## 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'}/_{∂xμ}
and
a^{μ}_{μ'} :=
^{∂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)
Γ^{ρ'}_{μ'ν'} =
Γ^{ρ}_{μν} a^{ρ'}_{ρ}
a^{μ}_{μ'}
a^{ν}_{ν'} −
a^{μ}_{μ'}
a^{ν}_{ν'}
^{∂ aρ'μ }/_{∂xν}

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 }/_{∂xμ}
=
a^{μ}_{μ'}
a^{ν}_{ν'}
^{∂ 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:

### 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