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Difference between revisions of "Subvariety, involutive"

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m (tex encoded by computer)
m (fixing spaces)
 
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be a vector space of dimension  $  2n $
 
be a vector space of dimension  $  2n $
 
and  $  \omega $
 
and  $  \omega $
a non-degenerate alternating  $  2 $-
+
a non-degenerate alternating  $  2 $-form on it. Given a subspace  $  W $
form on it. Given a subspace  $  W $
 
 
of  $  V $,  
 
of  $  V $,  
 
one defines (as usual)
 
one defines (as usual)
Line 47: Line 46:
 
is dense in  $  V $,  
 
is dense in  $  V $,  
 
then  $  V $
 
then  $  V $
is involutive if and only if for every two  $  C  ^ {1} $-
+
is involutive if and only if for every two  $  C  ^ {1} $-functions  $  f $,  
functions  $  f $,  
 
 
$  g $
 
$  g $
 
on  $  X $
 
on  $  X $
 
which vanish on  $  V $
 
which vanish on  $  V $
the Poisson bracket  $  \{ f, g \} $(
+
the Poisson bracket  $  \{ f, g \} $ (defined by the symplectic  $  2 $-form on  $  X $)  
defined by the symplectic  $  2 $-
 
form on  $  X $)  
 
 
also vanishes on  $  V $.
 
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=52047