From: palais@binah.cc.brandeis.edu
Subject:  Re: Covariant & Lie Derivative
Reply-To: palais@binah.cc.brandeis.edu
Organization: Brandeis University
Date: Tue, 17 Nov 1992 19:03:57 GMT

I tried twice before to post this message, in responseto message 15638 from
phfrom@nyx.uni-konstanz.de (Hartmut Frommert), but each time a transmission
problem completely garbled it and I'll try once again.  
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> 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  that the group of isometries of a Riemannian manifold
is a Lie group). 

   OK, to see that the group G of diffeomorphisms of a manifold M that 
preserves a connection C on TM is finite dimensional, it suffices to show
that G acts freely on the frame bundle F(M). (For then, if f is any frame 
of M, the orbit map, g |---> gf is an immersion of G into F(M), so 
dim (G) \le dim (F(M)) = dim (M) + dim (M)^2). 
This will follow if we can show that the subgroup of  G that fixes a
frame f (say at p) is the identity. But if g in G fixes f, it  fixes p and
Dg fixes the tangent space at p. Then, since g also fixes the connection
C,  it fixes every affine path starting at p, and it follows that g fixes
all points  near p. A little connectivity argument completes the proof. 
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