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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 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.

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.

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 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 assumed to be smooth; so the most general definition I can come up with is that

a family of $G$-orbits depending on not more than $m$ parameters is the set $G(f(M))\subset X$, where $M$ is some smooth manifold of dimension $\leq m$ 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 a 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.

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 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.

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 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.

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 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 assumed to be smooth; so the most general definition I can come up with is that

a family of $G$-orbits depending on not more than $m$ parameters is the set $G(f(M))\subset X$, where $M$ is some smooth manifold of dimension $\leq m$ 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 a 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.