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  "body": "\n## Definition\n\nLet $\\mathbb{D}$ be the unit disc in $\\C$ with its Poincaré distance $\\rho$.\nLet $X$ be a connected [[complex manifold|JSGWAN2RJV]] with [[Kobayashi pseudodistance|DY8QQ5GTTX]] $k_X$, normalised so that $k_{\\mathbb{D}} = \\rho$.\nA *complex geodesic* of $X$ is a [[holomorphic map|3MQTWKBBBT]] $\\varphi \\colon \\mathbb{D} \\to X$ such that\n$$k_X(\\varphi(a), \\varphi(b)) = \\rho(a, b) \\quad \\text{for all } a, b \\in \\mathbb{D}.$$\n\n## Remarks\n\nSince holomorphic maps decrease Kobayashi distances, the inequality $\\le$ always holds; the condition is the reverse inequality.\nSome authors use the Carathéodory distance $c_X$ in place of $k_X$; on bounded convex domains the two notions agree, since $c_X = k_X$ there.\nExample: in the unit ball $B \\subset \\C^n$, $\\varphi(\\zeta) = (\\zeta, 0, \\dots, 0)$ is a complex geodesic.\nCheck: the projection $\\pi(z) = z_1$ maps $B$ into $\\mathbb{D}$ and $\\pi \\circ \\varphi = \\id$, so $\\rho(a, b) = \\rho(\\pi\\varphi(a), \\pi\\varphi(b)) \\le k_B(\\varphi(a), \\varphi(b)) \\le \\rho(a, b)$.\nNon-example: $\\psi(\\zeta) = (\\zeta / 2, 0, \\dots, 0)$ is holomorphic into $B$, but $k_B(\\psi(a), \\psi(b)) = \\rho(a/2, b/2) \u003c \\rho(a, b)$ for $a \\ne b$, by the same argument and the Schwarz–Pick lemma for $\\zeta \\mapsto \\zeta/2$.\nA complex geodesic is injective, since $\\rho(a, b) \u003e 0$ for $a \\ne b$.\n",
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