Difference between revisions of "Subvariety, involutive"
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 17:01, 15 April 2012
(in symplectic geometry)
Let 
be a vector space of dimension 
and 
a non-degenerate alternating 
-form on it. Given a subspace 
of 
, one defines (as usual)
 |
One says that 
is an isotropic subspace if 
, that it is an involutive subspace (or co-isotropic subspace) if 
, and that it is a Lagrangian subspace if 
. Note that for 
to be involutive it is necessary that 
.
Now, let 
be a subvariety (possibly with singularities; or, more generally, an analytic subset) of a symplectic manifold 
. Let 
be the set of points of 
which have a neighbourhood in 
that is free of singular points. Then 
is an involutive subvariety of 
if for all 
the subspace 
of 
is involutive. The notions of an isotropic subvariety and a Lagrangian subvariety are defined analogously. If 
is dense in 
, then 
is involutive if and only if for every two 
-functions 
, 
on 
which vanish on 
the Poisson bracket 
(defined by the symplectic 
-form on 
) also vanishes on 
.
References
| [a1] | P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) MR0882548 Zbl 0643.53002 |
Subvariety, involutive. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subvariety,_involutive&oldid=24573