PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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Tensor products and contractions of smooth tensor fields are smooth

Statement

The tensor product of smooth tensor fields is smooth. If a smooth mixed tensor field has at least one contravariant and one covariant slot, then contracting its first contravariant slot against its first covariant slot is smooth.

Facts & Assumptions

Given: Smooth tensor fields $S$ and $T$ , and a smooth mixed tensor field $R$ with at least one covariant and one contravariant slot.

[L1]

Tensor-field smoothness is equivalent to smoothness of the local coefficient functions (Smoothness of a tensor field is equivalent to smooth coordinate components).

[L2]

Tensor product is bilinear, and contraction is basis-independent (Tensor product of multilinear tensors is associative and bilinear, Contraction is independent of the basis formula).

Proof

technique · direct

1.1

In any chart, [L1] identifies $S$ , $T$ , and $R$ with families of smooth coefficient functions. By [L2], the coefficients of $S \otimes T$ are finite sums of products of the coefficients of $S$ and $T$ . Those are smooth.

L1L2givenalgebra

1.2

In the same chart, [L2] writes each contracted coefficient of $R$ as a finite sum of coordinate coefficients of $R$ . Because the contraction formula is basis-independent, these chartwise definitions glue. The resulting coefficient functions are smooth by [L1].

L1L2givenalgebra

2.1

Therefore tensor products and the contraction defined above preserve smoothness.

step 1.1step 1.2∎

Depends on

Used by

Pullback of covariant tensors is smooth and functorial Proposition

Dependency tree · two levels

10 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)