My favorite torus game is Tetris on a Torus

https://www.benjoffe.com/code/games/torus/

very neat. I'd say this is kinda on top of a torus, and more of a cylinder from the perspective of the Tetris pieces. not sure how you'd play on the surface of a torus, since you need to wrap around vertically too.

E.g the pieces from the rows you eliminate reappear at random at the top row.