🚀 go-pugleaf

jbaez@riesz.mit.edu (John C. Baez) writes:
>anderson@fermi.phys.ualberta.ca (Warren G. Anderson) writes:
>>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.  [..]
>Perhaps the symplectic geometers and fans of gravity theories with
>asymmetric metric tensors can straighten this out in a jiffy.

Although not totally contained in that set :-)

Your distrust on the determination of a connection G via an asymmetric metric
g is appropriate. While the metricity condition, i.e. vanishing of 
nonmetricity

  0 == Q_{abc} = - g_{ab,c} + g_{db} G^d_{ac} + g_{ad} G^d_{bc}

determines the symmetric part of G, 

  G_{(ab)c} [= g_{d(a} G^d_{b)c}] = (1/2) g_{ab,c} ,

in case of a (pseudo)-orthogonal metric, and thereby in the case of 
vanishing torsion (and in a coordinate frame) the total of G, this is not
true for a non-symmetric g unless this contains a (tensorially invariant)
symmetric part that alone may serve as (pseudo)orthogonal/Riemannian metric,
while the skew-symmetric part may then be treated as independent tensor 
field. You then only have the metricity condition given above, restricting
some components of G, but usually not all. So the case of a symplectic 
manifold you mentioned thus leaves some part of the connection (the 
symmetric one after lowering an index with the symplectic metric j) 
undetermined, so that the connection has a part that is independent of j, 
which cannot be annulled by a reasonable condition like torsion freeness. 

Of course you can still postulate *by hand* that the connection should be 
given by a combination of metric components, but that restricts the theory.
At least the author does not see a reasonable argument for this procedure.

[For those who now quest how the connection G is defined: G comes from the 
covariant derivative of the tangent space base vectors:

    D_a e_b = G^c_{ba} e_c  ]
--
 Hartmut Frommert                 <phfrom@nyx.uni-konstanz.de>
 Dept of Physics, Univ of Constance, P.O.Box 55 60, D-W-7750 Konstanz, Germany
                                           -- Eat whale killers, not whales --