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

Famous quotes containing the words smooth and/or map:

    Nations are possessed with an insane ambition to perpetuate the memory of themselves by the amount of hammered stone they leave. What if equal pains were taken to smooth and polish their manners?
    Henry David Thoreau (1817–1862)

    Unless, governor, teacher inspector, visitor,
    This map becomes their window and these windows
    That open on their lives like crouching tombs
    Break, O break open,
    Stephen Spender (1909–1995)