Let $X$ be a smooth manifold and $G$ a Lie group acting on it. V. I. Arnold defines the *modality* of a point $x\in X$ as follows \[1\]:

>We say that a point $x$ has modality $m$ (under the given action) if a sufficiently small neighbourhood of $x$ in $X$ can be covered by finitely many families of orbits, depending on not more than $m$ parameters (and an arbitrarily small neighbourhood of $x$ intersects some $m$-parameter family of orbits).

(Afterwards he describes his classification of singularities of small modality for germs  at $0$ of functions $\mathbb C^n\to \mathbb C$. For $m=0$ this is an [ADE classification][1] into 5 families, while for $m=1$ there is a 3-index series of 1-parameter families together with 14 exceptional cases.)

OK, but what exactly is an $m$-parameter family of orbits? I suppose that all families are tacitly supposed to be smooth; so the most general definition I can come up with is that

>a smooth $m$-parameter family of $G$-orbits is the set $G(f(M))\subset X$, where $M$ is some smooth manifold of dimension $m$ and $f:M\to X$ a smooth map.

But maybe it's too general; one can add the requirement that

>$f$ is an injective immersion and the image of $f$ is $G$-invariant,

or else that

>$M$ is an $m$-dimensional disk and $f$ is a smooth embedding such that the intersection of $f(M)$ with any $G$-orbit has cardinality $0$ or $1$.

**Question.** Do these 3 definitions of a family of orbits give the same notion of modality? What is the right definition?

\[1\] V. I. Arnold, [Normal forms of functions in neighbourhoods of degenerate critical points][2].


  [1]: https://ncatlab.org/nlab/show/ADE+classification
  [2]: https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=4352&option_lang=eng