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)
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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=19173