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{\nopagenumbers
\parindent=0pt
~\vskip 2 truecm
\centerline{\bf Breaking of linear symmetries and Michel theory:}
\medskip
\centerline{\bf degeneracy, Grassmann manifolds, and invariant subspaces}
\footnote{}{{Last modified \giorno }}
\vskip 2 truecm
\bigskip
\bigskip
\centerline{Giuseppe Gaeta}
\medskip
\centerline{\it Dipartimento di Fisica, Universit\`a di Roma}
\centerline{\it 00185 Roma (Italy)}
\centerline{\tt giuseppe.gaeta@roma1.infn.it}
\bigskip
\bigskip
\bigskip
\vfill
{\bf Summary.} Michel's theory of symmetry breaking in its
original formulation has some difficulty in dealing with
problems with a linear symmetry, due to the degeneration in
the symmetry type implied by the linearity of group action.
One can usually circumvent this difficultyu by quotienting out
linear relations, but here we propose a more intrinsic, and
fully geometric, approach to the problem, making use of
Grassmann manifolds. In this way Michel theory can also be
applied to the determination of dynamically invariant
manifolds for equivariant nonlinear flows.
\vfill\eject
}
\pageno=1
\parskip=10pt
\parindent=0pt
\section{1. Introduction and motivations.}
The L. Michel's geometric theory of symmetry breaking \ref{1-5} allows to identify points
which are critical for {\it any } $G$-invariant potential defined on a smooth
$G$-manifold, where $G$ is a compact Lie group.
The fundamental result in this theory is stated by saying that a $G$-orbit is $G$-critical
if and only if it is isolated in its stratum; this means essentially that all nearby orbits
have different symmetry properties (see below for definitions, more detail, extensions and
limitations).
However, when $G$ acts in $\R^N$ via a linear representation,
it is unavoidable that the
only orbit which can possibly be isolated in its stratum is the
trivial one, made of the
origin alone, due precisely to the degeneration introduced by the
linearity of the group
action. In this case one can still extract useful information not
only by the complex of
Michel's theory -- which indeed provides a quite detailed
understanding of the orbit space
in terms of variational properties -- but also from the
simple enunciation of the main
theorem given above in non-rigorous way: this is usually
done, explicitely or implicitely
\ref{6-9}, by ``quotienting out'' the radial direction, i.e.
considering the unit sphere $S^{N-1}$ in $\R^N$ -- as on this one can
have orbits isolated in their stratum -- and then reintroducing in
the analysis the radial direction again.
This procedure has proved adequate to all the applications of Michel's
theory in the case of linear $G$-action, but is quite unsatisfactory
in that on the one side the easiest case (linear action in $\R^N$)
appears more difficult -- in principles if not in practice -- to deal
with than other ones, and also in that it requires to introduce
analytical constraints (i.e. consider the subset $|x| =1$) in the
context of a theory which is purely geometrical.
The purpose of this note is to analyze Michel's theory in this case,
namely linear $G$
action in $\R^N$, resorting to the natural geometric framework for
such a setting, i.e. in
terms of {\it Grassmann manifolds}.
This will turn out to have some useful fallouts, in that
we can extend the method and results of Michel to
the search for invariant submanifolds under an equivariant
dynamics (as these will be
points in the appropriate Grassmannian).
It would also be possible to consider the case of
tangent bundles to smooth $G$-manifolds $M$: in this frame, the
fibers $\T_x M$ are linear
spaces, on which we can consider the linearization
$(DG)(x)$ (if well defined), so that the analysis developed here
does actually extend beyond the linear frame.
It should also be mentioned that a different kind of motivation exists
for setting
Michel's theory in terms of Grassmannians: this has to do
with the theory of integrable
systems \ref{10,11}, and with recent work by
L. Martinez et al. \ref{12} in which they
identify integrable hierarchies associated
to singular strata of the infinite
Grassmannian. The connection with their work will not be touched upon here,
but will be discussed elsewhere.
\section{2. The geometry of symmetry breaking.}
In this section, I will briefly recall the basics of Michel's geometric theory of symmetry
breaking; for further detail, the reader is referred to the original papers \ref{1,2}, or
to some of the papers providing extensions and/or applications of it and including a
detailed exposition \ref{3-7,13-16}, see in particular \ref{5}.
We consider a smooth\footnote{$^1$}{In this note, smooth will always mean ${\cal
C}^\infty$.} manifold $M$, on which is defined an
action of a compact, connected, semisimple
Lie group $G$, so that $M$ is a $G$-manifold.
It should be mentioned that the compactness assumption could be
somewhat relaxed, in particular when we consider variational problems
defined on Sobolev spaces of sections of a fiber bundle \ref{17-19}; however, we prefer to
keep to as simple a (meaningful) setting as possible, in order to focus on the geometric
ideas. Similarly, the extension to non-connected groups would be obvious, but we will not
consider it for the same reason.
To any point $x \in M$ we associate an isotropy subgroup,
$$ G_x \ := \ \{ g \in G \ : \ g x = x \} \eqno(1) $$
and we will call the equivalence class of $G_x$ under
conjugation in $G$ the {\it isotropy type} of $x$.
Points on the same $G$-orbit have the same isotropy type, as for $y = gx$ we have
immediately
$$ G_y \ = \ g \, G_x \, g^{-1} \ . \eqno(2) $$
The set of points having the same isotropy type is called a {\it stratum}; thus the
stratum of $x$ is defined as
$$ \s (x) \ := \ \{ y \in M \ : \ G_y = h G_x h^{-1} \ , \ h \in G \} \ ; \eqno(3) $$
it should be stressed that isomorphism of isotropy subgroup is {\it not } enough for
belonging to the same stratum (for a simple example, consider $\R^2$ with $G
= Z_2^x \times Z_2^y$ given by reflections
across the $x$ and $y$ axes: nonzero points $p$
on the axes all have $G_p = Z_2$, but $Z_2^x$
is not conjugate to $Z_2^y$ in $G$). Under
the present hypotheses, strata are smooth submanifolds in $M$.
Let us now consider the {\it orbit space } $\Om := M/G$ (this is also called the
``configuration space'' in physical literature, to stress the fact that $G$-related states
should be seen as completely equivalent). To any orbit we can associate, thanks to (2), an
isotropy type: thus, an isotropy stratification can also be defined in $\Om$.
The space $M/G$ is not, except in highly exceptional cases, a manifold. However, at
least for $G$ compact, it is a {\it stratified manifold}, i.e. the disjoint union of
manifolds.
As an example of a stratified manifold, the reader can think of the cube: this is the union
of its interior (a three-dimensional manifold), the interior of the faces (six
two-dimensional manifolds), the interior of the edges (twelve one-dimensional manifolds),
and the vertices (six zero-dimensional manifolds). It happens that this example
illustrates several general features of stratified manifolds: in particular, there is a
unique generic stratum, open and dense, and strata of dimension $k$ are in the frontier of
strata of dimension $m > k$; both of these features are always true for stratified
manifolds. An analysis, or even a very sketchy summary, of stratified geometry would be
out of place here \ref{20}.
To avoid any confusion, we will call these strata {\it geometric strata}, or also Whitney
strata; the strata previously defined in terms of isotropy types will be called {\it
isotropy strata}, or also Michel strata. As we will mainly deal with the latter kind,
whenever I write just ``stratum'' it is understood I am referring to an isotropy stratum.
