On the differentiability conditions at spacelike infinity

We consider space-times which are asymptotically flat at spacelike infinity, i. It is well known that, in general, one cannot have a smooth differentiable structure at i, but have to use direction dependent structures. Instead of the oftenly used C-differentiabel structure, we suggest a weaker differential structure, a C + structure. The reason for this is that we have not seen any completions of the Schwarzschild space-time which is C in both spacelike and null directions at i. In a C + structure all directions can be treated equal, at the expense of logarithmic singularities at i. We show that, in general, the relevant part of the curvature tensor, the Weyl part, is free from these singularities, and that the (rescaled) Weyl tensor has a certain symmetry property. ∗Linköping University, Department of Mathematics, S-581 83 Linköping, Sweden