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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
Then f is a Riemannian submersion if and only if the isomorphism
is an isometry.
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:
In particular the lower bound for the sectional curvature of is at least as big as the lower bound for the sectional curvature of .
Generalizations and variations
- Gilkey, Peter B.; Leahy, John V.; Park, Jeonghyeong (1998), Spinors, Spectral Geometry, and Riemannian Submersions, Global Analysis Research Center, Seoul National University.