Thus, $\Om$ has a stratified structure both in terms of geometric strata and in terms of
isotropy strata. The two stratifications happen to be compatible: any isotropy stratum of
$\Om$ is the union of geometric strata (a more detailed analysis, aimed at
the study of symmetry breaking, in terms of algebraic invariants for $G$ and of algebraic
geometry is given e.g. in the works by Sartori and collaborators \ref{5,13-16}; see also
\ref{21-23}). This means in particular that isotropy strata are smooth submanifolds of
$\Om$.
In the study of equivariant dynamics, one also associates to a subgroup $H \sse G$ the
space of fixed points (of $M$) under $H$,
$$ M^H \ := \ \{ x \in M \ : \ gx = x \ \forall g \in H \} \ = \ \{ x \in M \ : \ H \sse
G_x \} \eqno(4) $$
(notice no $G$-conjugation enters in this definition); when we are specially interested in
the case $H = G_x$ for some point $x \in M$, we write
$$ M^{(x)} \ := \ \{ y \in M \ : \ G_y \sse G_x \} \ . \eqno(5) $$
Notice that $M^{(x)} \sse \s (x)$; the equality here corresponds to exceptional
situations, and requires in particular that $G_x$ is a normal subgroup of $G$, since $\om
(x) \sse \s (x)$, and $\om (x) \sse M^{(x)}$ requires that $g G_x g^{-1} = G_x$.
Let us now consider a $G$-invariant potential on $M$, i.e. a differentiable function $V :
M \to \R$ such that, for all $x \in M$ and all $g \in G$,
$$ V (g x) \ = \ V (x) \ ; \eqno(6) $$
this can also be seen as a potential $\Phi : \Om \to \R$ defined on $\Om$, as it is
constant along $G$-orbits. In particular, if $x$ is a critical point of $V$, so is any
point $ y = g x$; thus, critical points of $G$-invariant potentials come in $G$-orbits,
and we can speak of critical orbits.
{\tt Example 1.} Let us consider a trivial example in order to fix
ideas: let $M = \R$, and $G = \Z_2 = \{ e , h \}$, where
the nontrivial element acts as $h (x) = - x$. The orbits
are made of two points, except for the exceptional orbit
made of the origin $x = 0$ alone. The isotropy subgroup
of any point is just $G_x = \{ e \}$, except for the
origin, for which $G_0 = G$. Thus, the orbit space is
$\R_+ = \{ x \ge 0 \} = \{ 0 \} \cup \{ x > 0 \}$, this
decomposition providing the stratification. $\odot$
In this example, one cannot avoid noticing that the
special point $x=0$, which constitutes a stratum by
itself, also have the special property of being a
critical point of {\it any } $G$-invariant (that is,
even) potential; conversely, the only point which is
critical for any $G$-invariant potential is precisely
$x=0$. One could wonder if this is a curious property of
this trivial -- and very special -- example, or if it is possible to
have the same situation in more interesting cases.
It is remarkable that the latter hypothesis is true, so
that the geometrical theory of symmetry breaking can be
seen as a way to generalize this situation; the reader
tempted to think this is too trivial to give anything
interesting is warned that the physical properties of
particles in the $SU(3)$ theory of strong interactions
can be predicted on the basis of such a generalization
\ref{6,7,24}, and thus in a {\it model-independent } way.
We can now come back to discussing the general
structure of Michel theory. For any $x$ we consider the
linearization of $G_x$, which acts
naturally on the tangent
space $\T_x M$ and leaves the origin of this invariant.
(In more general case, one is
not guaranteed that the $G_x$-action can be linearized,
see \ref{17-19,25}; as we want to deal with linear action,
this point will not be of concern here.)
The tangent space $\T_x M$ has then different natural
splittings, according to different properties: first of
all, denoting by $\om$ the $G$-orbit of $x$, we have $
\T_x M \, = \, \T_x \om \, \oplus \, \N_x \om $ (here
$\N \om$ denotes the normal bundle to $\om$, seen as a
subbundle of $\T M$); moreover the normal space $\N_x
\om \ss \T_x M$ can be separated into a pointwise
$G_x$-invariant part $\N_x^{(0)} \om$ and a
complementary, globally invariant, part $\N_x^{(1)} \om
= \N_x \om \cap \T_x M^{(x)}$. Thus we write $$ \T_x M \
= \ \T_x \om \, \oplus \, \N_x^{(0)} \om \, \oplus \,
\N_x^{(1)} \om \eqno(7') $$
We can also split $\T_x M$ according to the
stratification, denoting by $\s$ the stratum of $x$, as
$$ \T_x M \ = \ \T_x \s \, \oplus \, \N_x \s \ ;
\eqno(7'') $$ indeed, as $ \om \sse \s$, it is clear
that $\T_x \om \sse \T_x \s$; notice that we cannot as
of now affirm the same for $\N_x^{(0)} \om$, since this
could have a larger isotropy subgroup; however, this is
not the case, as it follows by use of {\it slice
theory} \ref{26-28}.
A discussion of this would go beyonds our needs here, but let us however recall what a
slice is for the sake of completeness of this incomplete introduction.
A slice at $x$ is a local\footnote{$^2$}{We stress that in general it is not possible to
construct a global slice.} submanifold $S_x \sse M$ such that
$x \in S_x$ and: {\parskip=0pt\parindent=30pt
\item{\tt (i)} $S_x$ is transversal to the orbit $\om (x)$ and $\T_x S_x $ is
complementary to $\T_x \om (x)$ in $\T_x M$;
\item{\tt (ii)} $S_x$ is transversal to all the $G$ orbits which meet $S_x$;
\item{\tt (iii)} $S_x$ is (globally) invariant under $G_x$;
\item{\tt (iv)} For $y \in S_x$ and $g \in G$, $g y \in S_x$
implies $ g \in G_x$, i.e. $G_x$ is the maximal subgroup which
leaves $S_x$ globally invariant; this also implies $G_y \sse G_x$.
}
The $G$-orbit of $S_x$ gives a tubular neighbourhood $\U (x)$ of $\om (x)$, which has a
natural structure of a $G_x$ fiber bundle over $\om (x)$, the fiber over $y = gx$
corresponding to $S_y = g \( S_x \)$.
We can rewrite {\tt (i)} as $ \T_x M = \T_x S_x \oplus \T_x \om (x) $; we also write
$ S_x^0 \ = \ S_x \cap M^{(x)} $. Then it follows from properties {\tt (iii)} and {\tt
(iv)} of $S_x$, and the compactness of $G_x$, that
$ S^0_x = \s (x) \cap S_x$, and that
$ \s (x) \cap \U (x) = \bigcup_{y \in \om (x) } S^0_y $. These imply moreover, together
with $\om (x) \sse \s (x)$, that
$$ \T_x \s (x) \ = \ \T_x \om (x) \, \oplus \, \T_x S^0_x \ . \eqno(7''') $$
Let us now consider the gradient of an
invariant potential $V (x)$, which we will denote
by $f (x) = \nabla V (x) \in \T_x M$;
under the present hypotheses\footnote{$^3$}{Somewhat
surprisingly this is not true in full
generality, as can be seen by considering $M =
\R^2$, $G = \R$ acting by $g_t (x,y) =
(x + ty , y)$, and $V(x,y) = y$: it suffices then
to consider $\nabla V$ on the $x$ axis to see that this is not equivariant
\ref{18,25}.},
we are guaranteed this is an equivariant
vector field on $M$. In particular, if we consider
$g \in G_x$, we have
$$ f(x) \ = \ f( gx) \ = \ (Dg)_x \, f(x) \eqno(8) $$
so that necessarily $ f(x)$ is invariant
under $(DG_x )_x$ and thus $f(x) \in \T_x
M^{(x)}$.
