---
tag: B1S7A7J8H4
kind: definition
name: Infinitesimal stability
msc: [58K25, 57R45]
defines: C3TFCVE2YK
uses: [F45VQ47GPV, WY6K3P4YCG, 6P9PXQDSEN]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Definition

Let $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] and let $f \colon M \to N$ be a smooth map.
A *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]].
Let $df \colon TM \to TN$ be the [[differential|6P9PXQDSEN]] of $f$.
The map $f$ is *infinitesimally stable* if every vector field $w$ along $f$ has the following form.
There are a smooth vector field $\xi$ on $M$ and a smooth vector field $\eta$ on $N$ with
$$w = df \circ \xi + \eta \circ f.$$

## 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.