With[{r = 1}, 
  torus[{u_, 
     v_}] := {(3 + Cos[2 \[Pi] u]) Cos[
      2 \[Pi] v], (3 + Cos[2 \[Pi] u]) Sin[2 \[Pi] v], 
    r Sin[2 \[Pi] u]}];
Show[ParametricPlot3D[torus[{u, v}], {u, 0, 1}, {v, 0, 1}, 
  Mesh -> None],
 ParametricPlot3D[torus[{Sin[2 \[Pi] t], 0}], {t, 0, 1/2}],
 Graphics3D[{Arrowheads[0.05], 
   Arrow[Tube@{torus[{Sin[2 \[Pi] *.5], 0}], 
      torus[{Sin[2 \[Pi] *.51], 0}]}]}]]

Your torus takes two coordinates and outputs three coordinates. To map a curve onto the torus you describe it in (u,v) and then torus maps int onto the surface in 3D.

Sin[t] takes values between -1 and 1. If you want a whole circle you can multiply by Pi:

With[{r = 1}, 
  torus[{u_, v_}] := {(3 + Cos[u]) Cos[v], (3 + Cos[u]) Sin[v], 
    r Sin[u]}];
Show[ParametricPlot3D[torus[{u, v}], {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]},
   Mesh -> None], 
 ParametricPlot3D[torus[{Pi*Sin[t], 0}], {t, 0, 2 \[Pi]}], 
 Graphics3D[{Arrowheads[0.05], 
   Arrow[Tube@{torus[{Sin[2 \[Pi]*.49], 0}], 
      torus[{Sin[2 \[Pi]*.51], 0}]}]}]]

or else replace Sin[t] with simply t:

With[{r = 1}, 
  torus[{u_, v_}] := {(3 + Cos[u]) Cos[v], (3 + Cos[u]) Sin[v], 
    r Sin[u]}];
Show[ParametricPlot3D[torus[{u, v}], {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]},
   Mesh -> None], 
 ParametricPlot3D[torus[{t, 0}], {t, 0, 2 \[Pi]}, PlotStyle -> Black],
  Graphics3D[{Black, Arrowheads[0.05], 
   Arrow[Tube@{torus[{Sin[2 \[Pi]*.45], 0}], 
      torus[{Sin[2 \[Pi]*.51], 0}]}]}]]

Thank you very much. I will work with the draft and if any questions arise, may I get back to you?