If chess board were a globe, such that King from h1 cud move to a1, then there is no R&K v K #, righ

I think the question is self, explanatory. In this scenario there couldn't possibly be a Rook and King vs King #.

Yeah, a king would never has less than 8 free squares if both the top and bottom are connected and also the sides are connected.

For example on a1 the king could move to the usual 3 squares plus h1, h2, a8, b8, and h8 for a total of 8.

So to mate will always require opposing the king + rook to control 9 (8 free squares plus the one the king is standing on).

However the rook gains no additional mobility from a globe construction. In fact we can more easily visualize any attempt at checkmate by placing the pieces in the middle of a standard board... and we already know K+R mate cannot happen in the middle of the board.

Yes, but I have since been wondering if there is a Queen and King vs King # possible on this globe chess board.

I don't think there is, for the king is never confined to a 'final' rank beyond which there is no movement.

In fact I doubt any kind of a # would be possible on the board, barring the ones where several attacking pieces are delivering the mate or if the king's own pieces are blocking its ways of escape.