  In this paper, the Higgs-like approach is used to analyze the quantum
dynamics of a harmonic oscillator constrained on a circle. We obtain the
Hamiltonian of this system as a function of the Cartesian coordinate of the
tangent line through the gnomonic projection and then quantize it in the
standard way. We then recast the Hamiltonian in a shape-invariant form and
derive the spectrum energy of the confined harmonic oscillator on the circle.
With help of the f-deformed oscillator algebra, we construct the coherent
states on the circle and investigate their quantum statistical properties. We
find that such states show nonclassical features like squeezing and
sub-Poissonian statistics even in small curvatures of the circle.