{\tt Remark 1.} Notice that if $f (x)$ is invariant
under $(Dg)_x$, we have $$ f(x) \ = \ (Dg)_x \, f(x) \
= \ \phi^*g \, f(gx) \eqno(9) $$ with
$\phi_g : \T M
\to \T M$ the lift of $g:M \to M$ to $TM$ (so that $\phi_g^*$
maps $\T_{gx} M$ into $\T_x M$); thus we cannot conclude that
$g \in G_x$, and in general we can only affirm that, in obvious
notation, $G_x \sse G_{f(x)}$.
$\odot$
Obviously, $f (x) \in \T_x M^{(x)}$ implies also that
$f (x) \in \T_x \s (x)$. Actually, these statements --
and the discussion above -- hold for any
$G$-equivariant vector field $f$; when $f $ is the
gradient of an invariant potential $V$, we know in
addition that $f(x) \in \N_x \om (x)$,
as $V$ is constant along $\om$. (Notice that $\T_x M^{(x)}$
could have nonzero intersection with $\T_x \om (x)$,
corresponding to the existence of a nontrivial
centralizer $C_G [G_x ]$.)
Thus, we have shown that
$$ f(x) \ = \ (\grad V) (x) \ \in \ \N_x^{(0)}
\om(x) \ = \ \T_x S_x^0 \ \sse \ \T_x \s (x) \eqno(10) $$
This proof does essentially reproduce the argument of
Michel \ref{1}.
One could also proceed the other way round and prove that
if $\T_x \s (x) \not= \{ 0 \}$, then there is a nonzero
$f(x) = (\grad V) (x) \in \T_x \s (x)$, i.e. $\om (x)$
cannot be $G$-critical; the proof of this would require
again use of slice theory \ref{1}.
We have thus in particular that:
{\bf Theorem 1. } {\it If an orbit $\om$
is isolated in its stratum, then necessarily $(\grad V)
(x) = 0$ for all points $x \in \om$, and for {\rm any }
$G$-invariant potential $V$. Conversely, if $(\grad V)
(x) = 0$ for all points $x \in \om$, for {\rm all }
$G$-invariant potential $V$, then $\om$
is isolated in its stratum. }
This statement is also
known as Michel's Theorem, although Michel proved quite
more than this \ref{1-4}, as it is also clear from the
above reproduction of his discussion.
Notice that if we have a general equivariant $f(x)$ (that is, not
necessarily a gradient), having an orbit isolated in its stratum
only guarantees that $f(x) \in \T_x \om(x)$; recall that a group
orbit which is invariant under the dynamical flow is said to be a
{\it relative equilibrium}. Thus we have:
{\bf Theorem 2. } {\it If an orbit $\om$
is isolated in its stratum, then necessarily $f
(x) \in \T_x \om (x)$ for all points $x \in \om$, and for {\rm any }
$G$-equivariant vector field $f : M \to \T M$; thus it is a
relative equilibrium for any such $f$. Conversely, if $\om$ is a
relative equilibrium for {\rm all} $G$-equivariant vector fields on $M$,
then $\om$ is isolated in its stratum. }
\section{3. Linear group action.}
The discussion of the previous section applies to any
smooth action of a compact Lie group $G$ on a smooth
finite dimensional manifold $M$.
However, if we consider a linear $G$-action in $\R^n$,
it is clear that the only orbit which could possibly be
isolated in its stratum is the trivial one made of the
origin alone, $\{ O \}$. Indeed, the linearity of the
$G$-action implies that $x$ and $\la x$ (with $|x|
\not= 0$ and $\la \not= 0$ a real number) have the same
isotropy group: thus we have a degeneracy along radii,
and any nontrivial $G$-orbit $\om \in \Om$ belongs to a
set of orbits having the same symmetry properties and
which is at least one-dimensional.
In this case, as Michel's theory
also shows that $\grad V$ will be directed along the
stratum, and thus if the stratum is one-dimensional we
can still use Michel's theory and
reduce the problem to a one-dimensional one; this
leads to results of the kind of the ``Equivariant
branching lemma'' \ref{8,24,25,29-34}. However, it is somewhat
disturbing that the most simple situation would seem to
lead to a less powerful use of the geometrical theory of
symmetry breaking.
An alternative way of using the Michel's theorem in the
form given above would go simply through selecting a
sphere $S^{n-1} \ss \R^n$ (e.g. the unit sphere),
analysing $(\grad V) (x)$ for $x \in S^{n-1}$, and dealing
then separately with the radial direction. This approach is
also readily implemented, but again it is
unsatisfactory that we have to introduce anaytical
considerations (the constraint $|x|=1$) into an
otherwise completely geometric approach.
We are now going to settle this theory in a format
which is completely geometric and natural when dealing
with linear symmetries, i.e. in terms of {\it Grassmann
manifolds} \ref{35-40}
We will consider $M = \R^n$; all of our considerations
will extend immediately to ${\bf C}^n$. We recall that
$\Gr_{k,n} (\R )$ (in the following we just write
$\Gr_{k,n}$) is the space of $k$-dimensional linear
subspaces $\R^k$ of $\R^n$; this is given a standard
manifold structure, see e.g. \ref{35-40} or any textbook
on differential geometry; see also the appendix.
In particular, $\Gr_{n-1,n} $ is just the
projective space $\R P^{n-1}$.
Thus, a point $p_k \in \Gr_{k,n}$ can be
thought of as identified by $k$ linearly independent
vectors $\{ \xi_1 , ... \xi_k \}$, $\xi_i \in \R^n$, or
equivalently by the set $\P_k$ of vectors $\eta =
\sum_{i=1}^k a^i \xi_i $.
The {\it linear} $G$-action on $M = \R^n$, which we
denote by $L_g$ to emphsize its linearity, induces a
well-defined $G$-action on $\Gr_{k,n}$: indeed an
element $g \in G$ acts on $\eta$ as
$$ g (\eta ) \ = \ \sum_{i=1}^k \, a^i L_g \xi_i
\eqno(11) $$ and thus a linear $k$-dimensional subspace
$ \P_k \sse \R^n$ (i.e. a point $p_k \in \Gr_{k,n}$) is
transformed into a $k$-dimensional subspace
$L_g (\P_k ) \sse \R^n$ (i.e. into a point $\~g (p_k ) \in
\Gr_{k,n}$), independent of the choice of basis vectors
in $p_k$.
[It is
immediate to check that the linear independence of the
$\xi_i$'s, together with the group property, implies the
linear independence of the $(L_g \xi_i )$'s.]
Notice that if $L_g$ transforms vectors of
$\P_k$ into different vectors of $\P_k$, we have $L_g
(\P_k ) = \P_k$ and $\~g (p_k ) = p_k$: thus, the
representation $\{ \~g \}$ of $G$ on $\Gr_{k,n}$ can
well fail to be faithful or effective even if the representation $L$
of $G$ in $M$ is such.
The isotropy subgroup $G_p$ of $p \in \Gr_{k,n} $ is
the set of elements $g \in G$ such that $L_g$ transform
all vectors of $\P$ into (possibly different) vectors in
$\P$.
