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] (see also [2]):
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 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 guess it should be some map $f:M\to X/G$, where $M$ is an $m$-dimensional "something"; but what the requirements on $M$ and $f$ should be to get the right definition of modality?
Is it enough to assume that $M$ is an $m$-dimensional open disk and $f$ comes from a smooth embedding $\tilde f:M\to X$ such that the cardinality of the intersection of $\tilde f(M)$ with any $G$-orbit is either 0 or 1?
[1] V. I. Arnold, Normal forms of functions in neighbourhoods of degenerate critical points.
[2] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps, vol. 1.
