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In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. Throughout this article ''E'' → ''X'' is an oriented, real vector bundle of rank ''r''. == Formal definition == The Euler class ''e''(''E'') is an element of the integral cohomology group : constructed as follows. An orientation of ''E'' amounts to a continuous choice of generator of the cohomology : of each fiber ''F'' relative to the complement ''F''\''F''0 of its zero element ''F''0. From Thom isomorphism, this induces an orientation class : in the cohomology of ''E'' relative to the complement ''E''\''E''0 of the zero section ''E''0. The inclusions : where ''X'' includes into ''E'' as the zero section, induce maps : The Euler class ''e''(''E'') is the image of ''u'' under the composition of these maps. 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「Euler class」の詳細全文を読む スポンサード リンク
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