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 Γ^ρ_μν of a pseudo-Riemanian manifold M can be antisymmetric with respect to all coordinate bases, i.e. whether

(1)         Γ^ρ_μν = − Γ^ρ_νμ                 (antisymmetry of connection)

remains valid also under arbitrary local changes

(2)         x^μ' = x^μ'(x^μ)

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 Γ^ρ_μν → Γ^ρ'_μ'ν' to be specified here:

Let

(3)         a^μ'_μ := ^∂xμ'/_∂x^μ         and         a^μ_μ' := ^∂xμ/_∂x^μ'

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)         Γ^ρ'_μ'ν' = Γ^ρ_μν a^ρ'_ρ a^μ_μ' a^ν_ν' − a^μ_μ' a^ν_ν' ^{∂ aρ'_μ} /_∂x^ν

The first term on the right hand side does not disturb the symmetry behaviour: If Γ^ρ_μν is symmetric/antisymmetric in μ,ν then so is Γ^ρ_μν a^ρ'_ρ: in μ',ν' symmetric/antisymmetric respectively. However, the second term is of interest: Due to

(5)         a^ν_ν' a^μ_μ' ^{∂ aρ'_ν} /_∂x^μ = a^μ_μ' a^ν_ν' ^{∂ aρ'_μ} /_∂x^ν         (since         ^{∂ aρ'_ν} /_∂x^μ = ^{∂² xρ'} /_{∂x^μ∂x^ν} = ^{∂ aρ'_μ} /_∂x^ν )

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://www2.mathematik.tu-darmstadt.de/~bruhn/onEvansNote122.html

Links

Fundamental theorem of Riemannian geometry

Riemannian manifold

Pseudo-Riemannian manifold

Levi-Civita connection

Covariant derivative