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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 affine manifold ： ウィキペディア英語版 affine manifold In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset of $^n$, with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem. Equivalently, it is a manifold equipped with an atlas—called the affine structure—with all transition functions between charts affine (that is, have constant jacobian matrix);〔Bishop, R.L.; Goldberg, S.I. (1968), pp. 223–224.〕 two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts. In each affine coordinate domain the coordinate vector fields form a parallelization of that domain, so there is an associated connection on each domain. These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure. Note there is a link between linear connection (also called affine connection) and a web. ==Formal definition== An affine manifold $M\backslash ,$ is a real manifold with charts $\backslash psi\_i\backslash colon\; U\_i\backslash to^n$ such that $\backslash psi\_i\backslash circ\backslash psi\_j^\backslash in\; (^n)$ for all $i,\; j\backslash ,\; ,$ where $(^n)$ denotes the Lie group of affine transformations. An affine manifold is called complete if its universal covering is homeomorphic to $^n$. In the case of a compact affine manifold $M$, let $G$ be the fundamental group of $M$ and $\backslash tilde\; M$ be its universal cover. One can show that each $n$-dimensional affine manifold comes with a developing map $D\backslash colon\; \backslash to^n$, and a homomorphism $\backslash varphi\backslash colon\; G\backslash to\; (^n)$, such that $D$ is an immersion and equivariant with respect to $\backslash varphi$. A fundamental group of a compact complete flat affine manifold is called an affine crystallographic group. Classification of affine crystallographic groups is a difficult problem, far from being solved. The Riemannian crystallographic groups (also known as Bieberbach groups) were classified by Ludwig Bieberbach, answering a question posed by David Hilbert. In his work on Hilbert's 18-th problem, Bieberbach proved that any Riemannian crystallographic group contains an abelian subgroup of finite index. 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「affine manifold」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.