{\tt Remark 2.} We stress that there could be points -- or linear
subspaces -- in $\P$ with higher isotropy: thus, the
origin will always be invariant under all of $G$, but
belong to any linear subspace; and a subspace
$\P_1$ having $G_1 \ss G$ as isotropy group could
contain a smaller dimensional subspace $\P_0 \ss \P_1$
which admits $G_0$ as isotropy group, with $G_1 \ss G_0
\sse G$. As already stated in the definition above,
$G_p$ will thus be the larger subgroup of $G$ under
which {\it all } points in the subspace $\P_k$
represented by $p \in \Gr_{k,n}$ are mapped into points
of the same subspace. $\odot$
{\tt Remark 3.} Suppose now that all points of $\P_k$ are
mapped into points of a different subspace $\~{\P}_k$:
this means that the two subspaces belong to the same
$G$-orbit, and hence to the same isotropy stratum.
$\odot$
Not surprisingly, such a lift from $\R^n $ to
$\Gr_{k,n}$ is not possible when we consider a
(equivariant) nonlinear flow. Indeed, let us consider
$f : M \to \T M$ and the flow on $M$ induced by $e^{t
f}$; it is clear that {\it in general } a linear subspace
$\P_k \sse \R^n$ is {\it not } mapped into a (possibly
different) linear subspace by this, i.e.
$$ e^{t f} \, \sum_{i=1}^k \, a^i \xi_i \ \not= \
\sum_{i=1}^k b^i \, e^{t f}Ê\xi_i \ . \eqno(12) $$
In other words, if we look at the ``flow'' $\phi$ induced
by $f$ on $\Gr_{k,n}$, this is ill-defined, as a point
$p_k \in \Gr_{k,n}$ is in general taken into $p_n =
\Gr_{n,n} $ (i.e. the whole space $\R^n$).
It is also clear, however, in particular in view of the
discussion in sect.2, that the equivariance of $f$
can entail the $f$-invariance of certain linear
subspaces identified by symmetry properties.
{\tt Example 2.}
Thus, if $x_0 \in S^{n-1}$ is isolated in
its stratum (upon restriction to $S^{n-1}$), we know
that $f(x_0 ) \in \T_{x_0} \ell_0$, where $\ell_0$ is
the line through the origin and $x_0$. This is a linear
one-dimensional space, i.e. a point in $\Gr_{1,n}$, and
our discussion of Michel's theorem shows that indeed
$ e^{t \phi} \ell_0 \ \sse \ \ell_0$, for all $t>0$. $\odot$
{\tt Remark 4.} Notice that had we considered a {\it linear }
vector field $f$, this would have induced a (local) flow on
$\Gr_{k,n}$ (see remark 9 below). $\odot$
In summary, the discussion of this section shows how we can set Michel's
theorem in terms of Grassmann manifolds: indeed, the $G$-action
$\~g : \Gr \to \Gr$ is well defined, and -- as discussed in the next
section -- it can be used to determine not only $f$-invariant lines, as in the simple
example above, but more general linear subspaces which are invariant under any
$G$-equivariant flow $f : M \to \T M$.
\section{4. Isolated points in Grassmannian
stratification.}
Let us now make the argument mentioned in the previous
section more precise. We will
deal with equivariant flows $f: M \to \T M$ and, to
avoid unessential complications (i.e. not to have to
discuss possible degeneracies), we assume that $f (0) =
0$; the same effect would be obtained by requiring that
the $G$-action in $\R^n$ is effective, as in this case
the only point such that $G_x = G$ is the origin, and
Michel theorem ensures that $f(0)=0$.
First of all, consider $\Gr_{1,n}$. A point $p \in
\Gr_{1,n}$ isolated in its stratum corresponds to a
line $\ell_0 \ss \R^n$ isolated in its stratum, and we
know that in this case $f : \ell_0 \to \T \ell_0$, by
Michel's theorem; thus, points $p \in \Gr_{1,n}$
isolated in their stratum identify one-dimensional
linear subspaces $\P_1 $ of $\R^n$ which are invariant
under {\it any } $G$-equivariant flow.
{\tt Remark 5.} Notice that we do not have to worry about $\{ O \}$,
which is invariant by itself and included in any
linear subspace. In this respect, it is appropriate to
mention that $f : \s \to \T \s$ (here $\s$ any stratum
in $M$) does {\it not } guarantee that $\s$ is
invariant under $e^{t f}$ for all $t \in [0,\infty ]$:
we can only conclude that $ e^{t f} : \s \to \bar{\s }$
(this is the closure of $\s$). $\odot$
Let us now consider a higher dimensional subspace
$\P_k$, $k \ge 2$. Suppose that all points in $\P_k$
belong to a single stratum $\s_0$ (so that the isotropy
type of $p_k \in \Gr_{k,n}$ is obviously well defined),
and that $p_k$ is isolated in its stratum. Then, as we
know that for all points $x \in \P_k$ it must be $f (x)
\in \T_x \s_0$, we can conclude that $\P_k$ is invariant
under {\it any } $G$-equivariant flow.
The apparent problem with this discussion is that an invariant
subspace $\P_k$ could well contain points of higher
symmetry (even apart from the origin). However, as
mentioned above (remark 2), this is not really a problem as the
isotropy group of $p_k \in \Gr_{k,n}$ is defined as the
group which maps all points of the corresponding
subspace $\P_k \ss \R^n$ into points of the same
subspace. Thus, even in this case, the isotropy type of
$p_k$ (i.e. the conjugacy class of $G_p$ in $G$)
is well defined, and hence so is its stratum in
$\Gr_{k,n}$. We can again conclude that if $p_k$ is
isolated in its stratum, it corresponds to a linear
subspace $\P_k$ which is invariant under {\it any }
$G$-equivariant flow.
For the converse (i.e. that these are the only linear
subspaces invariant under all $G$-equivariant flows),
rather than dealing directly with slices in $\Gr_{k,n}$
one can use the manifold structure of $\Gr_{k,n}$, or
its relation with $O(n)/[O(k) \times O(n-k)]$ (see
the appendix): with this, we have to deal with a
finite dimensional (smooth, compact) $G$-manifold,
on which Michel's theorem applies.
We summarize our discussion in the following form, in
which the natural identification between a point $p
\in \Gr_{k,n}$ and a linear subspace $\P \sse \R^n$ is
understood.
{\bf Definition 1.} {\it The isotropy group of $p \in
\Gr_{k,n}$ is the larger subgroup $G_p \sse G$ such
that for all points $x \in \P$ and for all $g \in G_p$,
$y = L_g x \in \P$.}
{\bf Definition 2.} {\it The $(k,n)$ Grassmannian
isotropy stratum of $p \in \Gr_{k,n}$, denoted by
$\s_{k,n} (p )$ is the equivalence class of points $q
\in \Gr_{k,n}$ for which there is an element $g \in G$
such that $G_q = g G_p g^{-1}$.}
{\bf Theorem 3.} {\it If a point $p_k \in \Gr_{k,n}$ is
isolated in its (Grassmannian isotropy) stratum
$\s_{k,n} (p_k)$, then it corresponds to a linear subspace
$\P_k \sse \R^n$ which is invariant under {\rm any}
$G$-equivariant flow $f$. Conversely, if a linear subspace
$\P_k \sse \R^n$ is invariant under {\rm all} $G$-invariant flows
$f : \R^n \to \T \R^n$, then $p_k \in \Gr_{k,n}$ is isolated in
its (Grassmannian isotropy) stratum.}
{\tt Remark 6.} Notice that for the limit cases $k=0$
(corresponding to $\P_0 \equiv \{ O \}$) and $k=n$
(corresponding to $\P_n \equiv \R^n$) the theorem is
trivially true. $\odot$
\vfill\eject
\section{5. Isolated orbits in Grassmannian
stratification.}
The reader has possibly noticed that so far we have
only considered {\it points}, rather than {\it orbits},
isolated in their stratum.
