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In Lee, there is an exercise involving the pullback that I can't understand. If $M,N$ are smooth manifolds and $F:M\to N$ is smooth, I am asked to show that the pullback $F^*$ is a smooth bundle map $T^*N \to T^*M$ where $T^*N$ denotes the cotangent bundle on $N$ and similarly for $M$.

The problem is that I can't figure out how this map is supposed to be defined and I can't find the definition in the book. For example, it doesn't seem clear what $F^*$ of a covector at a point not in the image of $F$ should be. Strangely enough I haven't been able to find the definition online either. Does anyone know the relevant definition?

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up vote 3 down vote accepted

This is an error in the book: the statement is supposed to assume that $F\colon M \to N$ is a diffeomorphism.

Lee keeps an updated errata here:

At any rate, the definition of pullback on the individual fibers $T^*_{F(p)}N$ is given in the text (on page 136 of the First Edition):

$$F^*\colon T^*_{F(p)}N \to T^*_{p}M$$ $$\omega_{F(p)} \mapsto (F^*\omega)_p,$$ where $(F^*\omega)_{p}$ is defined by $(F^*\omega)_p(X_p) = \omega_{F(p)}(F_*X_p)$ for $X_p \in T_pM$.

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Thank you, that was very helpful. – Seth Nov 15 '12 at 2:32

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