The Legendre maps from two Lagrangians or from a Lagrangian and a p-form

The Legendre maps from two Lagrangians or from a Lagrangian and a p-form

Článek Scopus

Let FMm,n denote the category of fibered manifolds with mdimensional bases and n-dimensional fibres and their fibered local diffeomorphisms. We prove that if m, n and s are positive integers, then any FMm,n-natural operator C transforming tuples (λ1, λ2) of Lagrangians λ1, λ2: JsY → ∧m T∗M on FMm,n-objects Y → M into Legendre maps C(λ1, λ2): JsY → SsT M ⊗ V∗Y ⊗∧m T∗M on Y is of the form C(λ1, λ2) = c1Λ(λ1) + c2Λ(λ2), c1, c2 ∈ R, where Λ is the Legendre operator. We also prove that if m, n, s and p are positive integers, then any FMm,n-natural operator C transforming tuples (λ, η) of Lagrangians λ: JsY →∧m T∗M and p-forms η ∈ Ωp(M) into Legendre maps C(λ, η): JsY → SsT M ⊗ V∗Y ⊗∧m T∗M is of the form C(λ, η) = cΛ(λ), c ∈ R, where Λ is the Legendre operator.

Let FMm,n denote the category of fibered manifolds with mdimensional bases and n-dimensional fibres and their fibered local diffeomorphisms. We prove that if m, n and s are positive integers, then any FMm,n-natural operator C transforming tuples (λ1, λ2) of Lagrangians λ1, λ2: JsY → ∧m T∗M on FMm,n-objects Y → M into Legendre maps C(λ1, λ2): JsY → SsT M ⊗ V∗Y ⊗∧m T∗M on Y is of the form C(λ1, λ2) = c1Λ(λ1) + c2Λ(λ2), c1, c2 ∈ R, where Λ is the Legendre operator. We also prove that if m, n, s and p are positive integers, then any FMm,n-natural operator C transforming tuples (λ, η) of Lagrangians λ: JsY →∧m T∗M and p-forms η ∈ Ωp(M) into Legendre maps C(λ, η): JsY → SsT M ⊗ V∗Y ⊗∧m T∗M is of the form C(λ, η) = cΛ(λ), c ∈ R, where Λ is the Legendre operator.

Fibered manifolds | Lagrangian | Legendre map | Legendre operator | natural operator

Fibered manifolds | Lagrangian | Legendre map | Legendre operator | natural operator

DOUPOVEC, M.; KUREK, J.; MIKULSKI, W.

01.01.2025

Annales Universitatis Mariae Curie-Skłodowska. Sectio A, Mathematica

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Polská republika

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https://journals.umcs.pl/a/article/view/19841/12578