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  "body": "\n## Definition\n\nLet $M$ be a connected smooth manifold, let $E \\to M$ be a smooth vector bundle with a [[connection|5ZYRRVE8BW]] $\\nabla$, and let $p \\in M$.\nFor a piecewise smooth loop $\\gamma$ based at $p$, let $P_\\gamma : E_p \\to E_p$ be the [[parallel transport|TVAGN3D6W5]] along $\\gamma$.\nThe *holonomy group* of $\\nabla$ at $p$ is\n$$\\mathrm{Hol}_p(\\nabla) = \\set{P_\\gamma : \\gamma \\text{ a piecewise smooth loop based at } p} \\subset \\mathrm{GL}(E_p).$$\nThe *restricted holonomy group* $\\mathrm{Hol}^0_p(\\nabla)$ is the subset given by the loops that are null-homotopic.\n\n## Remarks\n\n$\\mathrm{Hol}_p(\\nabla)$ is a group, because $P_{\\gamma \\cdot \\delta} = P_\\delta \\circ P_\\gamma$ and $P_{\\gamma^{-1}} = P_\\gamma^{-1}$.\nIt is a Lie subgroup of $\\mathrm{GL}(E_p)$, and $\\mathrm{Hol}^0_p(\\nabla)$ is its identity component; the quotient is a quotient of $\\pi_1(M, p)$.\nChanging the base point along a curve $\\sigma$ from $p$ to $q$ conjugates the group: $\\mathrm{Hol}_q = P_\\sigma \\mathrm{Hol}_p P_\\sigma^{-1}$.\nExample: the flat connection on $T\\R^n$ has $P_\\gamma = \\id$ for every loop, so $\\mathrm{Hol}_p$ is trivial.\nFor the Levi-Civita connection of the round sphere $S^2$, $\\mathrm{Hol}_p = \\mathrm{SO}(2)$: the transport around a loop is the rotation by the enclosed area.\nFor the Levi-Civita connection of any Riemannian $n$-manifold, $\\mathrm{Hol}_p \\subset \\mathrm{O}(T_pM)$, and $\\mathrm{Hol}_p \\subset \\mathrm{SO}(T_pM)$ if $M$ is orientable.\nNon-example: the flat connection on the Möbius line bundle over the circle has trivial restricted holonomy, but $\\mathrm{Hol}_p = \\set{\\pm 1}$ is not trivial; so flatness alone does not force trivial holonomy.\n",
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