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Canonical tangent and cotangent splittings for products
Statement
For smooth manifolds and , there are canonical vector-space isomorphisms
Facts & Assumptions
Given: Smooth manifolds and a point .
Products of smooth manifolds come with smooth projections and (Products of smooth manifolds have a canonical product smooth structure).
The differential of a smooth map is a linear map on tangent spaces (The differential of a smooth map).
Cotangent pullback is defined by precomposition with the differential and is functorial (Pullback of a cotangent vector, Cotangent pullback is contravariantly functorial).
Proof
Define by . In product coordinates , the tangent basis at splits into the -coordinate derivations and the -coordinate derivations, so sends that basis to the direct-sum basis and is therefore an isomorphism.
Dualizing the isomorphism from step 1.1 gives an isomorphism . Concretely, it is the map induced by the inclusion maps of the product factors, and [F3] makes this construction canonical.
Therefore both the tangent and cotangent product splittings are canonical.
Depends on
Used by
- Ricci lower bound does not control every sectional curvature in dimension at least three Counterexample
- Coordinate circles give the alternating intersection matrix of a torus Example
- Curvature of a Riemannian product Example
- Latitude and meridian intersections on the torus Example
- The diagonal in the two-sphere has self-intersection two Example
- Characteristic numbers of products satisfy the Whitney-sum and Kunneth product formulas Lemma
- Graph-diagonal transversality is exactly fixed-point nondegeneracy Lemma
- The L-genus is an oriented rational bordism ring homomorphism Lemma
- The local intersection sign of graph against diagonal is sign det(I-Df) Lemma
- The normal bundle of the diagonal is canonically the tangent bundle Lemma
- A vector bundle projection is a surjective submersion Proposition
- Right-invariant fields carry the opposite Lie bracket Proposition
- The signature is multiplicative under Cartesian products Theorem
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)