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.
a (surjective) submersion, i.e., a fibered manifold.
is a sub-bundle of the tangent bundle of
Then, f is called a Riemannian submersion if and only if, for all
, the vector space isomorphism
[1] An example of a Riemannian submersion arises when a Lie group
acts isometrically, freely and properly on a Riemannian manifold
equipped with the quotient metric is a Riemannian submersion.
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
is the Lie bracket of vector fields and
is the projection of the vector field
In particular the lower bound for the sectional curvature of
is at least as big as the lower bound for the sectional curvature of
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