In terms of synthetic differential geometry, we give a variational characterization of the connection (parallelism) associated to a pseudo-Riemannian metric on a manifold.
Theory and Applications of Categories, Vol. 4, 1998, No. 9, pp 195-207 http://www.tac.mta.ca/tac/volumes/1998/n9/n9.dvi http://www.tac.mta.ca/tac/volumes/1998/n9/n9.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1998/n9/n9.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1998/n9/n9.ps