Having a diffeomorphism.

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  • Diffeomorphism — In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. The image of a… …   Wikipedia

  • Exotic sphere — In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n sphere. That is, M is a sphere from the point of view of all its… …   Wikipedia

  • Manifold — For other uses, see Manifold (disambiguation). The sphere (surface of a ball) is a two dimensional manifold since it can be represented by a collection of two dimensional maps. In mathematics (specifically in differential geometry and topology),… …   Wikipedia

  • Cusp (singularity) — An ordinary cusp on the curve x3–y2=0 In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve. The plane curve… …   Wikipedia

  • Schoen-Yau conjecture — In mathematics, the Schoen Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing Tung Yau.It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of …   Wikipedia

  • Riemannian geometry — Elliptic geometry is also sometimes called Riemannian geometry. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent… …   Wikipedia

  • Nonlinear dimensionality reduction — High dimensional data, meaning data that requires more than two or three dimensions to represent, can be difficult to interpret. One approach to simplification is to assume that the data of interest lies on an embedded non linear manifold within… …   Wikipedia

  • Exotic R4 — In mathematics, an exotic R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic. The first examples were found by Robion Kirby and Michael Freedman, by using the contrast between Freedman s theorems …   Wikipedia

  • Hopf fibration — In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3 sphere (a hypersphere in four dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it… …   Wikipedia

  • Differential structure — In mathematics, an n dimensional differential structure (or differentiable structure) on a set M makes M into an n dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential …   Wikipedia

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