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''(in symplectic geometry)''
 
''(in symplectic geometry)''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910601.png" /> be a vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910603.png" /> a non-degenerate alternating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910604.png" />-form on it. Given a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910605.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910606.png" />, one defines (as usual)
+
Let $  V $
 +
be a vector space of dimension $  2n $
 +
and $  \omega $
 +
a non-degenerate alternating $  2 $-form on it. Given a subspace $  W $
 +
of $  V $,  
 +
one defines (as usual)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910607.png" /></td> </tr></table>
+
$$
 +
W  ^  \perp  = \{ {x \in V } : {\omega ( x, w)= 0 \textrm{
 +
for  all  }  w \in W } \}
 +
.
 +
$$
  
One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910608.png" /> is an isotropic subspace if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s0910609.png" />, that it is an involutive subspace (or co-isotropic subspace) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106010.png" />, and that it is a Lagrangian subspace if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106011.png" />. Note that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106012.png" /> to be involutive it is necessary that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106013.png" />.
+
One says that $  W $
 +
is an isotropic subspace if $  W \subset  W  ^  \perp  $,  
 +
that it is an involutive subspace (or co-isotropic subspace) if $  W \supset W  ^  \perp  $,  
 +
and that it is a Lagrangian subspace if $  W = W  ^  \perp  $.  
 +
Note that for $  W $
 +
to be involutive it is necessary that $  \mathop{\rm dim} ( W) \geq  n $.
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106014.png" /> be a subvariety (possibly with singularities; or, more generally, an analytic subset) of a [[Symplectic manifold|symplectic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106015.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106016.png" /> be the set of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106017.png" /> which have a neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106018.png" /> that is free of singular points. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106019.png" /> is an involutive subvariety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106020.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106021.png" /> the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106023.png" /> is involutive. The notions of an isotropic subvariety and a Lagrangian subvariety are defined analogously. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106024.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106026.png" /> is involutive if and only if for every two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106027.png" />-functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106029.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106030.png" /> which vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106031.png" /> the Poisson bracket <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106032.png" /> (defined by the symplectic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106033.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106034.png" />) also vanishes on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091060/s09106035.png" />.
+
Now, let $  V $
 +
be a subvariety (possibly with singularities; or, more generally, an analytic subset) of a [[Symplectic manifold|symplectic manifold]] $  X $.  
 +
Let $  \mathop{\rm Reg} ( V) $
 +
be the set of points of $  V $
 +
which have a neighbourhood in $  V $
 +
that is free of singular points. Then $  V $
 +
is an involutive subvariety of $  X $
 +
if for all $  p \in  \mathop{\rm Reg} ( V) $
 +
the subspace $  V _ {p} $
 +
of $  X _ {p} $
 +
is involutive. The notions of an isotropic subvariety and a Lagrangian subvariety are defined analogously. If $  \mathop{\rm Reg} ( V) $
 +
is dense in $  V $,  
 +
then $  V $
 +
is involutive if and only if for every two $  C  ^ {1} $-functions $  f $,  
 +
$  g $
 +
on $  X $
 +
which vanish on $  V $
 +
the Poisson bracket $  \{ f, g \} $ (defined by the symplectic $  2 $-form on $  X $)  
 +
also vanishes on $  V $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) {{MR|0882548}} {{ZBL|0643.53002}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) {{MR|0882548}} {{ZBL|0643.53002}} </TD></TR></table>

Latest revision as of 15:36, 11 February 2022


(in symplectic geometry)

Let $ V $ be a vector space of dimension $ 2n $ and $ \omega $ a non-degenerate alternating $ 2 $-form on it. Given a subspace $ W $ of $ V $, one defines (as usual)

$$ W ^ \perp = \{ {x \in V } : {\omega ( x, w)= 0 \textrm{ for all } w \in W } \} . $$

One says that $ W $ is an isotropic subspace if $ W \subset W ^ \perp $, that it is an involutive subspace (or co-isotropic subspace) if $ W \supset W ^ \perp $, and that it is a Lagrangian subspace if $ W = W ^ \perp $. Note that for $ W $ to be involutive it is necessary that $ \mathop{\rm dim} ( W) \geq n $.

Now, let $ V $ be a subvariety (possibly with singularities; or, more generally, an analytic subset) of a symplectic manifold $ X $. Let $ \mathop{\rm Reg} ( V) $ be the set of points of $ V $ which have a neighbourhood in $ V $ that is free of singular points. Then $ V $ is an involutive subvariety of $ X $ if for all $ p \in \mathop{\rm Reg} ( V) $ the subspace $ V _ {p} $ of $ X _ {p} $ is involutive. The notions of an isotropic subvariety and a Lagrangian subvariety are defined analogously. If $ \mathop{\rm Reg} ( V) $ is dense in $ V $, then $ V $ is involutive if and only if for every two $ C ^ {1} $-functions $ f $, $ g $ on $ X $ which vanish on $ V $ the Poisson bracket $ \{ f, g \} $ (defined by the symplectic $ 2 $-form on $ X $) also vanishes on $ V $.

References

[a1] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) MR0882548 Zbl 0643.53002
How to Cite This Entry:
Subvariety, involutive. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subvariety,_involutive&oldid=24573