The notion of isometric submersion is extended to Finsler spaces and it is used to construct examples of Finsler metrics on complex and quaternionic projective spaces all of whose geodesics are (geometrical) circles.
Paiva, J., & Duran, C. (2001). Isometric submersions of Finsler manifolds. Proceedings of the American Mathematical Society, 129(8), 2409-2417. https://doi.org/10.1090/S0002-9939-01-05910-X (Original work published 2001)