This was done in order to discuss first the situation
which is simplest to analyze and visualize; however, we are now
going to consider $G$-orbits in $\Gr_{k,n}$ and strata
in $\Gr_{k,n} / G$.
As stressed above (see sect.3), the $G$-action on
$\R^n$, being linear, induces an action on $\Gr_{k,n}$.
Thus we have $\~G_k : \Gr_{k,n} \to \Gr_{k,n}$, and we
can consider $G$-orbits (i.e. orbits under this
$G$-action) in $\Gr_{k,n}$.
It is easy to see, e.g. using the canonical
correspondence between points $p \in \Gr_{k,n}$ and
$k$-dimensional linear subspaces $\P \sse \R^n$, that
$ q = \~g p $ implies that $G_q = g G_p g^{-1} $. Thus
the $G$-orbit of $p$ is a submanifold of $\s_{k,n}
(p)$. This means that -- as it was the case for the
manifold $M$ and the orbit space $M/G$ -- the
isotropy stratification in $\Gr_{k,n}$ can be lifted to
an isotropy stratification of the orbit space $\Om_{k,n}
:= \Gr_{k,n} / G$.
It should be clear that, as $\Gr_{k,n}$ is a smooth
manifold, all the discussion of sect.2 immediately
applies to it. Thus, in particular, the concept of an
orbit $\om \in \Om$ isolated in its stratum is well
defined.
{\tt Remark 7.}
Notice that an orbit in $\Gr_{k,n}$ -- and hence also a
stratum, which is the union of $G$-orbits -- corresponds
to the union of linear subspaces, and thus to a
submanifold in $\R^n$. For all points $x \in \R^n$ which
lie in subspaces $\P$ which are part of a stratum $\s$
corresponding to an isotropy type represented by $G_0
\ss G$, the isotropy subgroup $G_x$ is conjugate in $G$
to a group $G_1 $ such that
$$ G_0 \ \sse \ G_1 \ . \eqno(15) $$
We stress that, contrary to the case of stratification
in $M$ or in $M/G$, in (15) $G_1$ can be strictly
greater than $G_0$.
This is due to the fact that the isotropy subgroup of
$p$ (i.e. of a linear subspace $\P$) is the
intersection of the isotropy subgroups of all points $x
\in \P$: thus a point $x_0 \in \P$ (as already remarked
in sects. 3 and 4) can have an isotropy group $G_0$ strictly
containing $G_p$. $\odot$
It is remarkable that, thanks to the general theorems
on geometric and isotropy stratifications, this
situation is not any more difficult to deal with than the
standard one.
Indeed, we know that points in $M$ having an isotropy
type strictly greater than the one of the stratum
$\s_0$, will lie in strata which are on the border of
$\s_0$. Thus, if we consider, as we have to do in our
case, the union of $\s_0$ and of the strata with
strictly higher isotropy type, this will just be the
closure $\={\s_0}$ of $\s_0$. We denote this as
$\s_0 \cup_{i=1}^b \s_0^i $, where it is understood that
the strata $\s_0^i$ belong to $\pa \s_0$.
Let us come back to the orbit $\om = G (q) \sse
\Gr_{k,n}$ and the linear spaces $\P$ corresponding to
points $p \in \om$. We know that all points $x$
belonging to such $\P$ will lie in $\={\s_0}$, where
$\s_0$ is the stratum in $\R^n$ with isotropy type
equal to that of $\s_{k,n} (q)$ (i.e. corresponds to
``generic'' points in these $\P$). The vector field
$f(x)$ is always tangent to $\s (x)$, and thus we are
guaranteed that for all points $x \in \P$ we have,
in a somewhat folkloristic notation,
$$ f(x) \ \in \ \T_x (\s_0 ) \cup_{i=1}^b \T_x (\s_0^i
) \eqno(16) $$
(obviously only one of these tangent spaces really
exists, as $x$ belongs to one and only one stratum in
$\R^n$). We are therefore guaranteed that
$$ e^{t f} \ : \ \s_{k,n} (p) \to \s_{k,n} (p)
\ . \eqno(17) $$
In other words, using the bordering relations among
strata in $\R^n$ which corresponds to group/subgroup
relations \ref{1-5}, we have proved that the
isotropy Grassmannian strata $\s_{k,n} \sse \Gr_{k,n}$
are also $f$-invariant for any equivariant $f$.
The same holds for the isotropy Grassmannian strata in
the orbit space $\Om_{k,n} = \Gr_{k,n} / G$;
thus, an orbit $\om \in \Om_{k,n}$
which is isolated in its stratum is invariant under any
$G$-equivariant flow $f$.
Again, for the converse one can use the standard manifold
structure of $\Gr_{k,n}$ discussed in the appendix, and make
use of Michel theorem.
We summarize our discussion in a way similar to the one
adopted at the end of the previous section 4.
{\bf Lemma 1.} {\it The $(k,n)$ Grassmannian
isotropy stratification of $~\Gr_{k,n}$ extends to an
isotropy stratification of the orbit space $\Om_{k,n} :=
\Gr_{k,n} / G $.}
{\bf Theorem 4.} {\it If an orbit $\om \in \Om_{k,n} =
\Gr_{k,n} /G$ is isolated in its [$(k,n)$ Grassmannian isotropy]
stratum $\S_{k,n} (\om )$, then it is invariant under {\rm any}
$G$-equivariant flow $f$. Conversely, if $\om \in \Om_{k,n}$ is
invariant under {\rm all} the $G$-equivariant flows, then it is
isolated in its [$(k,n)$ Grassmannian isotropy] stratum.}
\section{6. Variational problems and Grassmann
manifolds.}
We have seen in sect.2 how Michel theorem allows to
identify the points (the orbits) in $M$ which are
critical for any $G$-invariant potential. It is natural
to ask if the ``Michel theory on Grassmann manifolds''
developed here could give some similar result.
First of all, it is clear that, unless $V$ depends only
on angular coordinates (and thus is either trivial or
singular in the origin), we cannot define a potential
on $G_{k,n}$, as $V(x)$ will take different values on
different points of $\R^n$ lying in the subspace $\P$
which corresponds to $p \in \Gr_{k,n}$; notice that
this also applies to the simplest case of $\Gr_{1,n}$,
and of invariant lines $\ell_0$ as seen above.
Thus, we can only conclude that, as $(\grad V) = f$ is
an equivariant vector field on $M$ and a point $p$
(an orbit $\om$) in $\Gr_{k,n}$ isolated in its stratum
is invariant under an equivariant flow, at such point
the gradient is necessarily directed along the space
$\P$ corresponding to $p$ (along the spaces $\P$
corresponding to points of the orbit).
Let us first consider the case of a point $p$ isolated
in its stratum: in this case, we can consider the
restriction $V_p$ of $V: \R^n \to \R$ to the subspace
$\P$ corresponding to the point $p \in \Gr_{k,n}$ (i.e.
$V_p : \P \to \R$).
The above discussion and theorem 3 show that critical
points of $V_p$ are guaranteed to be also critical
points of $V$. Obviously, minima (or maxima) of $V_p$
could be saddle points for $V$.
Notice that we can use topological tool (e.g. Morse
theory, or simpler ones) in order to guarantee the
existence of a certain number and kind of critical
points for $V_p$, even in the case $V$ depends smoothly
on external parameters. The simplest example of this
use is provided, as already mentioned, by the
{\it equivariant branching lemma}
\ref{8,24,25,29-34}, dealing with the case of an invariant line
$\ell_0$ or -- in the present language -- with isolated
points in the $(1,n)$ isotropy Grassmannian
stratification of $\Gr_{k,n}$.
