Is the angular degree of freedom on a circle considered a gauge degree?

[plasmon]

Is it okay to classify the angular degree of freedom of a point on a circle as a gauge degree

of freedom?

(i). One can add 2Pi to angle without changing the position of point.

(ii). U(1) symmetry exp (i theta)

 

[tom.stoer]

Typically this corresponds to global translational invariance; in order to be gauge trf. it should be a local symmetry.