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  "body": "\n## Definition\n\nLet $\\pi : E \\to M$ be a smooth [[fibre bundle|CX4M8K61AB]], and let $TE$ be the [[tangent bundle|WY6K3P4YCG]] of $E$.\nThe *vertical bundle* is $V = \\ker d\\pi \\subset TE$, where $d\\pi$ is the [[differential|6P9PXQDSEN]] of $\\pi$.\nIts fibre $V_e$ is the tangent space at $e$ of the fibre through $e$.\nAn *Ehresmann connection* on $E$ is a smooth subbundle $H \\subset TE$, the *horizontal bundle*, such that $T_eE = H_e \\oplus V_e$ for every $e \\in E$.\n\n## Remarks\n\nSince $d\\pi_e$ is onto with kernel $V_e$, it restricts to an isomorphism $H_e \\to T_{\\pi(e)}M$.\nSo each tangent vector of $M$ has a unique horizontal lift at each point of the fibre above it.\nExample: for the product bundle $E = M \\times F$ with $\\pi$ the first projection, $H_{(p, y)} = T_pM \\times \\set{0}$ is an Ehresmann connection; $V_{(p,y)} = \\set{0} \\times T_yF$, and the two are complementary.\nNon-example: $H = TE$ fails the condition when the fibres have positive dimension, because $H_e \\cap V_e = V_e \\ne \\set{0}$.\nEquivalently, an Ehresmann connection is a smooth vector-valued $1$-form $\\Phi$ on $E$ with $\\Phi \\circ \\Phi = \\Phi$ and image $V$; then $H = \\ker \\Phi$.\nHorizontal lifts of curves exist locally; if the fibres are not compact they may fail to exist for all times.\nOn a vector bundle, the Ehresmann connections invariant under the scalings of the fibres and compatible with addition correspond to connections in the sense of covariant derivatives.\n",
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