Notice that when $\P$ contains subspaces $\P_0$ with
higher isotropy, one could apply similar considerations,
and topological tools, to $\P / \P_0$; a recent -- and
physically relevant -- example of application of this
method is provided by \ref{41}.
The same kind of reduction approach would also apply to
the case of orbits $\om \sse \Gr_{k,n}$ isolated in
their stratum $\S_{k,n}$. In this case, it should be
observed that the set of points $x$ belonging to linear
subspaces $p \in \om$ can be quotiented in two ways,
i.e. by the $G$ action in $\R$, and by the action of
$\R$ acting in $\R^n$ as multiplication (i.e.
homogeneous dilation, taking a point $x_0 \not=0$ into
the line $\ell_0 = \{ a x_0 \}$). Now $V$ is constant
along $G$-orbits in $\R^n$, and thus one can reduce to
considering the restriction $V_p$ to the linear
subspace $\P$ corresponding to {\it any one } point $p
\in \om$. In other words, we can in this way quotient
out the $G$ action, and we are then reduced to the
previous case.
It should be mentioned, however, that the kind of
``symmetry reduction'' results which one can obtain in
this way could also be obtained, more directly, using the
{\it symmetric criticality principle} of Palais
\ref{18} or, as mentioned in sect.3, by restricting to $S^{n-1}$
and dealing separately with the radial direction.
\section{7. Examples.}
In this section, we present some simple examples to
illustrate applications of the theorem obtained above.
As far as I know, the result obtained in example 6 is new.
{\tt Example 3.} Let us consider $M = \R^2$, with the
action of $G = \Z_2^{(x)} \times \Z_2^{(x)}$,
corresponding to reflections, respectively, in $x$ and
in $y$. The orbit space correspond to one (say the
first) quadrant,
$$ \Om \approx \{ (x,y) \, : \ x \ge 0 ,\, y \ge 0 \} \
; \eqno(18) $$
points $\xi = (x,y)$ with $xy \not= 0 $ have isotropy
group $G_\xi = \{ e \}$, points on the $x$ axis $\xi =
(x,0)$ have isotropy group $G_\xi = \Z_2^{(y)}$, and
points on the $y$ axis $\xi = (0,y)$ have isotropy group
$G_\xi = \Z_2^{(x)}$. Notice that $\Z_2^{(x)}$ and
$\Z_2^{(y)}$ are not conjugated in $G$, so that they do
not correspond to the same isotropy type, and thus the
two axes are not in the same stratum.
In this case the only interesting grassmannian is
$\Gr_1$, so we have to look at points $p \in \Gr_1$,
i.e. at one-dimensional linear subspaces $\P_1 = (\la
x_0 , \la y_0 )$, where $x_0^2 + y_0^2 \not= 0$.
If $x_0 y_0 \not= 0$, the isotropy subgroup $G_p$ is just
$\{ e \}$, as it follows from definition 1 in section 4
(notice $\P_1$ always contains $\{ 0 \}$ which has a
larger isotropy, cfr. remark 7). For $y_0 = 0$, $\P_1$
corresponds to the $x$ axis and thus $G_{p_x} =
\Z_2^{(y)}$, and similarly for $x_0 = 0$ we have the
$y$ axis and $G_{p_y} = \Z_2^{(x)}$.
This means in particular that $p_x$ and $p_y$ are both
isolated in their stratum (actually in both cases their
stratum does not contain any other point), and
according to theorem 3 they correspond to invariant
subspaces under any $G$-equivariant flow on $M$.
It is immediate to check that indeed the
$G$-equivariance for a vector field $X = f(x,y) \pa_x +
g(x,y) \pa_y $ implies
$$ \eqalign{
f(x,-y) \, = \, ~~ f(x,y) \ \ & \ \ g(x,-y) \, = \, -
g(x,y) \cr
f(-x,y) \, = \, - f(x,y) \ \ & \ \ g(- x,y) \,
= \, ~~ g(x,y) \ \ , \cr } \eqno(19) $$
and thus
$$ f(x,y) \ = \ \a (x^2,y^2) \, x \ \ ; \ \ g(x,y) \ = \ \b
(x^2 , y^2 ) \, y \ . \eqno(20) $$
Thus, the axes are invariant lines, as stated. $\odot$
\bigskip
{\tt Example 4.} Let us consider $M = \R^3$ with $G =
SO(2)$ acting as rotations around the $z$ axis. Now the
orbit space is identified by the $z$ coordinate, $\Om =
\R^1$. Points on the $z$ axis $\xi = (0,0,z)$ have
isotropy $G_\xi = G$, and points $\xi = (x,y,z)$ with
$x^2 + y^2 \not= 0$ have isotropy group $G_\xi = \{ e
\}$.
The $G$-equivariant vector fields are given by $X =
f(x,y,z) \pa_x + g(x,y,z) \pa_y + h(x^2 + y^2,z)
\pa_z$ such that
$$ \eqalign{
y f_x \, - \, x f_y \ = & \ ~~g \cr
y g_x \, - \, x g_y \ = & \ - f \ ; \cr} \eqno(21) $$
this yields immediately
$$ \eqalign{
f(x,y,z) \ =& \ \a (x^2 + y^2 , z) \,
x - \b (x^2 + y^2 , z) \, y \cr
g(x,y,z) \ =& \ \b (x^2 + y^2 , z) \, x + \a (x^2 + y^2 ,
z) \, y \ . \cr} \eqno(22) $$
If we look at $\Gr_1$, all points $p \in
\Gr_1$ corresponding to linear subspaces $\P_1 = ( \la
x_0 , \la y_0 , \la z_0 )$ with $x_0^2 + y_0^2 \not= 0$
have isotropy $G_{p_0} = \{ e \}$, while $p_z$
corresponding to $\P_1 = (0,0 , \la )$ has isotropy
group $G_{p_z} = G$. This point is isolated in its
Grassmannian isotropy stratum and thus it corresponds to
an invariant linear subspaces, as it is indeed immediate
to check from (22).
Looking now at $\Gr_2$, all points $p \in \Gr_2$ have
trivial isotropy group $G_p = \{ e \}$; thus there is
no point isolated in its stratum, and therefore no
two-dimensional linear subspace invariant under all
$G$-equivariant flows; again it is easy to check this
is the case, i.e. that for any two-dimensional subspace
$\P_2$ we can find a vector field of the form prescribed
by (22) such that $\P_2$ is not invariant. $\odot$
\bigskip
{\tt Example 5.} Let us now look at the case where $M$ is the Lie
algebra $\G$ of the group $G = SU(3)$, so that $M = \R^8
\approx su(3)$, on which $G$ acts by the adjoint action:
i.e., for $g \in G$, $A \in M$, $g: A \to g A g^{-1}$.
We recall that $M$ is the space of three dimensional
traceless hermitian matrices.
We will use as a basis in $M$ the Gell-Mann lambda
matrices $\la_1 , ... \la_8$ \ref{42,2-7,24}; with these, the
Cartan subalgebra $\G_0 = u(1) \oplus u(1)$ generating the
center $Z = U(1) \times U(1)$ of $G$ is spanned by $\la_3$
and $\la_8$; in physical terms, these correspond
respectively to the third direction of isospin $I_3$ and
to the hypercharge $Y$.
