Fibrado tangente

Em matemática, o fibrado tangente de uma variedade diferenciável M é a união disjunta^{[nota 1]} de todos os espaços tangentes de M. Em símbolos,

TM=\bigsqcup _{p\in M}T_{p}M=\bigcup _{p\in M}\left\{p\right\}\times T_{p}M.

onde $T_{p}M$ denota o espaço tangente de $M$ no ponto $p\in M$ . Um elemento de $TM$ pode ser pensado como um par $(p,v):p\in M,~v\in T_{p}M$ . Assim, existe uma projeção natural

\pi :TM\to M,\pi (p,v):=p\in M.

Construção do fibrado tangente via classes de equivalências

Seja $M^{n}$ uma variedade suave de dimensão $n$ . Fixe um altas maximal ${\mathcal {A}}$ em $M$ . Para cada $p\in M$ , considere o conjunto $I_{p}:=\bigcup _{U}U\times \mathbb {R} ^{n},\varphi :U\to \mathbb {R} ^{n}$ , onde $(U,\varphi )\subset {\mathcal {A}}:p\in U$ . Introduzimos em $I_{p}$ a seguinte relação de equivalência: sejam $U,V$ tais que $p\in U\cap V$ . Então, $(p,v)\sim (p,w)\Leftrightarrow d\varphi _{U}\circ \circ \varphi _{V}^{-1}(v)=w$ . O espaço das classes de equivalência por essa relação coincide com o espaço tangente a $p\in M$ . Assim, o fibrado tangente associado a $M$ é obtido como espaço das classes da relação definida em $\bigcup _{p}I_{p}$ por: $(U,p,v)\sim (V,q,w)\Leftrightarrow p=q,d\varphi _{U}\circ \varphi _{V}^{-1}(v)=w.$

Notas

↑ ^a ^b A união disjunta garante que para quaisquer dois pontos x₁ e x₂ da variedade M os espaços tangentes T₁ e T₂ não têm vetores em comum. Isso é ilustrado graficamente na imagem para o fibrado tangente da circunferência S¹: todas as tangentes à circunferência estão no plano da circunferência. Para torná-las disjuntas, é preciso alinhá-las em um plano perpendicular.

Referências

John M. Lee, Introduction to Smooth Manifolds, (2003) Springer-Verlag, New York. ISBN 0-387-95495-3.
Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X
M. De León, E. Merino, J.A. Oubiña, M. Salgado, A characterization of tangent and stable tangent bundles, Annales de l'institut Henri Poincaré (A) Physique théorique, Vol. 61, no. 1, 1994, 1-15 [1]

Ligações externas

«Tangent Bundle» (em inglês). - MathWorld
«Tangent Bundle» (em inglês). - PlanetMath

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