is a vector field on a surface always parallel at each point to some vector?

[lavinia]

Take a surface with a Levi-Cevita connection on its tangent circle bundle and pull this connection down to the surface using a vector field of unit length. ( Such a vector field may not exist globally but ignore this and consider the region where is does exist.) At each point of the surface the pulled back connection 1-form has a non-trivial kernel which is a line in the tangent space at that point. Is this line a direction in which the vector field is parallel?