On the differentiability conditions at spacelike infinity

We consider spacetimes which are asymptotically Bat at spacelike infinity, i(0). It is well known that, in general, one cannot have a smooth different iable structure at i(0), but rather one has to use direction-dependent stru ctures there. Instead of the usual C->1-differentiable structure, we sugges t a weaker differential structure, a C1+ structure. The reason for this is that there do not appear to be any completions of the Schwarzschild spaceti me which is C->1 in both spacelike and null directions at i(0). In a C1+ st ructure all directions can be treated on an equal footing, at the expense o f logarithmic singularities at i(0). We show that, in general, the relevant part of the curvature tensor, the Weyl parr, is free from these singularit ies, and that the (rescaled) Weyl tensor has a certain symmetry property.