The Reeb vector field of a contact form #
Let α be a contact form on a finite-dimensional manifold M. At each point x, the two-form
dα_x is nondegenerate on the hyperplane ξ_x = ker α_x, so it has a one-dimensional kernel
transverse to ξ_x. The Reeb vector field R_α (TauCeti.SmoothForm.reebVectorField) is the
vector field spanning this kernel and normalized by α:
α(R_α) = 1, ι_(R_α) dα = 0.
These two equations determine R_α uniquely
(TauCeti.SmoothForm.IsContactForm.eq_reebVectorField_iff), and it is enough to test the second
one against vectors of ξ
(TauCeti.SmoothForm.IsContactForm.eq_reebVectorField_iff_kerDistribution).
At each point the tangent space splits as ξ_x ⊕ ℝ R_α(x).
Unlike the contact distribution ξ, the Reeb field depends on the contact form, not only on its
kernel. For a nowhere vanishing function f, the rescaling law
(TauCeti.SmoothForm.IsContactForm.reebVectorField_eq_inv_smul_add_iff) reads
R_(f α) = R_α / f + Y,
where Y is the unique vector field in ξ with dα(Y, w) = df(w) / f² for all w ∈ ξ. For a
constant c ≠ 0 this gives R_(c α) = R_α / c
(TauCeti.SmoothForm.IsContactForm.reebVectorField_smul), and in particular R_(-α) = -R_α.
All statements here are pointwise linear algebra at each point, applying
Module.Dual.reebVector to the covector α_x and the bilinear form dα_x.
Main declarations #
TauCeti.SmoothForm.reebVectorField: the Reeb vector field of a one-form.TauCeti.SmoothForm.IsContactForm.apply_reebVectorField,TauCeti.SmoothForm.IsContactForm.mextDeriv_reebVectorField_left: the Reeb equations.TauCeti.SmoothForm.IsContactForm.eq_reebVectorField_iff: uniqueness.TauCeti.SmoothForm.IsContactForm.isCompl_kerDistribution_span_reebVectorField: the splittingT_x M = ξ_x ⊕ ℝ R_α(x).TauCeti.SmoothForm.IsContactForm.reebVectorField_eq_inv_smul_add_iff: the rescaling law.
References #
- H. Geiges, An Introduction to Contact Topology, Cambridge University Press, 2008, Section 1.1.
The Reeb vector field of a one-form α: at each point x, the tangent vector R with
α_x(R) = 1 and dα_x(R, w) = 0 for every w ∈ ker α_x. For a contact form on a
finite-dimensional manifold it exists and is unique
(TauCeti.SmoothForm.IsContactForm.eq_reebVectorField_iff); it is the Reeb vector
Module.Dual.reebVector of the covector α_x for the bilinear form dα_x.
Instances For
Characterization of the Reeb vector field, tested on the contact distribution: R_α(x) is
the unique tangent vector v with α(v) = 1 and dα(v, w) = 0 for all w ∈ ker α_x.
A contact form takes the value 1 on its Reeb vector field.
The Reeb vector field lies in the kernel of dα: ι_(R_α) dα = 0.
The Reeb vector field lies in the kernel of dα: dα vanishes when its second argument is
the Reeb vector field.
Characterization of the Reeb vector field: R_α(x) is the unique tangent vector v with
α(v) = 1 and ι_v dα = 0.
The Reeb vector field of a contact form vanishes nowhere.
The Reeb vector field is transverse to the contact distribution.
The tangent space of a contact manifold splits as the contact hyperplane plus the Reeb line,
T_x M = ξ_x ⊕ ℝ R_α(x).
Rescaling law for the Reeb vector field. Let α be a contact form and β = f α for a
function f, C¹ and nonzero at x. Then R_β(x) = R_α(x) / f(x) + y exactly when y lies in
ξ_x = ker α_x and dα(y, w) = df(w) / f(x)² for all w ∈ ξ_x. In particular
R_β(x) - R_α(x) / f(x) is the unique such vector.
Multiplying a contact form by a nonzero constant c divides its Reeb vector field by c.
The Reeb vector field of -α is -R_α.