An Analogue of the ${\pi}$-strategy in Pursuit and Evasion Differential Games with many Pursuers on a Surface

In the present work we study the simple motion differential games of several objects on the surface with positive curvature. Using Jacobi field (Gromoll, Klingenberg, Meyer, 1968; Thorpe, 1979) it is constructed analogue of the strategy of parallel approach (Petrosjan, 1965), characterized with the property that straight lines through positions of the pursuer and the evader remains close to each other during the game. This strategy is used to solve a pursuit problem with many pursuers, in which the maximal speeds of all players are equal (similar pursuit game without phase constraints was studied by B. N. Pshenichniy, 1976). Moreover, it is proved that if the evader has advantage over pursuers in speed, then the evader can avoid contact with pursuers (in case players move on the plane the game was examined by F. L. Chernous'ko, 1976, and an evasion strategy in direction was constructed by him).