Cotangent Space - The Pullback of A Smooth Map

The Pullback of A Smooth Map

Just as every differentiable map f : MN 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 XxTxM. 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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