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In mathematics, the vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors that are tangent to the fibers. More precisely, if ''π'' : ''E'' → ''M'' is a smooth fiber bundle over a smooth manifold ''M'' and ''e'' ∈ ''E'' with ''π''(''e'') = ''x'' ∈ ''M'', then the vertical space V''e''''E'' at ''e'' is the tangent space T''e''(''E''''x'') to the fiber ''E''''x'' containing ''e''. That is, ''V''''e''''E'' = T''e''(E''π''(''e'')). The vertical space is therefore a vector subspace of T''e''''E'', and the union of the vertical spaces is a subbundle V''E'' of T''E'': this is the vertical bundle of ''E''. ==Formal definition== Let ''π'':''E''→''M'' be a smooth fiber bundle over a smooth manifold ''M''. The vertical bundle is the kernel V''E'' := ker(d''π) of the tangent map d''π'' : T''E'' → T''M''.〔 (page 77)〕 Since dπe is surjective at each point ''e'', it yields a ''regular'' subbundle of T''E''. Furthermore the vertical bundle V''E'' is also integrable. An Ehresmann connection on ''E'' is a choice of a complementary subbundle to V''E'' in T''E'', called the horizontal bundle of the connection. 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「Vertical bundle」の詳細全文を読む スポンサード リンク
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