Dirac-geodesics with curvature term

Dirac-geodesics with curvature term are Dirac-harmonic maps with curvature term from one-dimensional domains. In this paper, we first describe the structure of Dirac-geodesics with curvature term in surfaces and give solutions on the unit 2-sphere and the hyperbolic plane, and then we give the structure of solutions in warped product spaces. Finally, we define the heat flow of Dirac-geodesics with curvature term and prove the global existence and sub-convergence of the heat flow into any closed surfaces and space forms. Our results provide the first examples and general existence theorems of Dirac-geodesics with curvature term.