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In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched ''n''-manifold is covered by ''n''-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable ''n''-disk in R''n''. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks (''n'' = 1) and branched surfaces (''n'' = 2) and play prominent role in the geometry of three-manifolds after Thurston. == Definition == Let ''K'' be a metrizable space, together with: # a collection of closed subsets of ''K''; # for each ''U''''i'', a finite collection of closed subsets of ''U''''i''; # for each ''i'', a map ''π''''i'': ''U''''i'' → ''D''''i''''n'' to a closed ''n''-disk of class ''C''''k'' in R''n''. These data must satisfy the following requirements: # ∪''j'' ''D''''ij'' = ''U''''i'' and ∪''i'' Int ''U''''i'' = ''K''; # the restriction of ''π''''i'' to ''D''''ij'' is a homeomorphism onto its image ''π''''i''(''D''''ij'') which is a closed class ''C''''k'' ''n''-disk relative to the boundary of ''D''''i''''n''; # there is a cocycle of diffeomorphisms of class ''C''''k'' (''k'' ≥ 1) such that ''π''''l'' = ''α''''lm'' · ''π''''m'' when defined. The domain of ''α''''lm'' is ''π''''m''(''U''''l'' ∩ ''U''''m''). Then the space ''K'' is a branched ''n''-manifold of class ''C''''k''. The standard machinery of differential topology can be adapted to the case of branched manifolds. This leads to the definition of the tangent space ''T''''p''''K'' to a branched ''n''-manifold ''K'' at a given point ''p'', which is an ''n''-dimensional real vector space; a natural notion of a ''C''''k'' differentiable map ''f'': ''K'' → ''L'' between branched manifolds, its differential ''df'': ''T''''p''''K'' → ''T''''f''(''p'')''L'', the germ of ''f'' at ''p'', jet spaces, and other related notions. 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「Branched manifold」の詳細全文を読む スポンサード リンク
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