Infinitesimal stability

Remarks

The 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 $\eta$. So the condition says that every first-order deformation of $f$ is tangent to the orbit of $f$ under changes of coordinates. Example: a submersion $f$ is infinitesimally stable. Indeed, 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 $\eta = 0$. Non-example: a constant map $f \colon S^1 \to \R$ with value $c$ is not infinitesimally stable. Here $df = 0$ 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.