🚀 go-pugleaf

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.