The problem of optimal approach in locally Euclidean spaces

A differential game of optimal approach with simple motions when players move in locally Euclidean spaces is studied. The game-end moment is fixed, and the game payment is a distance between the pursuer and the evader at the game-end moment. The value of game is obtained in the explicit form for any initial positions of players. Moreover, the differential game of optimal approach for the denumerable number of pursuers and one evader in the Euclidean space is solved. All pursuers are controlled by one parameter.