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.
Formal definition
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]
Examples
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.
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, named for Barrett O'Neill:
![{\displaystyle K_{N}(X,Y)=K_{M}({\tilde {X}},{\tilde {Y}})+{\tfrac {3}{4}}|[{\tilde {X}},{\tilde {Y}}]^{V}|^{2}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
.
Generalizations and variations
See also
Notes
- ^ 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
References
- 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
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