🚀 go-pugleaf

In article <1992Nov13.213840.10075@galois.mit.edu> jbaez@riesz.mit.edu
(John C. Baez) writes:
>In article <1992Nov12.172748.16273@kakwa.ucs.ualberta.ca>
>anderson@fermi.phys.ualberta.ca (Warren G. Anderson) writes:
>>In article <1992Nov11.062853.22717@galois.mit.edu> jbaez@riesz.mit.edu
>>(John C. Baez) writes:
>>> 2) The covariant derivative only requires a tangent vector at one point
>>> of the manifold.  The price you pay is this: to define it you need to
>>> choose a connection on the (tangent bundle of) the manifold.  Of course,
>>> such a connection - the Levi-Civita connection - comes for free if your
>>> manifold has a Riemann metric on it.  
>>
>>Or even a pseudo-Riemannian metric. In fact, wouldn't any way of identifying 
>>the tangent space with it's dual be enough?
>
>Agreed, pseudo-Riemannian is fine and of course that's what you have in GR.
>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