Cotangent lifts of diffeomorphisms #
A diffeomorphism h : M → N lifts to a diffeomorphism of cotangent bundles,
T*M → T*N, (x, p) ↦ (h x, p ∘ (dh_x)⁻¹),
where the inverse of the differential dh_x is the differential of h⁻¹ at h x
(Diffeomorph.cotangent). It is characterized by covering h
(Diffeomorph.proj_cotangent) and by preserving the pairing of covectors with tangent vectors:
the lifted covector, evaluated on dh_x v, is p v (Diffeomorph.cotangent_snd_apply_mfderiv).
It is linear on fibres (Diffeomorph.cotangent_mk_smul), so it sends exactly the zero covectors
to zero covectors (Diffeomorph.cotangent_snd_eq_zero_iff).
Smoothness of the lift is the smoothness of pulling covectors back along a map,
TauCeti.CotangentBundle.contMDiff_mk_comp_mfderiv.
The construction is functorial: cotangent lifts preserve identity maps, inverses and composition
(Diffeomorph.cotangent_refl, Diffeomorph.cotangent_symm, Diffeomorph.cotangent_trans),
and a diffeomorphism is determined by its lift (Diffeomorph.cotangent_injective,
Diffeomorph.cotangent_inj).
It is the cotangent counterpart of the tangent lift Diffeomorph.tangent. The cotangent lift of
a diffeomorphism preserves the Liouville form and the canonical symplectic form of the cotangent
bundle; see TauCeti.Geometry.Symplectic.Cotangent.Bundle.Lift.
By functoriality, a group G acting on M by smooth maps (ContMDiffConstSMul I ∞ G M) acts on
T*M by the cotangent lifts of these maps,
g • (x, p) = (g • x, p ∘ (dg_x)⁻¹)
(TauCeti.CotangentBundle.instMulAction), again by smooth maps
(TauCeti.CotangentBundle.instContMDiffConstSMul). For a smooth action of a Lie group the lifted
action is jointly smooth on G × T*M (TauCeti.CotangentBundle.instContMDiffSMul), since the
differential of x ↦ g⁻¹ • x depends smoothly on g. This cotangent-lifted action covers the
action on M (TauCeti.CotangentBundle.proj_smul), preserves the pairing of covectors with
tangent vectors (TauCeti.CotangentBundle.smul_snd_apply_mfderiv) and the zero section, and the
cotangent lift of each x ↦ g • x is the action of g on T*M
(Diffeomorph.cotangent_constSmul). Cotangent lifts are equivariant for these actions: the
cotangent lift of a diffeomorphism intertwines the lifted actions exactly when the diffeomorphism
intertwines the actions on the base (Diffeomorph.cotangent_smul,
Diffeomorph.forall_cotangent_smul_iff).
References #
- A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001, Sections 2 and 22.
- J. E. Marsden, T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Texts in Applied Mathematics 17, Springer, 1999, Section 12.1 (cotangent lifts of group actions).
- J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2013, Chapter 11.
The cotangent lift of a smooth diffeomorphism h : M → N: the smooth diffeomorphism
T*M → T*N sending a covector p at x to the covector p ∘ (dh_x)⁻¹ at h x. Its inverse is
the cotangent lift of h.symm.
Instances For
The cotangent lift sends a covector p at x to p ∘ (dh_x)⁻¹ at h x, where
(dh_x)⁻¹ is the differential of h.symm at h x.
The cotangent lift of h covers h.
The lifted covector evaluates a tangent vector w at h x as the original covector
evaluates (dh_x)⁻¹ w.
The cotangent lift preserves the pairing of covectors with tangent vectors: the lifted
covector evaluated on dh_x v is the original covector evaluated on v.
The cotangent lift of h sends the zero covector at x to the zero covector at h x.
Cotangent lifts are linear on fibres: rescaling a covector before lifting it rescales the lifted covector by the same scalar.
The cotangent lift of h sends a covector to zero exactly when the covector is zero.
The inverse of a cotangent lift is the cotangent lift of the inverse diffeomorphism.
The cotangent lift of the identity diffeomorphism is the identity diffeomorphism of the cotangent bundle.
Cotangent lift commutes with composition of smooth diffeomorphisms.
A smooth diffeomorphism is determined by its cotangent lift, which covers it.
Two smooth diffeomorphisms have the same cotangent lift exactly when they are equal.
The cotangent lift of a smooth group action: an element g acts on T*M by the cotangent
lift of the diffeomorphism x ↦ g • x, sending a covector p at x to the covector
p ∘ (dg_x)⁻¹ at g • x (TauCeti.CotangentBundle.smul_eq_mk).
The cotangent-lifted action of g is the cotangent lift of the diffeomorphism x ↦ g • x.
The cotangent-lifted action sends a covector p at x to the covector p ∘ (dg_x)⁻¹ at
g • x, where (dg_x)⁻¹ is the differential of x ↦ g⁻¹ • x at g • x.
The cotangent-lifted action covers the action on the base.
The covector g • p evaluates a tangent vector w at g • x as p evaluates
(dg_x)⁻¹ w, the differential of x ↦ g⁻¹ • x at g • x applied to w.
The cotangent-lifted action preserves the pairing of covectors with tangent vectors: the
covector g • p evaluated on dg_x v is p v.
The cotangent-lifted action preserves the zero section.
The cotangent lift of a smooth group action is a group action.
The cotangent lift of a smooth group action is again an action by smooth maps.
The cotangent lift of a smooth Lie group action is smooth. If a Lie group G acts
smoothly on M, then (g, p) ↦ g • p is smooth on G × T*M: the covector g • p is p
pulled back along the differential of x ↦ g⁻¹ • x, which depends smoothly on g.
The cotangent lift of the diffeomorphism x ↦ g • x is the diffeomorphism z ↦ g • z of the
cotangent-lifted action.
Cotangent lifts of equivariant diffeomorphisms are equivariant. If h : M → N commutes
with the action of g, its cotangent lift commutes with the cotangent-lifted action of g.
Equivariance of cotangent lifts. The cotangent lift of a diffeomorphism h : M → N
commutes with the cotangent-lifted action of g exactly when h commutes with the action of g
on the base.