A remark on antisymmetric connections

Gerhard W. Bruhn, Darmstadt University of Technology

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:

Let

(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 = ∂ 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:

A coordinate transformation (2) preserves symmetry of the connection in the two lower indices while antisymmetry is NOT preserved in general.

Therefore the answer to our introductory question is negative: A connection antisymmetric in all possible coordinate bases cannot exist.


References

[1] M.W. Evans, ON THE SYMMETRY OF THE CONNECTION IN RELATIVITY AND ECE THEORY,
      http://www.aias.us/documents/uft/a122ndpaper.pdf

[2] S.M. Carroll, Lecture Notes on General Relativity, Chapter 3

[3] G.W. Bruhn, Commentary on Evans' web note #122,
      http://www.mathematik.tu-darmstadt.de/~bruhn/onEvansNote122.html



Links

Fundamental theorem of Riemannian geometry

Riemannian manifold

Pseudo-Riemannian manifold

Levi-Civita connection

Covariant derivative