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A mathematical rose with k petals is an immersion of the circle in the plane with a single k-tuple point; k can be any odd number, but if even must be a multiple of 4, so the figure 8, with k = 2, is not a rose. The Klein bottle, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded.
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]
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.
Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds that preserves the (pseudo)-Riemannian metrics. Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length of curves (cf. Nash embedding theorem ).
An animation of Boy's surface. In geometry, Boy's surface is an immersion of the real projective plane in three-dimensional space.It was discovered in 1901 by the German mathematician Werner Boy, who had been tasked by his doctoral thesis advisor David Hilbert to prove that the projective plane could not be immersed in three-dimensional space.
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
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.
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.