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A smooth embedding is an injective immersion f : M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any point x ∈ M there is a neighbourhood, U ⊆ M, of x such that f : U → N is an embedding, and conversely a local embedding is an ...
The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class C k, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if M is a compact manifold, and with n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an isometric embedding ƒ: M → R n (also analytic or of class C k). [15]
Basic results include the Whitney embedding theorem and Whitney immersion theorem. Riemann surface for the function f(z) = √ z, shown as a ramified covering space of the complex plane. In complex geometry, ramified covering spaces are used to model Riemann surfaces, and to analyze maps between surfaces, such as by the Riemann–Hurwitz formula.
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for >, any smooth -dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean-space, and a (not necessarily one-to-one) immersion in ()-space.
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.. The inverse function theorem implies that a smooth map : is a local diffeomorphism if and only if the derivative: is a linear isomorphism for all points .
The notion of a closed immersion is local in the sense that f is a closed immersion if and only if for some (equivalently every) open covering = the induced map : is a closed immersion. [ 5 ] [ 6 ] If the composition Z → Y → X {\displaystyle Z\to Y\to X} is a closed immersion and Y → X {\displaystyle Y\to X} is separated , then Z → Y ...
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 .