ParametricPlot3D[{(3 + Cos[v]) Sin[u], (3 + Cos[v]) Cos[u],
  Sin[v]}, {u, -Pi, Pi}, {v, 0, 2 Pi},
 RegionFunction -> (-Pi + Pi/16 <= #4 <= Pi - Pi/16 &)]

ParametricPlot3D[{(3 + Cos[v]) Sin[u], (3 + Cos[v]) Cos[u],
  Sin[v]}, {u, -Pi, Pi}, {v, 0, 2 Pi}, MeshFunctions -> {#4 &},
 Mesh -> {{-Pi + Pi/16, Pi - Pi/16}},
 MeshShading -> {None, Automatic, None}]