# pullback of (r,s)-tensor fields

Let $$F:M \to N$$ be a smooth map between smooth manifolds and let $$w\in T_s^0(N)$$ be a $$(0,s)$$-tensor field on $$N$$. Then let us define the pullback of $$w$$ by $$F$$ as the $$(0,s)$$-tensor field $$F^*w\in T_s^0(M)$$ on $$M$$ defined, for any $$p\in M$$, by $$F^*w|_p := dF^*_p(w|_{F(p)})$$ where $$(\forall v_1,\dots,v_s \in T_pM) \quad dF^*_p(w|_{F(p)})(v_1,\dots,v_s) :=w|_{F(p)}(dF_pv_1,\dots, dF_pv_s).$$

Thus $$F^* : T_s^0(N) \to T_s^0(M)$$. Is there a way to show that this is actually just a specific case of the following generation of the pullback of $$(r,s)$$-tensor fields by $$F$$: $$F^* :=(F^{-1})_* : T_s^r(N) \to T_s^r(M)?$$

• But you need a diffeomorphism in the general case. Look carefully at the definition. Jan 5 at 5:19

## 1 Answer

In your last line, you need $$F^{-1}$$ to be well-defined (at least differentiable), which is not the case in general.

The way you defined the pull-back operations is the most common one, and your last line should be the definition of push-forward when $$F$$ is a diffeomorphism, for example.