Dear vanesh, I have decided to move the anti-gravitation discussion into a new thread. At least in this regard josh is right, it's gotten a bit off-topic in the old one. For those who want to follow, discussion is about the paper Anti-Gravitation see also garrett's smart comments on my blog No, it has not. (Actually, that offends me because that is what I criticize most about the works by Frederic Henry-Couannier etal. They do have two metrics, which is imo completely unphysical. They should at least try to interpret this). The manifold has one metric. The different set of connection coefficients corresponds to the different properties of the quantity to be transported, not to any different properties of the manifold. If it's too implausible to think about anti-gravitation, think about transporting a spinor. It has a well defined transport-behaviour. The connection you use for this however, is not that of a vector. It's different, because it transforms under another representation than the vector. Same with the anti-gravitating particle. It has a different transformation behaviour under general diffeomorphism than a usual particle. Therefore, the covariant derivative (and the connection coefficients appearing in it) are different. They can be derived directly from the metric and the transformation law of the particle, but they are not identical to the usual Christoffelsymbols. Best, B.