The Liouville vector field of a cotangent bundle #
The cotangent bundle T*M = TauCeti.CotangentBundle I M carries the Liouville form λ = p dq and
the canonical symplectic form ω = -dλ. Its fibres are vector spaces, and the Euler vector field
of the bundle (TauCeti.vectorBundleEulerField) is the radial field Z = p ∂_p: in the bundle
chart at a covector with coordinates (q, p) it is (0, p).
This file proves that Z is the Liouville vector field of the primitive -λ of ω:
contracting ω with Z gives ι_Z ω = -λ
(TauCeti.CotangentBundle.symplecticForm_interior_vectorBundleEulerField), and therefore
L_Z ω = ω by Cartan's formula
(TauCeti.CotangentBundle.isLiouvilleVectorField_vectorBundleEulerField). This is the global form
of TauCeti.isLiouvilleVectorField_cotangentLiouvilleField on the linear cotangent space, and the
sign convention is the same: with ω = -dλ, the Liouville primitive is -λ and its Liouville field
is p ∂_p. Since Z is vertical, λ vanishes on it
(TauCeti.CotangentBundle.liouvilleForm_interior_vectorBundleEulerField). The flow of Z is the
fibre dilation by e^t (TauCeti.isMIntegralCurve_vectorBundleEulerField), under which ω scales
by e^t (TauCeti.CotangentBundle.mpullback_vectorBundleScalarDiffeomorph_symplecticForm).
A diffeomorphism ψ : T*M → T*N with ψ^*λ_N = λ_M also preserves ω, so it carries the radial
field of T*M to that of T*N
(Diffeomorph.mfderiv_apply_vectorBundleEulerField_of_mpullback_liouvilleForm). Hence it commutes
with their flows: ψ sends c p to c ψ(p) for every c > 0
(Diffeomorph.apply_mk_smul_of_mpullback_liouvilleForm). This positive homogeneity is the first
step in identifying such a ψ with the cotangent lift of a diffeomorphism of the bases, which is
done in TauCeti.Geometry.Symplectic.Cotangent.Bundle.Classification.
Main results #
TauCeti.CotangentBundle.liouvilleForm_interior_vectorBundleEulerField:ι_Z λ = 0.TauCeti.CotangentBundle.symplecticForm_interior_vectorBundleEulerField:ι_Z ω = -λ.TauCeti.CotangentBundle.isLiouvilleVectorField_vectorBundleEulerField:L_Z ω = ω.Diffeomorph.mfderiv_apply_vectorBundleEulerField_of_mpullback_liouvilleForm: a diffeomorphism of cotangent bundles preserving the Liouville forms preserves the radial fields.Diffeomorph.apply_mk_smul_of_mpullback_liouvilleForm: such a diffeomorphism commutes with the positive fibre dilations.
References #
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Chapter 3 (cotangent bundles, Liouville vector fields and hypersurfaces of contact type).
- J.-F. Barraud, M. Damian, V. Humilière, A. Oancea, Floer homology with DG coefficients. Applications to cotangent bundles, arXiv:2404.07953, §1.2.
The radial field p ∂_p is vertical, so the Liouville form vanishes on it: ι_Z λ = 0.
The radial field p ∂_p is the Liouville field of -λ. Contracting the canonical
symplectic form ω = -dλ of the cotangent bundle with its Euler vector field gives minus the
Liouville form: ι_Z ω = -λ.
The radial field p ∂_p is a Liouville vector field of the canonical symplectic form of
the cotangent bundle: L_Z ω = ω, for the Euler vector field Z of T*M.
Diffeomorphisms preserving the Liouville form #
Liouville-preserving diffeomorphisms preserve the radial field. A diffeomorphism
ψ : T*M → T*N pulling the Liouville form λ_N back to λ_M carries the radial field p ∂_p of
T*M to that of T*N. It preserves ω = -dλ as well, and the radial fields are the vector fields
Z with ι_Z ω = -λ.
Liouville-preserving diffeomorphisms commute with positive fibre dilations. Let
ψ : T*M → T*N be a diffeomorphism pulling the Liouville form λ_N back to λ_M. If ψ sends
the covector p at x to the covector q at y, then it sends c p to c q for every c > 0.
The dilations by e^t are the flow of the radial fields, which ψ preserves.