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  "body": "\n## Definition\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] and let $f \\colon M \\to N$ be a smooth map.\nA *vector field along $f$* is a smooth map $w \\colon M \\to TN$ with $w(p) \\in T_{f(p)} N$ for all $p \\in M$, where $TN$ is the [[tangent bundle|WY6K3P4YCG]].\nLet $df \\colon TM \\to TN$ be the [[differential|6P9PXQDSEN]] of $f$.\nThe map $f$ is *infinitesimally stable* if every vector field $w$ along $f$ has the following form.\nThere are a smooth vector field $\\xi$ on $M$ and a smooth vector field $\\eta$ on $N$ with\n$$w = df \\circ \\xi + \\eta \\circ f.$$\n\n## Remarks\n\nThe two terms are the derivatives at $t = 0$ of $f \\circ \\varphi_t$ and $\\psi_t \\circ f$, where $\\varphi_t$ and $\\psi_t$ are the flows of $\\xi$ and $\\eta$.\nSo the condition says that every first-order deformation of $f$ is tangent to the orbit of $f$ under changes of coordinates.\nExample: a submersion $f$ is infinitesimally stable.\nIndeed, each $df_p$ is surjective, so a choice of complement of $\\ker df_p$ depending smoothly on $p$ gives a vector field $\\xi$ with $df \\circ \\xi = w$, and one takes $\\eta = 0$.\nNon-example: a constant map $f \\colon S^1 \\to \\R$ with value $c$ is not infinitesimally stable.\nHere $df = 0$ and $\\eta \\circ f$ is the constant $\\eta(c)$, so a nonconstant function $w$ on $S^1$, viewed as a vector field along $f$, is not of the required form.\n",
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