Torus

r  =  {(3 +  Cos[v]) Cos[u], (3  +   Cos[v]) Sin[u],  
          Sin[v]};

w  =  Cross[D[r, u], D[r, v]] // Simplify

w/(Sqrt[w . w]) // Simplify

n = {Cos[v] Cos[u], Cos[v] Sin[u], Sin[v]};

torus = 
      ParametricPlot3D[r,  {u, 0, 2  π}, {v, 0, 2  π}, 
        Axes -> False, Boxed -> False]

torusnormals  =  
      Graphics3D[{Thickness[ .01], 
          Table[Line[{r,  r  +  n}],  {u, π/4, 
              2  π + π/4, π/2},  {v, 0, 
              2  π, π/3}]}]

Show[torus, torusnormals]

Moebius Band

r  =  {2  Cos[u]  +  v Cos[u/2],  
          2  Sin[u]  +  v Cos[u/2],  v Sin[u/2]};

moebius =  
      ParametricPlot3D[
        r, {v, - .5,  .5}, {u, 0, 
          2  π}, 
PlotRange -> {{-3, 
              3}, {-3, 3}, {-3, 3}}, 
        ViewPoint -> {-1.002,  2.404,  2.160}, Axes -> False, 
        Boxed -> False]

w  =  Cross[D[r, u], D[r, v]] // Simplify

n  =  w/(Sqrt[w . w]) // Simplify

moebiusnormals[stop_]  :=  
      Graphics3D[{Thickness[ .01], 
          Table[Line[{r,  r  +  n}],  {u, 0, stop, π/4},  {v, 0, 
              0}]}]

Do[Show[moebius, moebiusnormals[t]], {t, 0, 
        4  π, π/4}]