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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold, and let $\\pi : E \\to M$ be a smooth real vector bundle.\nLet $\\Gamma(E)$ be the space of smooth [[sections|WWX7H5W1RP]] of $E$, and let $\\mathfrak{X}(M)$ be the space of smooth vector fields on $M$.\nA *connection* on $E$ is a map\n$$\\nabla : \\mathfrak{X}(M) \\times \\Gamma(E) \\to \\Gamma(E), \\quad (X, s) \\mapsto \\nabla_X s,$$\nsuch that for all $X, Y \\in \\mathfrak{X}(M)$, $s, t \\in \\Gamma(E)$, $f, h \\in C^\\infty(M)$ and $a, b \\in \\R$:\n\n1. $\\nabla_{fX + hY} s = f \\nabla_X s + h \\nabla_Y s$.\n2. $\\nabla_X (a s + b t) = a \\nabla_X s + b \\nabla_X t$.\n3. $\\nabla_X (f s) = (X f) \\, s + f \\nabla_X s$.\n\n## Remarks\n\nExample: on the trivial bundle $E = M \\times \\R^r$, a section is a smooth map $s = (s^1, \\dots, s^r)$, and $\\nabla_X s = (X s^1, \\dots, X s^r)$ is a connection; clause 3 is the product rule $X(f s^i) = (Xf) s^i + f X s^i$.\nNon-example: on the [[tangent bundle|WY6K3P4YCG]], the map $(X, Y) \\mapsto [X, Y]$ satisfies clause 2 but fails clause 1, since $[fX, Y] = f[X, Y] - (Yf) X$.\nBy clause 1, $(\\nabla_X s)(p)$ depends only on $X_p$, so $\\nabla$ can be seen as a map $\\Gamma(E) \\to \\Gamma(T^*M \\otimes E)$.\nIn a local frame $e_1, \\dots, e_r$ of $E$, $\\nabla e_j = \\sum_i \\omega^i_j \\otimes e_i$ for a matrix $(\\omega^i_j)$ of $1$-forms, and locally $\\nabla = d + \\omega$.\nThe difference of two connections is $C^\\infty(M)$-linear in both arguments, so it is a section of $T^*M \\otimes \\mathrm{End}(E)$; thus connections form an affine space over this space of sections.\nComplex vector bundles are treated the same way with complex scalars.\n",
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