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Homotopy Principle
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas. The theory was started by Yakov Eliashberg, Mikhail Gromov and Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to homotopy, particularly for immersions. The first evidence of h-principle appeared in the Whitney–Graustein theorem. This was followed by the Nash–Kuiper isometric ''C''1 embedding theorem and the Smale–Hirsch immersion theorem. Rough idea Assume we want to find a function ''ƒ'' on R''m'' which satisfies a partial differential equation of degree ''k'', in co-ordinates (u_1,u_2,\dots,u_m). One can rewrite it as :\Psi(u_1,u_2,\dots,u_m, J^k_f)=0 where J^k_f stands for all parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p'', namely a normal vector to ''X'' at ''p''. The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator. Gauss first wrote a draft on the topic in 1825 and published in 1827. There is also a Gauss map for a link, which computes linking number. Generalizations The Gauss map can be defined for hypersurfaces in R''n'' as a map from a hypersurface to the unit sphere ''S''''n'' &min ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): :Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged. The theorem is "remarkable" because the starting ''definition'' of Gaussian curvature mak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus. Gaussian curvature is an ''intrinsic'' measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the ''Theorema egregium''. Gaussian curvature is named after Carl Friedrich Gauss, who published the ''Theorema egregium'' in 1827. Informal definition At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash-Kuiper Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word '' eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere inside out in this way (with possible self-intersections) without cutting or tearing it or creating any crease. This is surprising, both to non-mathematicians and to those who understand regular homotopy, and can be regarded as a veridical paradox; that is something that, while being true, on first glance seems false. More precisely, let :f\colon S^2\to \R^3 be the standard embedding; then there is a regular homotopy of immersions :f_t\colon S^2\to \R^3 such that ''ƒ''0 = ''ƒ'' and ''ƒ''1 = −''ƒ''. History An existence proof for crease-free sphere eversion was first created by . It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Short Map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 1964). They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that ''X'' and ''Y'' are metric spaces and ƒ is a function from ''X'' to ''Y''. Thus we have a metric map when, for any points ''x'' and ''y'' in ''X'', : d_(f(x),f(y)) \leq d_(x,y) . \! Here ''d''''X'' and ''d''''Y'' denote the metrics on ''X'' and ''Y'' respectively. Examples Let us consider the metric space ,1/2/math> with the Euclidean metric. Then the function f(x)=x^2 is a metric map, since for x\ne y, , f(x)-f(y), =, x+y, , x-y, <, x-y, . Category of metric maps The |
Nash–Kuiper Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treated b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Embedding Theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (''C''1) embeddings and the second for embeddings that are analytic or smooth of class ''Ck'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other. The first theorem has a very simple proof but leads to some counterintuitive conclusions, while the second theorem has a technical and counterintuitive proof but leads to a less surprising result. The ''C''1 theorem was published in 1954, the ''Ck''-theorem in 1956. The real analytic theorem was first treated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendrian Knot
In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into which is tangent to the standard contact structure In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. ... on It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane. Two Legendrian knots are equivalent if they are isotopic through a family of Legendrian knots. There can be inequivalent Legendrian knots that are isotopic as topological knots. Many inequivalent Legendrian knots can be distinguished by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots. More s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holonomic (robotics)
Holonomic (introduced by Heinrich Hertz in 1894 from the Greek ὅλος meaning whole, entire and νόμ-ος meaning law) may refer to: Mathematics * Holonomic basis, a set of basis vector fields such that some coordinate system exists for which e_k = * Holonomic constraints, which are expressible as a function of the coordinates x_j\,\! and time t\,\! * Holonomic module in the theory of D-modules * Holonomic function, a smooth function that is a solution of a linear homogeneous differential equation with polynomial coefficients Other uses * Holonomic brain theory, model of cognitive function as being guided by a matrix of neurological wave interference patterns See also * Holonomy in differential geometry * Holon (other) * Nonholonomic system A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
