Canonical cotangent forms of a product #
Under the identification Φ : T*(M × N) ≃ T*M × T*N of
TauCeti.CotangentBundle.prodDiffeomorph, the Liouville form of a product is the sum of the
Liouville forms of the factors, pulled back along the two projections:
Φ^* (pr₁^* λ_M + pr₂^* λ_N) = λ_(M × N)
(TauCeti.CotangentBundle.mpullback_prodDiffeomorph_liouvilleForm).
Indeed Φ covers the identity of M × N, so the projections of T*M and T*N composed with
pr₁ ∘ Φ and pr₂ ∘ Φ are the components of the projection of T*(M × N); and a covector on the
product evaluates a tangent vector as the sum of its two restrictions evaluated on the components.
Since exterior differentiation commutes with pullback, the canonical symplectic form ω = -dλ of
the product is the product form (TauCeti.SmoothTwoForm.prod) of the canonical symplectic forms
of the factors:
Φ^* (pr₁^* ω_M + pr₂^* ω_N) = ω_(M × N)
(TauCeti.CotangentBundle.mpullback_prodDiffeomorph_symplecticForm).
So Φ is a symplectomorphism from T*(M × N) to the product of the symplectic manifolds T*M
and T*N.
Both identities equivalently hold along the inverse diffeomorphism, where they read
(Φ⁻¹)^* λ_(M × N) = pr₁^* λ_M + pr₂^* λ_N and (Φ⁻¹)^* ω_(M × N) = pr₁^* ω_M + pr₂^* ω_N.
Differences of the pulled-back forms play the same role for correspondences: the twisted product
pr₁^* ω_M - pr₂^* ω_N on T*M × T*N is -d(pr₁^* λ_M - pr₂^* λ_N)
(TauCeti.CotangentBundle.twistedProd_symplecticForm).
References #
- A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001, Section 2.
The Liouville form of a product is the sum of the Liouville forms of the factors. The
identification T*(M × N) ≃ T*M × T*N pulls pr₁^* λ_M + pr₂^* λ_N back to λ_(M × N).
The canonical symplectic form of a product is the product of the canonical symplectic forms
of the factors. The identification T*(M × N) ≃ T*M × T*N pulls the product form
pr₁^* ω_M + pr₂^* ω_N back to ω_(M × N).
A primitive of the twisted product of cotangent bundles. On T*M × T*N, the twisted
product pr₁^* ω_M - pr₂^* ω_N of the canonical symplectic forms is -dθ for the one-form
θ = pr₁^* λ_M - pr₂^* λ_N.