sage: S1 = Manifold(1, 'S^1') # the circle
sage: U = S1.open_subset('U') # the complement of one point
sage: Xt.<t> =  U.chart('t:(0,2*pi)') # the standard angle coordinate
sage: V = S1.open_subset('V') # the complement of the point t=pi
sage: Xu.<u> = V.chart('u:(0,2*pi)') # the angle t-pi
sage: S1.declare_union(U, V)
sage: e = S1.vector_frame('e') # a global vector frame (makes S^1 parallelizable)
sage: R2 = Manifold(2, 'R^2', start_index=1) # Euclidean space R^2
sage: X2.<x,y> = R2.chart() # Cartesian coordinates
sage: F = S1.diff_map(R2, {(Xt,X2): [cos(t), sin(t)],
....:                      (Xu,X2): [-cos(u), -sin(u)]}) # embedding S^1 --> R^2
sage: dx = X2.coframe()[1] # the 1-form dx on R^2
sage: a = F.pullback(dx) ; a
1-form on the 1-dimensional differentiable manifold S^1
sage: a._components
{Coordinate frame (V, (d/du)): 1-index components w.r.t. Coordinate frame (V, (d/du)),
 Coordinate frame (U, (d/dt)): 1-index components w.r.t. Coordinate frame (U, (d/dt))}