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]
Examplesedit
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.
Propertiesedit
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:
^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
Referencesedit
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
riemannian, submersion, differential, geometry, branch, mathematics, submersion, from, riemannian, manifold, another, that, respects, metrics, meaning, that, orthogonal, projection, tangent, spaces, contents, formal, definition, examples, properties, generaliz. 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 Contents 1 Formal definition 2 Examples 3 Properties 4 Generalizations and variations 5 See also 6 Notes 7 ReferencesFormal definition editLet M g and N h be two Riemannian manifolds and f M N displaystyle f M to N nbsp a surjective submersion i e a fibered manifold The horizontal distribution K ker df displaystyle K mathrm ker df perp nbsp is a sub bundle of the tangent bundle of TM displaystyle TM nbsp which depends both on the projection f displaystyle f nbsp and on the metric g displaystyle g nbsp Then f is called a Riemannian submersion if and only if for all x M displaystyle x in M nbsp the vector space isomorphism df x Kx Tf x N displaystyle df x K x rightarrow T f x N nbsp is isometric i e length preserving 1 Examples editAn example of a Riemannian submersion arises when a Lie group G displaystyle G nbsp acts isometrically freely and properly on a Riemannian manifold M g displaystyle M g nbsp The projection p M N displaystyle pi M rightarrow N nbsp to the quotient space N M G displaystyle N M G nbsp equipped with the quotient metric is a Riemannian submersion For example component wise multiplication on S3 C2 displaystyle S 3 subset mathbb C 2 nbsp by the group of unit complex numbers yields the Hopf fibration Properties editThe 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 KN X Y KM X Y 34 X Y V 2 displaystyle K N X Y K M tilde X tilde Y tfrac 3 4 tilde X tilde Y V 2 nbsp where X Y displaystyle X Y nbsp are orthonormal vector fields on N displaystyle N nbsp X Y displaystyle tilde X tilde Y nbsp their horizontal lifts to M displaystyle M nbsp displaystyle nbsp is the Lie bracket of vector fields and ZV displaystyle Z V nbsp is the projection of the vector field Z displaystyle Z nbsp to the vertical distribution In particular the lower bound for the sectional curvature of N displaystyle N nbsp is at least as big as the lower bound for the sectional curvature of M displaystyle M nbsp Generalizations and variations editFiber bundle Submetry co Lipschitz mapSee also editFibered manifold Geometric topology ManifoldNotes edit Gilkey Peter B Leahy John V Park Jeonghyeong 1998 Spinors Spectral Geometry and Riemannian Submersions Global Analysis Research Center Seoul National University pp 4 5References editGilkey 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 nbsp nbsp This differential geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Riemannian submersion amp oldid 1200366824, wikipedia, wiki, book, books, library,