Nov 29, 2008
A recent web publication [1, Abstract and conclusion (15)?Þ?(16)] leads to the question whether a linear connection Γρμν of a pseudo-Riemanian manifold M can be antisymmetric with respect to all coordinate bases, i.e. whether
(1) Γρμν = − Γρνμ (antisymmetry of connection)
remains valid also under arbitrary local changes
(2) xμ' = xμ'(xμ)
of the coordinate basis.
The answer can easily be given by considering the transformation behaviour of the connection coefficients as reported here from a private communication by W.A. Rodrigues Jr.:
Any coordinate transformation (2) causes a transformation of the connection coefficient Γρμν → Γρ'μ'ν' to be specified here:
(3) aμ'μ := ∂xμ'/∂xμ and aμμ' := ∂xμ/∂xμ'
denote the transformation coefficients of the coordinate transformation (2). Then, as is well known (see introductory textbooks e.g. [2, p.56]), the connection transforms as follows:
(4) Γρ'μ'ν' = Γρμν aρ'ρ aμμ' aνν' − aμμ' aνν' ∂ aρ'μ /∂xν
The first term on the right hand side does not disturb the symmetry behaviour: If Γρμν is symmetric/antisymmetric in μ,ν then so is Γρμν aρ'ρ: in μ',ν' symmetric/antisymmetric respectively. However, the second term is of interest: Due to
(5) aνν' aμμ' ∂ aρ'ν /∂xμ = aμμ' aνν' ∂ aρ'μ /∂xν (since ∂ aρ'ν /∂xμ = ∂² xρ' /∂xμ∂xν = ∂ aρ'μ /∂xν )
this term is term is always symmetric in μ',ν'. So if antisymmetry is wanted then this term does not play with and spoils the wanted antisymmetry in general:
Therefore we have the following result:
Therefore the answer to our introductory question is negative: A connection antisymmetric in all possible coordinate bases cannot exist.
 M.W. Evans, ON THE SYMMETRY OF THE CONNECTION IN RELATIVITY
AND ECE THEORY,
 S.M. Carroll, Lecture Notes on General Relativity, Chapter 3
 G.W. Bruhn, Commentary on Evans' web note #122,
Fundamental theorem of Riemannian geometry