🚀 go-pugleaf

>In article <1992Nov13.213840.10075@galois.mit.edu> jbaez@riesz.mit.edu
>(John C. Baez) writes:
>>As for other cases, I'm suspicious.  Take a look at the proof of the
>>existence and uniqueness of the Levi-Civita connection and see what
>>happens if your metric is replaced by a nondegenerate *skew-symmetric*
>>bilinear form on the tangent bundle.  I'm afraid something will go
>>wrong.  Why?  If nothing did, every symplectic maniold would be blessed
>>with a natural connection analogous to the Levi-Civita connection.  If
>>such a thing existed I should have heard about it, but I haven't.  Of
>>course, it's possible that I am missing out on this crucial facet of
>>symplectic geometry!!
>>
>>Perhaps the symplectic geometers and fans of gravity theories with
>>asymmetric metric tensors can straighten this out in a jiffy.

>
>
>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.
>
>
>-JAMES DOLAN

A good reference for this sort of result is
`Transformation groups in differential geometry' by S. Kobayashi

Offhand, there is a possibility that one might require compactness
of the structural group to make things work: one approach for
generalizing the Riemannian result is to define a Riemannian
metric on the total space of the frame bundle in such a way
that connection-preserving maps of the manifold lift to
isometries of the frame bundle.  I seem to recall seeing
something like this in Kobayashi and Nomizu.

DRW