The Mumford-Shah model for planar image segmentation chas been extended to variational models for image segmentation on implicit surfaces in this paper. The closed surfaces are denoted as zero-level set of signed distance functions, and the open surfaces are expressed as interaction sets of zero-level set and binary functions. With the use of intrinsic gradients and intrinsic divergences, the Mumford-Shah model for image segmentation on implicit surfaces is formulated. In order to improve their generality, the paper uses a generalized smoothness term which can be exemplified for different types. In order to improve computation efficiency, the study designs the Split Bregman algorithms via introducing auxiliary variables and Bregman iterative parameters for the proposed models. Numerical examples validate the models and algorithms finally.