  In this paper, we consider the non-singular extension problem of
circle-valued Morse functions on closed orientable surfaces. The problem is
when can a submersion from a collaring of a closed orientable surface to a
circle, which is a circle-valued Morse function on the boundary, be extended to
a submersion from a 3-dimensional manifold to a circle. We provide necessary
and sufficient conditions for the existence of a non-singular extension of a
circle-valued Morse function as the main theorem. After that, we consider a
special case that the surface is a torus as one application of the theorem.
