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        "quote": "嘉当处理曲率的方法完全用微分形式来表达。设 $e _ { i } ( i = 1 , \\ldots , n )$ 是向量场的局部基，在开集 $U \\subseteq M$ 上张成 ${ \\mathcal { T } } ( U )$，而 $\\{ \\varepsilon ^ { i } \\}$ 是 ${ \\mathcal { T } } ^ { * } ( U )$ 的对偶基，使得 $\\langle e _ { i } , \\varepsilon ^ { j } \\rangle = \\delta _ { i } ^ { j }$。例如，在坐标卡 $( U ; x ^ { i } )$ 中，我们可以取 $e _ { i } = \\partial _ { x ^ { i } }$ 和 $\\varepsilon ^ { j } = \\mathrm { d } x ^ { j }$，但对于任意基 $\\{ e _ { i } \\}$，这样的坐标系存在当且仅当对所有 $i , j = 1 , \\ldots , n$ 都有 $[ e _ { i } , e _ { j } ] = 0$。我们通过下式定义 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:47cb9b8a-f841-4b73-85bc-a777473aa609\" data-concept-route=\"/concept-identities#concept-47cb9b8a-f841-4b73-85bc-a777473aa609\" id=\"first-d612eb4de01c68cd76e7-zh\">联络一形式(connection 1-form)</u> $\\omega _ { \\textit { j } } ^ { i } : { \\mathcal { T } } ( U ) \\to { \\mathcal { F } } ( U )$",
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        "quote": "Cartan’s approach to curvature is expressed entirely in terms of differential forms. Let $e _ { i } ( i = 1 , \\ldots , n )$ be a local basis of vector fields, spanning ${ \\mathcal { T } } ( U )$ over an open set $U \\subseteq M$ and $\\{ \\varepsilon ^ { i } \\}$ the dual basis of ${ \\mathcal { T } } ^ { * } ( U )$ , such that $\\langle e _ { i } , \\varepsilon ^ { j } \\rangle = \\delta _ { i } ^ { j }$ . For example, in a coordinate chart $( U ; x ^ { i } )$ we may set $e _ { i } = \\partial _ { x ^ { i } }$ and $\\varepsilon ^ { j } = \\mathrm { d } x ^ { j }$ , but such a coordinate system will exist for an arbitrary basis $\\{ e _ { i } \\}$ if and only if $[ e _ { i } , e _ { j } ] = 0$ for all $i , j = 1 , \\ldots , n$ . We define the <u class=\"concept-term\" data-concept-identity=\"urn:uuid:47cb9b8a-f841-4b73-85bc-a777473aa609\" data-concept-route=\"/concept-identities#concept-47cb9b8a-f841-4b73-85bc-a777473aa609\" id=\"first-c6b3fcebb0af1cf3e6c4-en\">connection 1-forms</u> $\\omega _ { \\textit { j } } ^ { i } : { \\mathcal { T } } ( U ) \\to { \\mathcal { F } } ( U )$ by",
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