From: palais@binah.cc.brandeis.edu
Newsgroups: sci.physics,sci.math
Subject: Re: Covariant vs. Lie Derivative in Gen. Rel.?
Date: 17 Nov 92 17:45:28 GMT
Reply-To: palais@binah.cc.brandeis.edu
Organization: Brandeis University

> rscott@libws3.ic.sunysb.edu (Robert Scott) writes:

>>ISN'T IT EASY TO SHOW THAT THE LIE ALGEBRA OF SYMMETRIES OF AN
>>AFFINE CONNECTION ON A CONNECTED FINITE-DIMENSIONAL MANIFOLD IS
>>FINITE-DIMENSIONAL?  THUS THE LIE ALGEBRA OF INFINITESIMAL
>>SYMPLECTOMORPHISMS OF A SYMPLECTIC MANIFOLD IS FAR TOO BIG TO
>>PRESERVE AN AFFINE CONNECTION.

>You confuse me a bit :)

>If I understand right, the covariant derivative must only preserve the
>symplectic metric j with components

>  j_{ab} = j_{[ab]} = - j_{ba} ,

>not the set of all symplectic transformations, to obtain a metricity
>condition analogous to the Riemannian: There the Levi-Civita connection only
>leaves (pseudo)-orthogonal g invariant, not all elements of the Lorentz
>group. The symplectic metricity condition reads explicitely

>  0 == - j_{ab;c} = - j_{ab,c} + j_{db} G^d_{ac} + j_{ad} G^d_{bc} .
                                                   ^^^^^^
                                                 = - j_{da}

>The connection is then a 1-form with values in the symplectic Lie
>algebra which is defined as the set of all generators of transformations
>that leave j invariant, as the Riemannian is contained in the (local,
>[pseudo]-orthogonal) invariance group of g. As I tried to line out in a
>previous posting, I do not see a reasonable argument for uniquely fixing a
>symplectic-metric connection, which happens to be possible for the
>Riemannian by the demand of vanishing torsion, so that there remain
>connection components independent of j.

>BTW: The symplectic Lie algebra Sp(D,j) is of course finite-dimensional at
>  each point, and the j-metric connection is contained in it by def. It is
>  only infinite-dimensional (like the space of any scalar, vector, tensor,
>  spinor, or isotensor fields, i.e. the corresponding bundle) when viewed
>  non-locally, i.e. in some neighborhood, etc. Dimension of Sp(D,j) is
>  D*(D+1)/2  if D is the dimension of the manifold under consideration.
--
===============
   There is great confusion going on here, and I hope the following will help 
clear it up, rather than increase the confusion! 

  First, Robert Scott was replying to a remark of John Baez, saying that he did 
not believe that there was any way to construct canonically a connection (or 
covariant equivalently a covariant derivative) from a symplectic structure). 
Scott's answer (while not new) was exactly on target. His point was that if 
there was a canonical connection on a symplectic manifold (like the "Levi-Civita"
connection on a Riemannian manifold) then just as an isometry of a Riemannian 
manifold preserves the Levi-Civita connection, any symplectomorphism of a 
symplectic manifold would preserve this canonical connection. But then he says 
it would follow that the  group of syplectomorphisms would be isomorphic to 
a subgroup of the group of automorphisms of a connection---which is well-known 
to be finite dimensional (proof sketched below) while it is also well known that 
the group of symplectomorphisms is always infinite dimensional (the Lie algebra
is isomorphic to the algebra of smooth functions (modulo constants) under Poisson
bracket). (Of course the same argument applies to Riemannian manifolds and shows