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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Whitney embedding theorem ： ウィキペディア英語版 Whitney embedding theorem In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any smooth real -dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the (), if . This is the best linear bound on the smallest-dimensional Euclidean space that all -dimensional manifolds embed in, as the real projective spaces of dimension cannot be embedded into real -space if is a power of two (as can be seen from a characteristic class argument, also due to Whitney). *The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided . Whitney similarly proved that such a map could be approximated by an immersion provided . This last result is sometimes called the weak Whitney immersion theorem. ==A little about the proof== The general outline of the proof is to start with an immersion with transverse self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections. If has boundary, one can remove the self-intersections simply by isotoping into itself (the isotopy being in the domain of ), to a submanifold of that does not contain the double-points. Thus, we are quickly led to the case where has no boundary. Sometimes it is impossible to remove the double-points via an isotopy—consider for example the figure-8 immersion of the circle in the plane. In this case, one needs to introduce a local double point. Once one has two opposite double points, one constructs a closed loop connecting the two, giving a closed path in . Since is simply connected, one can assume this path bounds a disc, and provided one can further assume (by the weak Whitney embedding theorem) that the disc is embedded in such that it intersects the image of only in its boundary. Whitney then uses the disc to create a 1-parameter family of immersions, in effect pushing across the disc, removing the two double points in the process. In the case of the figure-8 immersion with its introduced double-point, the push across move is quite simple (pictured). This process of eliminating opposite sign double-points by pushing the manifold along a disc is called the Whitney Trick. To introduce a local double point, Whitney created a family of immersions which are approximately linear outside of the unit ball, but containing a single double point. For such an immersion is defined as :$\backslash begin$ \alpha_1 : \mathbf^1 \to \mathbf^2 \\ \alpha_1(t_1)=\left(\frac, t_1 - \frac\right) \end Notice that if is considered as a map to i.e.: :$\backslash alpha\_1(t\_1)\; =\; \backslash left(\; \backslash frac,t\_1\; -\; \backslash frac,0\backslash right)$ then the double point can be resolved to an embedding: :$\backslash beta\_1(t\_1,a)\; =\; \backslash left(\backslash frac,t\_1\; -\; \backslash frac,\backslash frac\backslash right).$ Notice and for then as a function of , is an embedding. Define :$\backslash alpha\_2(t\_1,t\_2)\; =\; \backslash left(\backslash beta\_1(t\_1,t\_2),t\_2\backslash right)\; =\; \backslash left(\backslash frac,t\_1\; -\; \backslash frac,\backslash frac,\; t\_2\; \backslash right).$ can similarly be resolved in , this process ultimately leads one to the definition: :$\backslash alpha\_m(t\_1,t\_2,\backslash cdots,t\_m)\; =\; \backslash left(\backslash frac,t\_1\; -\; \backslash frac,\; \backslash frac,\; t\_2,\; \backslash frac,\; t\_3,\; \backslash cdots,\; \backslash frac,\; t\_m\; \backslash right),$ where :$u=(1+t\_1^2)(1+t\_2^2)\backslash cdots(1+t\_m^2).$ The key properties of is that it is an embedding except for the double-point . Moreover, for large, it is approximately the linear embedding . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Whitney embedding theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.