian submersion if and only if, for all , the vector space isomorphism is isometric, i.e., length-preserving.^{[1]}

An example of a Riemannian submersion arises when a Lie group acts isometrically, freely and properly on a Riemannian manifold . The projection to the quotient space equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on by the group of unit complex numbers yields the Hopf fibration.

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**, named for Barrett O'Neill:

where are orthonormal vector fields on , their horizontal lifts to , is the Lie bracket of vector fields and is the projection of the vector field to the vertical distribution.

In particular the lower bound for the sectional curvature of is at least as big as the lower bound for the sectional curvature of .

- Fiber bundle
- Submetry
- co-Lipschitz map

**^**Gilkey, Peter B.; Leahy, John V.; Park, Jeonghyeong (1998),*Spinors, Spectral Geometry, and Riemannian Submersions*, Global Analysis Research Center, Seoul National University, pp. 4–5

- Gilkey, Peter B.; Leahy, John V.; Park, Jeonghyeong (1998),
*Spinors, Spectral Geometry, and Riemannian Submersions*, Global Analysis Research Center, Seoul National University. - Barrett O'Neill.
*The fundamental equations of a submersion.*Michigan Math. J. 13 (1966), 459–469. doi:10.1307/mmj/1028999604