Spatial-temporal optical vortices (STOVs) have recently become the focus of newly structured optical fields. In this paper, their propagation on a 2D curved surface named the constant Gaussian curvature surface (CGCS) is studied. Using the matrix optics approach, we provide the analytical solution of the STOV propagation under the paraxial approximation on the CGCS with positive curvature. One method of creating timers is made possible by the spatiotemporal distribution direction of STOV light intensity, which swings like a pendulum throughout the evolution, in contrast to propagation on a flat surface. This swing, however, stops when the curved surface's curvature radius matches the light's Rayleigh distance. Besides, the transverse orbital angular momentum of STOV is deduced, and we find that the intrinsic and extrinsic OAM periodically exchange, but the total transverse OAM is always zero during the propagation on CGCS. It aids in controlling the transverse extrinsic orbital angular momentum of STOV in nontrivial space.