EXTRINSIC SPHERE AND UMBILICAL SUBMANIFOLDS IN FINSLER SPACES

Based on a definition for circle in Finsler space, recently proposed by one of the present authors and Z. Shen, a natural definition of extrinsic sphere in Finsler geometry is given and it is shown that a connected submanifold of a Finsler manifold is totally umbilical and has non-zero parallel mean curvature vector field, if and only if its circles coincide with circles of the ambient manifold. Finally, some examples of extrinsic sphere in Finsler geometry, particularly in Randers spaces are given.