TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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Canonical tangent and cotangent splittings for products

Statement

For smooth manifolds $M$ and $N$ , there are canonical vector-space isomorphisms $T_{(p, q)} (M \times N) ≅ T_{p} M \oplus T_{q} N and T_{(p, q)}^{*} (M \times N) ≅ T_{p}^{*} M \oplus T_{q}^{*} N .$

Facts & Assumptions

Given: Smooth manifolds $M, N$ and a point $(p, q) \in M \times N$ .

[F1]

Products of smooth manifolds come with smooth projections $π_{M}$ and $π_{N}$ (Products of smooth manifolds have a canonical product smooth structure).

[F2]

The differential of a smooth map is a linear map on tangent spaces (The differential of a smooth map).

[F3]

Cotangent pullback is defined by precomposition with the differential and is functorial (Pullback of a cotangent vector, Cotangent pullback is contravariantly functorial).

Proof

technique · direct

1.1

Define $Φ : T_{(p, q)} (M \times N) \to T_{p} M \oplus T_{q} N$ by $Φ (v) : = (d π_{M} (v), d π_{N} (v))$ . In product coordinates $(x, y)$ , the tangent basis at $(p, q)$ splits into the $x$ -coordinate derivations and the $y$ -coordinate derivations, so $Φ$ sends that basis to the direct-sum basis and is therefore an isomorphism.

F1F2given

2.1

Dualizing the isomorphism from step 1.1 gives an isomorphism $T_{(p, q)}^{*} (M \times N) ≅ T_{p}^{*} M \oplus T_{q}^{*} N$ . Concretely, it is the map $ω \mapsto ((ι_{M})^{*} ω, (ι_{N})^{*} ω)$ induced by the inclusion maps of the product factors, and [F3] makes this construction canonical.

F3step 1.1

3.1

Therefore both the tangent and cotangent product splittings are canonical.

step 1.1step 2.1∎

Depends on

Used by

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Dependency tree · two levels

23 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)