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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold, let $E \\to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\\nabla$, and let $\\gamma : [a, b] \\to M$ be a piecewise smooth curve.\nLet $D_t$ be the [[covariant derivative along $\\gamma$|5FZDQQHQKM]].\nA section $V$ of $E$ along $\\gamma$ is *parallel* if $D_t V = 0$.\nFor each $v \\in E_{\\gamma(a)}$ there is a unique parallel section $V$ along $\\gamma$ with $V(a) = v$.\nThe *parallel transport* along $\\gamma$ is the map\n$$P_\\gamma : E_{\\gamma(a)} \\to E_{\\gamma(b)}, \\quad v \\mapsto V(b).$$\n\n## Remarks\n\nExistence and uniqueness hold because, in a local frame, $D_t V = 0$ is a linear system of ordinary differential equations, whose solutions exist on the whole interval.\n$P_\\gamma$ is a linear isomorphism, its inverse is the transport along the reversed curve, and the transport along a concatenation is the composition.\nExample: for the flat connection on $T\\R^n$, $D_t V = \\dot V$, so parallel sections are constant and $P_\\gamma$ is the identity of $\\R^n$ for every curve $\\gamma$.\nNon-example of path independence: take the unit sphere $S^2$ with its Levi-Civita connection.\nTransport around the boundary of an octant, a geodesic triangle with three right angles, rotates tangent vectors by $\\pi/2$, the area of the octant.\nFor the Levi-Civita connection of a Riemannian metric, $P_\\gamma$ is an isometry between tangent spaces.\nThe connection is recovered by $\\nabla_{\\gamma'(0)} s = \\frac{d}{dt}\\big|_{t = 0} P_{\\gamma|_{[0, t]}}^{-1} s(\\gamma(t))$.\n",
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