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Pullback of covariant tensors is smooth and functorial

Statement

If $F : M \to N$ is smooth and $T$ is a smooth covariant tensor field on $N$ , then $F^{*} T$ is a smooth covariant tensor field on $M$ . Moreover,

({id}_{M})^{*} T = T, (G \circ F)^{*} T = F^{*} (G^{*} T)

for every composable smooth map $G$ .

Facts & Assumptions

Given: Smooth maps $F : M \to N$ and $G : N \to P$ , and a smooth covariant tensor field $T$ on the target.

[F1]

Pullback of a covariant tensor field is defined by inserting the differential into every slot (The pullback of a covariant tensor field).

[L1]

Tensor pullback commutes with the corresponding fibrewise linear operations, and tensor products and contractions of smooth tensor fields are smooth (Linear pullback respects tensor products and permutations, Tensor products and contractions of smooth tensor fields are smooth).

[L2]

Differentials satisfy the chain rule (The chain rule for differentials of smooth maps).

Proof

technique · direct

1.1

In local coordinates, [F1] expresses each coefficient of $F^{*} T$ as a finite sum of the coefficients of $T \circ F$ multiplied by partial derivatives of $F$ . Those are smooth, so [L1] implies that $F^{*} T$ is smooth.

F1L1givenalgebra

1.2

The identity map has identity differential, so [F1] gives $({id}_{M})^{*} T = T$ .

F1given

1.3

For $v_{1}, \dots, v_{k} \in T_{p} M$ , [F1] and [L2] give $((G \circ F)^{*} T)_{p} (v_{1}, \dots, v_{k}) = T_{G (F (p))} (d G_{F (p)} d F_{p} v_{1}, \dots, d G_{F (p)} d F_{p} v_{k}),$ which is exactly $(F^{*} (G^{*} T))_{p} (v_{1}, \dots, v_{k})$ .

F1L2givenalgebra

2.1

Therefore pullback of covariant tensors is smooth and functorial.

step 1.1step 1.2step 1.3∎

Depends on

Used by

A general mixed tensor field does not have a pullback by every smooth map False statement
A diffeomorphism pulls back tensor fields and forms isomorphically Proposition
Pullback of forms is smooth functorial and preserves wedges Proposition

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)