The orbit space $\Om = M / G$ is two dimensional
(corresponding to the two independent eigenvalues of the
traceless matrices $A \in M$), and can be represented as
the $(\la_3 , \la_8 )$ plane.
As well known \ref{4,7,24}, in $\Om$ there are three
strata: $a)$ the origin, with isotropy $G$; $b)$ the three
special lines $\ell_0 = \a \la_8$ and $\ell_\pm = \a
(\la_3 \pm \la_8 / \sqrt{3} )$, where $\a \in \R
\backslash \{ 0 \}$, with isotropy $G_0 \simeq G_\pm
\simeq SU(2) \times U(1)$; $c)$ the open and dense
(generic) stratum, including all other points, with
isotropy $U(1)$. As for the orbits, generic ones are
seven-dimensional, while those corresponding to $\om \in
\ell_i$ are four dimensional and obviously the origin of
$M$ corresponds to a zero-dimensional orbit \ref{4,7,24};
explicit computations are reported in detail in \ref{24}.
We can now apply the theorems obtained above; considering
$\Gr_{4,8}$ and using theorem 3, we obtain that the
four-dimensional spaces corresponding to $G (\ell_i )$
are invariant for any flow in $M = \G = su(3)$ which
is equivariant under the adjoint action of $G = SU(3)$.
Similarly, considering $\Gr_{4,8} / G = \Om_{4,8}$ and
using theorem 4, we would come to the same conclusion.
Notice that in the physically relevant \ref{2-7} case
where the equivariant flow $f$ is the gradient flow of an
invariant potential $V$ (i.e. $f = - \nabla V$) and we look
for critical points of $V$, and thus zeroes of $f$, this
result in itself allows only to reduce to a
four-dimensional problem. Actually, quotienting out the
$G$-action we are reduced to one-dimensional problems
along the lines $\ell_i$ (since $\nabla V$ is obviously
orthogonal to $G$-orbits when $V$ is $G$-invariant); if
$V(x)$ is known to be convex for $|x| \to \infty$ we can
immediately guarantee the existence of critical orbits on
the lines $\ell_i$, and if the origin corresponds to an
unstable critical point we can also guarantee the
existence of nontrivial critical orbits on the $\ell_i$'s
\ref {1-8,24}. $\odot$
\bigskip
{\tt Example 6.} (The Nahm equations)
As a final and, again, physically relevant example, we
consider the Nahm equations \ref{38,43-46}. From the point of
view of use here, the Nahm equations describe a dynamical
system in which the dynamical variables $A_i$ ($i =
1,2,3$) are the spatial components of a connection form
$\a = A_\mu {\rm d} x^\mu$ on the adjoint bundle to a $G$
principal bundle over $\R^{4}$ (we assume $x^0$ represents
the time coordinate $t$); thus we have $M = \R^{3k} = \G
\times \G \times \G$, where $\G \simeq \R^k$ is the Lie
algebra of the group $G$.
The Nahm equations are written in the form
$$ {d A_i \over dt}Ê\ + \ [A_0 , A_i ] \ = \
(1/2) \, \epsilon_{ijk} \, [A_j , A_k ] \ ; \eqno(23) $$
The gauge group $\Ga$ modelled on $G$ [of smooth
$G$-valued functions $g(t)$] acts on solutions to
the Nahm equation as
$$ \eqalign{
g (A_0 ) \ =& \ g \, A_0 \, g^{-1} \ - \ g_t \,
g^{-1} \cr
g (A_i ) \ =& \ g \, A_i \, g^{-1} \ . \cr} \eqno(24) $$
Thus by a gauge fixing we can reduce the Nahm equations to
$$ {d A_i \over dt} \ = \ {1 \over 2} \, \epsilon_{ijk} \,
[A_j , A_k ] \ , \eqno(25) $$
(notice $A_0$ does not enter in the equations any more).
Notice that this flow is still $G$-equivariant, the
$G$-action being now simply given by
$$ g (A_i) \ = \ g A_i g^{-1} \eqno(26) $$
i.e. the adjoint action of $G$ on its Lie algebra.
We also have a discrete symmetry corresponding to
permutations of the three $\G$ factors in $M$, but we
will not use this here.
In the case $G = SU(3)$, for which $M = \R^{24}$ we can
use the results obtained above in example 5.
One is in particular interested in a reduction to the
special subspaces $\ell_i$ considered in example 5; it is
natural to ask if initial data with $A_i \in \ell_{p_i}$
(where $p_i = 0,+,-$) will originate a flow which remains
in such subspaces. A direct proof of this would require
involved computations, but we can answer (affirmatively)
this question with very little effort by considering
$\Om = \Gr_{12,24} / G$ and in this the orbits
corresponding to $$ \om_{ijk} \ = \ \ell_{p_i} \, \times
\, \ell_{p_j} \, \times \, \ell_{p_k} \ \ss \ \G \times \G
\times \G \ \ . \eqno(27) $$
These orbits are isolated in their stratum, as it follows
from the results recalled above \ref{1-7}, and thus they
are also invariant under any equivariant flow -- as
stated in theorem 4 -- and thus in particular under the
flow given by the Nahm equations. $\odot$
\vfill\eject
\section{8. Discussion.}
In this final section, we wish to present briefly some
remarks, mainly in view of extensions and/or
applications of the approach presented here.
{\tt Remark 8.} First of all, we notice that although we
have preferred to deal with $\R^n$ and with real
Grassmann manifolds, all our discussion would be the
same when dealing with ${\bf C}^n$ and with complex
Grassmann manifolds. Actually, the real case could in
principle present difficulties not present in the
complex one, and that is why we preferred to treat it
explicitely. $\odot$
{\tt Remark 9.} In her thesis, M. Koenig has shown that
one can study the stratification of a manifold (and its
orbit space) by considering linear fields
alone \ref{47-49} (in this respect, see also the work
of Sartori and collaborators \ref{13-16}, who follow a
different but related approach). As mentioned in remark 4,
if the vector fields $f: M \to \T M$ (where $M = \R^n$)
we consider are
linear, we can also extend them to vector fields $\phi :
\Gr_{k,n} \to \T \Gr_{k,n}$. It seems thus reasonable to
expect that applying the Koenig's approach in this case
one could obtain better results concerning the
stratification of Grassmann manifolds, invariant
subspaces, and maybe also relevant applications to the
variational case. $\odot$
{\tt Remark 10.} Although in this note we only deal with
linear $G$-action in $\R^n$, it appears that this
approach can also be extended to a more general setting,
i.e. to a general, possibly nonlinear, action of a group
$G$ (possibly non-compact) on a smooth manifold $M$
(possibly infinite dimensional). In this case, provided
the $G$-action is such to be linearizable at all points
of $M$, or at least at all $G$-invariant points $M^{G}$,
one can study the linearization $(DG)_x$ acting on $\T_x
M \approx \R^n$ and repeat the same analysis conducted
here. The invariant linear subspaces identified in this
way would then be tangent subspaces to invariant
(possibly, only local) submanifolds in $M$. $\odot$
{\tt Remark 11.} Further extension is also possible to infinite
dimensional situations, provided the (now, infinite dimensional)
manifold $M$ and the group (now, possibly noncompact) $G$ and its
action satisfy suitable hypotheses. In particular, the same results
would continue to hold if we were considering a Sobolev space of sections
of a fiber bundle with $G$ compact, or even for $G$ non-compact provided
some other conditions are met. Suitable conditions are discussed e.g.
