differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics...

\mathbb {R} ^{n}\subset \mathbb {R} ^{m+n}} Local diffeomorphisms Riemannian submersions The projection in a smooth vector bundle or a more general smooth...

Sub-Riemannian manifold Riemannian submanifold Riemannian metric Riemannian circle Riemannian submersion Riemannian Penrose inequality Riemannian holonomy Riemann...

induced Riemannian metric on the quotient space, which is naturally diffeomorphic to M. The Hopf fibration S3 → S2 is a Riemannian submersion relative...

Riemann curvature tensor Riemannian manifold Riemannian submersion is a map between Riemannian manifolds which is submersion and submetry at the same...

inequality Riemannian polyhedron Riemannian singular value decomposition Riemannian submanifold Riemannian submersion Riemannian volume form Riemannian wavefield...

established the more powerful fact that Sharafutdinov's retraction is a Riemannian submersion, and even a submetry. Cheeger & Ebin 2008, Chapter 8; Petersen 2016...

projection is when W is a subspace of V. In Riemannian geometry, this is used in the definition of a Riemannian submersion. Centering matrix, which is an example...

leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem. A basic example of...

metric, in a manner that makes the projection from S to M into a Riemannian submersion. (For example, it follows that there exist Sasaki–Einstein metrics...

τ : T M → M {\displaystyle \tau \colon \mathrm {T} M\to M} is a Riemannian submersion. The metric on each tangent space T p ⊂ T M {\displaystyle \mathrm...

students. He made a foundational contribution to the theory of Riemannian submersions, showing how geometric quantities on the total space and on the...

the de Rham decomposition theorem. Alternatively, the theory of Riemannian submersions may be invoked. As a consequence of their splitting theorem, Cheeger...

leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem. A basic example of...

who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman abruptly quit his...

given a (non-Riemannian) pseudo-Riemannian structure; there are topological restrictions on doing so. The terminology for pseudo-Riemannian manifold varies...

class Chern class Pontrjagin class spin structure differentiable map submersion immersion Embedding Whitney embedding theorem Critical value Sard's theorem...

non-negatively curved Riemannian manifold onto its soul. It was first used by Sharafutdinov to show that any two souls of a complete Riemannian manifold with...

\wedge \beta +(-1)^{k}\alpha \wedge d\beta .} On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism...

{\displaystyle B.} The map π , {\displaystyle \pi ,} called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The...

G} the standard Riemannian fibration ϕ : G / H → G / K {\displaystyle \phi :G/H\to G/K} is a harmonic morphism. Riemannian submersions with minimal fibres...

theorem on submersions, a proper submersion of manifolds is a fiber bundle, hence codimension/relative dimension 0 immersions/submersions behave like...

Einstein metrics with prescribed conformal infinity Robert Bryant, Riemannian Submersions as PDE Greg Galloway, Stability of marginally trapped surfaces with...

is a submersion allowing the following Definition. Let M and Q be manifolds of dimension n and q≤n respectively, and let f : M→Q be a submersion, that...

special kinds of smooth mappings between manifolds, namely immersions and submersions, and the intersections of submanifolds via transversality. More generally...

is nonzero). The statement in the first case is a special case of the submersion theorem. These variants are restatements of the inverse functions theorem...

) is surjective, f {\displaystyle f} is said to be a submersion (or, locally, a "local submersion"); and if D f {\displaystyle Df} (or, locally, D f x...

function theorem § Implicit function theorem. Another consequence is the submersion theorem. A partition of an interval [ a , b ] {\displaystyle [a,b]} is...

natural projection π : P → P / G {\displaystyle \pi :P\to P/G} is a smooth submersion, and P {\displaystyle P} is a smooth principal G {\displaystyle G} -bundle...

Glossary of general topology Glossary of algebraic topology Glossary of Riemannian and metric geometry. See also: List of differential geometry topics Words...