翻訳と辞書 Words near each other ・ Hilda Lindley House ・ Hilbert's nineteenth problem ・ Hilbert's ninth problem ・ Hilbert's Nullstellensatz ・ Hilbert's paradox of the Grand Hotel ・ Hilbert's problems ・ Hilbert's program ・ Hilbert's second problem ・ Hilbert's seventeenth problem ・ Hilbert's seventh problem ・ Hilbert's sixteenth problem ・ Hilbert's sixth problem ・ Hilbert's syzygy theorem ・ Hilbert's tenth problem ・ Hilbert's theorem ・ Hilbert's theorem (differential geometry) ・ Hilbert's Theorem 90 ・ Hilbert's third problem ・ Hilbert's thirteenth problem ・ Hilbert's twelfth problem ・ Hilbert's twentieth problem ・ Hilbert's twenty-first problem ・ Hilbert's twenty-fourth problem ・ Hilbert's twenty-second problem ・ Hilbert's twenty-third problem ・ Hilbert, West Virginia ・ Hilbert, Wisconsin ・ Hilbert–Bernays paradox ・ Hilbert–Bernays provability conditions ・ Hilbert–Burch theorem Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hilbert's theorem (differential geometry) ： ウィキペディア英語版 Hilbert's theorem (differential geometry) In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface $S$ of constant negative gaussian curvature $K$ immersed in $\backslash mathbb^$. This theorem answers the question for the negative case of which surfaces in $\backslash mathbb^$ can be obtained by isometrically immersing complete manifolds with constant curvature. Hilbert's theorem was first treated by David Hilbert in, "Über Flächen von konstanter Krümmung" (Trans. Amer. Math. Soc. 2 (1901), 87-99). A different proof was given shortly after by E. Holmgren, "Sur les surfaces à courbure constante négative," (1902). ==Proof== The proof of Hilbert's theorem is elaborate and requires several lemmas. The idea is to show the nonexistence of an isometric immersion :$\backslash varphi\; =\; \backslash psi\; \backslash circ\; \backslash exp\_p:\; S\text{'}\; \backslash longrightarrow\; \backslash mathbb^$ of a plane $S\text{'}$ to the real space $\backslash mathbb^$. This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and Spivak. ''Observations'': In order to have a more manageable treatment, but without loss of generality, the curvature may be considered equal to minus one, $K=-1$. There is no loss of generality, since it is being dealt with constant curvatures, and similarities of $\backslash mathbb^$ multiply $K$ by a constant. The exponential map $\backslash exp\_p:\; T\_p(S)\; \backslash longrightarrow\; S$ is a local diffeomorphism (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an inner product in the tangent space of $S$ at $p$: $T\_p(S)$. Furthermore, $S\text{'}$ denotes the geometric surface $T\_p(S)$ with this inner product. If $\backslash psi:S\; \backslash longrightarrow\; \backslash mathbb^$ is an isometric immersion, the same holds for :$\backslash varphi\; =\; \backslash psi\; \backslash circ\; \backslash exp\_o:S\text{'}\; \backslash longrightarrow\; \backslash mathbb^$. The first lemma is independent from the other ones, and will be used at the end as the counter statement to reject the results from the other lemmas. Lemma 1: The area of $S\text{'}$ is infinite. ''Proof's Sketch:'' The idea of the proof is to create a global isometry between $H$ and $S\text{'}$. Then, since $H$ has an infinite area, $S\text{'}$ will have it too. The fact that the hyperbolic plane $H$ has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form. To obtain these ones, the hyperbolic plane can be defined as the plane with the following inner product around a point $q\backslash in\; \backslash mathbb^$ with coordinates $(u,v)$ :$E\; =\; \backslash left\backslash langle\; \backslash frac,\; \backslash frac\; \backslash right\backslash rangle\; =\; 1\; \backslash qquad\; F\; =\; \backslash left\backslash langle\; \backslash frac,\; \backslash frac\; \backslash right\backslash rangle\; =\; \backslash left\backslash langle\; \backslash frac,\; \backslash frac\; \backslash right\backslash rangle\; =\; 0\; \backslash qquad\; G\; =\; \backslash left\backslash langle\; \backslash frac,\; \backslash frac\; \backslash right\backslash rangle\; =\; e^$ Since the hyperbolic plane is unbounded, the limits of the integral are infinite, and the area can be calculated through :$\backslash int\_^\; \backslash int\_^\; e^\; du\; dv\; =\; \backslash infty$ Next it is needed to create a map, which will show that the global information from the hyperbolic plane can be transfer to the surface $S\text{'}$, i.e. a global isometry. $\backslash varphi:\; H\; \backslash rightarrow\; S\text{'}$ will be the map, whose domain is the hyperbolic plane and image the 2-dimensional manifold $S\text{'}$, which carries the inner product from the surface $S$ with negative curvature. $\backslash varphi$ will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces, :$\backslash psi:T\_p(H)\; \backslash rightarrow\; T\_(S\text{'})$. That is :$\backslash varphi\; =\; \backslash exp\_\; \backslash circ\; \backslash psi\; \backslash circ\; \backslash exp\_p^$, where $p\backslash in\; H,\; p\text{'}\; \backslash in\; S\text{'}$. That is to say, the starting point $p\backslash in\; H$ goes to the tangent plane from $H$ through the inverse of the exponential map. Then travels from one tangent plane to the other through the isometry $\backslash psi$, and then down to the surface $S\text{'}$ with another exponential map. The following step involves the use of polar coordinates, $(\backslash rho,\; \backslash theta)$ and $(\backslash rho\text{'},\; \backslash theta\text{'})$, around $p$ and $p\text{'}$ respectively. The requirement will be that the axis are mapped to each other, that is $\backslash theta=0$ goes to $\backslash theta\text{'}=0$. Then $\backslash varphi$ preserves the first fundamental form. In a geodesic polar system, the Gaussian curvature $K$ can be expressed as :$K\; =\; -\; \backslash frac\})\_\; +\; K\backslash cdot\; \backslash sqrt\; =\; 0$ Since $H$ and $S\text{'}$ have the same constant Gaussian curvature, then they are locally isometric (Minding's Theorem). That means that $\backslash varphi$ is a local isometry between $H$ and $S\text{'}$. Furthermore, from the Hadamard's theorem it follows that $\backslash varphi$ is also a covering map. Since $S\text{'}$ is simply connected, $\backslash varphi$ is a homeomorphism, and hence, a (global) isometry. Therefore, $H$ and $S\text{'}$ are globally isometric, and because $H$ has an infinite area, then $S\text{'}=T\_p(S)$ has an infinite area, as well. $\backslash square$ Lemma 2: For each $p\backslash in\; S\text{'}$ exists a parametrization $x:U\; \backslash subset\; \backslash mathbb^\; \backslash longrightarrow\; S\text{'},\; \backslash qquad\; p\; \backslash in\; x(U)$, such that the coordinate curves of $x$ are asymptotic curves of $x(U)\; =\; V\text{'}$ and form a Tchebyshef net. Lemma 3: Let $V\text{'}\; \backslash subset\; S\text{'}$ be a coordinate neighborhood of $S\text{'}$ such that the coordinate curves are asymptotic curves in $V\text{'}$. Then the area A of any quadrilateral formed by the coordinate curves is smaller than $2\backslash pi$. The next goal is to show that $x$ is a parametrization of $S\text{'}$. Lemma 4: For a fixed $t$, the curve , is an asymptotic curve with $s$ as arc length. The following 2 lemmas together with lemma 8 will demonstrate the existence of a parametrization $x:\backslash mathbb^\; \backslash longrightarrow\; S\text{'}$ Lemma 5: $x$ is a local diffeomorphism. Lemma 6: $x$ is surjective. Lemma 7: On $S\text{'}$ there are two differentiable linearly independent vector fields which are tangent to the asymptotic curves of $S\text{'}$. Lemma 8: $x$ is injective. ''Proof of Hilbert's Theorem:'' First, it will be assumed that an isometric immersion from a complete surface with negative curvature$S$ exists: $\backslash psi:S\; \backslash longrightarrow\; \backslash mathbb^$ As stated in the observations, the tangent plane $T\_p(S)$ is endowed with the metric induced by the exponential map $\backslash exp\_p:\; T\_p(S)\; \backslash longrightarrow\; S$ . Moreover, $\backslash varphi\; =\; \backslash psi\; \backslash circ\; \backslash exp\_p:S\text{'}\; \backslash longrightarrow\; \backslash mathbb^$ is an isometric immersion and Lemmas 5,6, and 8 show the existence of a parametrization $x:\backslash mathbb^\; \backslash longrightarrow\; S\text{'}$ of the whole $S\text{'}$, such that the coordinate curves of $x$ are the asymptotic curves of $S\text{'}$. This result was provided by Lemma 4. Therefore, $S\text{'}$ can be covered by a union of "coordinate" quadrilaterals $Q\_$ with $Q\_\; \backslash subset\; Q\_$. By Lemma 3, the area of each quadrilateral is smaller than $2\; \backslash pi$. On the other hand, by Lemma 1, the area of $S\text{'}$ is infinite, therefore has no bounds. This is a contradiction and the proof is concluded. $\backslash square$ 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「Hilbert's theorem (differential geometry)」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. 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