by Palais in \ref{18}; see also \ref{25}. $\odot$
{\tt Remark 12.} As shown in section 2, Michel theory has a natural
setting in terms of equivariant -- not necessarily gradient --
vector fields, in which case isolated orbits correspond to relative
equilibria for any equivariant vector field. For infinite
dimensional cases, it is known \ref{50} that one can have
common solutions to all the differential equations with a common
symmetry (in a given function space). It is not difficult to see,
comparing the discussion given here and the one in \ref{50}
that such an extension would also go through to identify relative
equilibria, i.e. invariant manifolds, in this wider case. However,
we prefer to postpone such a discussion to a more complete treatment
of the infinite dimensional case. $\odot$
{\tt Remark 13.} As mentioned in the Introduction,
recently Manas, Martinez Alonso and Medina noticed that a
relation exists between singular strata in the infinite
Grassmannian and integrable systems, which they call the
``hidden KdV hierarchies''; these were known before
\ref{51,52}, but their relation to singular strata was not
noticed; on the other side, Adler and van Moerbeke already
noticed connection between specific features of functions
associated to the KP hierarchy and to the multiperiodic
Toda flows \ref{53,54}. This was indeed a main motivation
for the present work, and we hope this can be used in the
future as a first step towards an analysis of these
questions related to the singular strata in the infinite
Grassmannian. $\odot$
{\tt Remark 14.} The main motivation for the present work
was provided by the analysis of the degeneration implied by linearity of
group actions in Michel's theory, and our whole discussion has been
based on linear concepts, first of all the very one of Grassmann
manifold. Thus asking if a ``nonlinear version'' of the theory
developed here is possible, could at first appear to be contradictory,
and -- even worse -- unuseful.
However our discussion, beside providing a proper geometrical setting for
dealing with the ``linear degeneration'' in Michel's theory, also showed that
the latter can be of use for the determination of flow-invariant (linear)
subspaces for equivariant flows; such an application of Michel's theory (which,
as far as I know, went unnoticed so far) calls naturally for an extension to
the general case, i.e. the one where the $G$-action is not necessarily linear,
the linear subspaces with same isotropy type are replaced by smooth
submanifolds with this property, and even more the linear invariant subspaces
are replaced by invariant smooth manifolds.
As mentioned above (see remark 10), one could proceed locally, i.e. considering
the linear space $\T_x M$ and the linearization $(D G)_x$ of the $G$-action on
it, at each point $x \in M$. However, a global formulation would obviously be
more attractive; this would require to use {\it extended jet bundles}
\ref{55-57} rather than Grassmann manifolds, and would be out of place here, so
that we defer such a discussion to a later contribution. $\odot$
\section{Acknowledgements.}
I would like to thank Paola Morando for useful E-discussion concerning both the
present work and its possible generalizations and extensions, and the Theory
Group of Rome I (``La Sapienza'') Physics Department for kind hospitality.
\vfill\eject
\section{Appendix. Some basic properties of Grassmann
manifolds.}
\def\dim{{\rm dim}\, }
\def\Mb{{\bf M}}
Grassmann manifolds are a standard tool in Differential
Geometry and many of its applications, including
physical and/or physically relevant ones; on the other
side, they have been rarely (or not at all) used in
the study of Symmetry Breaking or in Nonlinear Dynamics:
thus, it seems appropriate to collect here some basic
facts about them, for the benefit of readers not
familiar with the subject.
We will mainly follow \ref{38}; details on Grassmann
Manifolds are to be found in any book on Differential
Geometry, e.g. \ref{35-40}. We will discuss real
Grassmann manifolds; the construction and results for
complex ones are equivalent.
Consider $\R^n$; consider in here the set of
$k$-dimensional linear subspaces $\R^k \ss \R^n$.
This set can be given a structure of smooth manifold as
described below. We denote this set by $\Gr_{k,n}$, in
anticipation of the introduction of this structure,
with which it will actually be the Grassmann manifold.
Denote by $\P^k$ a given $k$-dimensional linear
subspace of $\R^n$, spanned by (say, orthogonal)
vectors $\{ \xi_1 , ... , \xi_k \}$ (where $\xi_i \in
\R^n$); we can obtain any other $k$-dimensional linear
subspace by acting on $\P^k$ by an element of $O (n)$,
the orthogonal group in $n$ dimensions. Actually, it is
clear that elements of the $O(k) \ss O(n)$ subgroup
corresponding to transformation of $\P^k$ into itself
will map $\P^k$ into itself and should be quotiented
out; similarly, elements of the $O(n-k)$ subgroup
acting in the space orthogonal to $\P^k$ will have no
influence whatsoever. Thus,
$$ \Gr_{k,n} \ \approx \ O(n) \, / \, [ \, O(k) \,
\times \, O(n-k) \, ] \ . \eqno(A1) $$
The orbit space $O(n)/[O(k) \times O(n-k)]$ is a smooth
manifold, as follows easily from general theorems in Lie
group theory (see e.g. \ref{5}) and thus (A1) shows
immediately that $\Gr_{k,n}$ is a manifold once equipped
with the manifold structure of $O(n)/[O(k) \times
O(n-k)]$ (see e.g. \ref{37-39} for details); its dimension
is therefore
$$ \dim \Gr_{k,n}Ê\ = \ \dim O(n) \, - \, \dim O(k) \, -
\, \dim O(n-k) \ = \ k (n-k) \ . \eqno(A2) $$
For a construction of $\Gr_{k,n}$ based on linear
algebra rather than Lie groups, we can proceed as
follows. Consider the set $\Mb_k$ of $(k \times
n)$ matrices of maximal rank $k \le n$. To the $k$
linearly independent vectors $\xi_i$ of components
$(\xi_i^1 , ... \xi_i^n )$ identifying $\P^k$ we
associate a matrix $M \in \Mb_k$ by $ M_{ij} \ = \
\xi_i^j$.
Consider now $G_k = {\rm GL} (k,\R )$: for any $g \in
G_k$, the matrix $M' = g M$ represents the same
$\P^k$, as it just provides a change of basis in $\P^k$.
Thus $$ \Gr_{k,n} \ \approx \ \Mb_k \, / \, G_k \ .
\eqno(A3) $$
Consider now $M \in \Mb_k$; we can write it as a
block matrix $ M \ = \ \( A_1 \, \vert \, A_* \)$,
where $A_1$ is a $(k \times k)$ matrix -- say for
simplicity corresponding to the first $k$ columns of $M$
-- of maximal rank (this is always possible as $M$ is
of rank $k$), and $A_*$ is a $[k \times (n-k)]$ matrix.
Then, acting by $A_1^{-1} \in G_k$ we can always obtain
as representative of the $G_k$ coset to which $M$
belongs, the matrix $$ \widetilde{M} \ = \ \( I \, |
\, A_1^{-1} \, A_* \) \ . \eqno(A4) $$ Thus, $\Mb_k /
G_k$ corresponds to the real matrices $A_1^{-1} \,
A_*$; notice these are just the matrices $A_*$ under a
change of variables, and that no restriction whatsoever
existed on the $A_*$; thus $\Mb_k / G_k$ corresponds to
the space of $[ k \times (n-k)]$ real matrices, which is
isomorphic to $\R^{k(n-k)}$, and can be given its
manifold structure.
\vfill\eject
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\parindent=10pt
\parskip=0pt
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\font\petitrm = cmr8
\font\petitit = cmsl8
\font\petitbf = cmbx8
\baselineskip=10pt
\def\tit#1{{\petitrm ``#1''}}
\def\tit#1{{``#1''}}
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\bye