In the present work we consider a curve embedded in a three-dimensional Riemannian manifolf moving in the binormal direction proportional to its curvature. We study how an appropiate orthonormal frame, the Frenet-Serret frame, along the curve evolves in order to deduce that a nonlinear Schrodinger-type equation rules the motion of the curve. Although there exist no conserved quantities, we establish. as one of our main results, local and global existence of solutions for the derived Cauchy problem on a unit circle (corresponding to closed curves ) and on the real line (corresponding to open curves). We also study the motion by mean curvature of a surface embedded in the four-dimensional Euclidean space. Introducing the language of gauge fields as an appropriate framework for presenting the structural properties of the surface and the evolution equations of its geometric quantities, we derive that the complex mean curvature of the evolving surface satisfies a nonlinear Schrodinger-type equation.