Extrinsic sphere and totally umbilical submanifolds in Finsler spaces

Based on a definition for circle in Finsler spaceý, ýrecently proposed by one of the present authors and Zý. ýShený, ýa 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 fieldý, ýif and only if its circles coincide with circles of the ambient manifoldý.
Finallyý, ýsome examples of extrinsic sphere in Finsler geometryý, ýparticularly in Randers spaces are given.