<p>This paper develops a mathematical framework that extends topology optimization for mass and heat transfer onto variable 2-manifolds, which are curved surfaces defined as the design domains allowed to evolve rather than remain fixed during the optimization process. Consequently, the design freedom of topology optimization is increased by incorporating the design domains themselves into the design space. The variable curved surfaces expressed as the implicit 2-manifolds are homeomorphously defined on predetermined fixed base manifolds. The concept of fiber bundle is used to describe the pattern of a surface structure together with the implicit 2-manifold as an integrated ensemble defined on the base manifold. Therefore, this topology optimization on variable 2-manifolds is developed to optimize the matching between the patterns of surface structures and the implicit 2-manifolds defined with the patterns. It is implemented based on the porous medium model by using the material distribution method, where the material density is used to interpolate the impermeability of the porous medium filled on the implicit 2-manifolds. Two sets of design variables are defined for the patterns of the surface structures and the implicit 2-manifolds, respectively. They are regularized by two surface-PDE filters and a parameter is introduced to the surface-PDE filter of the implicit 2-manifolds to control the variable magnitude. The topology optimization problems are analyzed by using the continuous adjoint method to derive the gradient information of the design objectives and constraints. They are then solved by using the gradient based iterative procedures numerically implemented based on the surface finite element method. To permit the use of linear surface elements for the consideration of computational cost, the variational formulations of the surface Navier–Stokes equations, surface convection-diffusion equation and surface convective heat-transfer equation are stabilized by using the Brezzi-Pitkäranta, Petrov-Galerkin and general least square techniques, respectively. The adjoint equations are derived for the stabilized variational formulations of those governing equations. In the numerical results, the variable amplitude in the surface-PDE filter of the implicit 2-manifolds, Reynolds number, Péclet number, pressure drop and dissipation power of the surface flow are investigated to demonstrate the increased design freedom and extended design space achieved by topology optimization on variable 2-manifolds for mass and heat transfer in surface flow.</p>