The Pullback of A Smooth Map
Just as every differentiable map f : M → N between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces
every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction:
The pullback is naturally defined as the dual (or transpose) of the pushforward. Unraveling the definition, this means the following:
where θ ∈ Tf(x)*N and Xx ∈ TxM. Note carefully where everything lives.
If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward. Let g be a smooth function on N vanishing at f(x). Then the pullback of the covector determined by g (denoted dg) is given by
That is, it is the equivalence class of functions on M vanishing at x determined by g o f.
Read more about this topic: Cotangent Space
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