In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.
Let M and N be differentiable manifolds and be a differentiable map between them. The map f is a submersion at a point if its differential
is a surjective linear map. In this case p is called a regular point of the map f, otherwise, p is a critical point. A point is a regular value of f if all points p in the are regular points. A differentiable map f that is a submersion at each point is called a submersion. Equivalently, f is a submersion if its differential has constant rank equal to the dimension of N.
A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of f at p is not maximal. Indeed, this is the more useful notion in singularity theory. If the dimension of M is greater than or equal to the dimension of N then these two notions of critical point coincide. But if the dimension of M is less than the dimension of N, all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim M). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.
Given a submersion between smooth manifolds of dimensions and , for each there are surjective charts of around , and of around , such that restricts to a submersion which, when expressed in coordinates as , becomes an ordinary orthogonal projection. As an application, for each the corresponding fiber of , denoted can be equipped with the structure of a smooth submanifold of whose dimension is equal to the difference of the dimensions of and .
The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).
For example, consider given by The Jacobian matrix is
This has maximal rank at every point except for .
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion. Let M and N be differentiable manifolds and be a differentiable map between them. The map f is a submersion at a point if its differential is a surjective linear map. In this case p is called a regular point of the map f, otherwise, p is a critical point.
Differentiable manifolds are a certain class of topological spaces which, in a way we will make precise, locally resemble R^n. We introduce the key concepts of this subject, such as vector fields, dif
La géométrie riemannienne est un (peut-être le) chapitre central de la géométrie différentielle et de la géométriec ontemporaine en général. Le sujet est très riche et ce cours est une modeste introdu
This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex
Covers the basics of linear algebra, including rank, differentiable applications, and submersions.
Covers tangent spaces and submersions in differential geometry, emphasizing vector spaces and differentiable structures.
Explores making tangent spaces linear, defining tangent vectors without an embedding space and their operations, as well as the equivalence of different tangent space notions.