24.39.61.178: /* Kripke's interpretation */ revised second case of "quus" to match Kripke source material (and make better mathematical sense)

2013-08-07T20:16:18Z

Kripke's interpretation: revised second case of "quus" to match Kripke source material (and make better mathematical sense)

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In [[differential geometry]], a branch of [[mathematics]], a '''Riemannian submersion''' is a [[Submersion (mathematics)|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
:<math>f:M\to N</math>
a submersion.

Then ''f'' is a Riemannian submersion if and only if the isomorphism
:<math>df : \mathrm{ker}(df)^{\perp} \rightarrow TN</math>
is an [[isometry]].

==Examples==

An example of a Riemannian submersion arises when a [[Lie group]] <math>G</math> acts isometrically, [[free action|freely]] and [[proper action|properly]] on a Riemannian manifold <math>(M,g)</math>.
The projection <math>\pi: M \rightarrow N</math> to the [[quotient space]] <math>N = M /G</math> equipped with the quotient metric is a Riemannian submersion.
For example, component-wise multiplication on <math>S^3 \subset \mathbb{C}^2</math> 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''':
:<math>K_N(X,Y)=K_M(\tilde X, \tilde Y)+\tfrac34|[\tilde X,\tilde Y]^V|^2</math>
where <math>X,Y</math> are orthonormal vector fields on <math>N</math>, <math>\tilde X, \tilde Y</math> their horizontal lifts to <math>M</math>, <math>[*,*]</math> is the [[Lie brackets]]{{Disambiguation needed|date=October 2012}} and <math>Z^V</math> is the projection of the vector field <math>Z</math> to the [[vertical distribution]].

In particular the lower bound for the sectional curvature of <math>N</math> is at least as big as the lower bound for the sectional curvature of <math>M</math>.

==Generalizations and variations==
*[[Fiber bundle]]
*[[Submetry]]
*[[co-Lipschitz map]]

==References==
*{{Citation|title=Spinors, Spectral Geometry, and Riemannian Submersions|first1=Peter B.|last1=Gilkey|first2=John V.|last2=Leahy|first3=Jeonghyeong|last3=Park|url=http://www.emis.de/monographs/GLP/|year=1998|publisher=Global Analysis Research Center, Seoul National University}}.

[[Category:Riemannian geometry]]
[[Category:Maps of manifolds]]

Private language argument - Revision history

24.39.61.178: /* Kripke's interpretation */ revised second case of "quus" to match Kripke source material (and make better mathematical sense)