Isotropic, coisotropic and Lagrangian maps into a symplectic manifold #
Let σ be a fiberwise nondegenerate smooth two-form on M, so that every tangent space
T_y M carries the linear symplectic form hσ.symplecticFormAt y. A map f : N → M is
isotropic, coisotropic or Lagrangian when, at every point x of N, its tangent
image df_x(T_x N), the range of the differential, is an isotropic, coisotropic or Lagrangian
subspace of T_(f x) M in the sense of TauCeti.SymplecticForm.IsIsotropic and its siblings.
The differential is mfderiv, which is 0 where f is not differentiable; there the tangent
image is zero, which is isotropic, so the predicates are only meaningful for differentiable maps.
The source is an arbitrary charted space and f need not be injective, so these predicates apply
to embedded and immersed submanifolds alike, through any parametrization.
The predicates are invariant under the natural operations:
- they depend only on the tangent images, hence not on the parametrization: precomposing
fwith a diffeomorphism of the source does not change them; - a diffeomorphism
ψ : M → M₂pulling a nondegenerate two-formσ₂back toσcarries isotropic, coisotropic and Lagrangian maps forσto those forσ₂, and conversely.
A smooth map is isotropic exactly when it pulls σ back to zero
(TauCeti.SmoothTwoForm.IsNondegenerate.isIsotropicMap_iff_mpullback_eq_zero), which is the
form in which isotropy of graphs and sections is usually checked.
When M is finite-dimensional and the differential is injective at some point of N (so N
is nonempty), the linear dimension bounds give 2 dim N ≤ dim M for isotropic maps and
2 dim N = dim M for Lagrangian ones, and an isotropic immersion from a manifold of half the
dimension is Lagrangian. When M and N are finite-dimensional, a coisotropic map from a
nonempty N satisfies dim M ≤ 2 dim N without any injectivity. From an empty source every
map satisfies all three predicates vacuously, and no dimension constraint follows.
A Lagrangian immersion (TauCeti.SmoothTwoForm.IsNondegenerate.IsLagrangianImmersion)
into a finite-dimensional manifold is a smooth immersion with isotropic tangent images from a
manifold of half the dimension; when the source is nonempty, it is equivalently an immersion
which is a Lagrangian map (TauCeti.SmoothTwoForm.IsNondegenerate.isLagrangianImmersion_iff).
It is a separate notion from an embedded Lagrangian submanifold: its image may have
self-intersections.
The basic example: in a finite-dimensional symplectic vector space with its constant symplectic
form, a continuous linear map is a Lagrangian map exactly when its range is a Lagrangian subspace
(TauCeti.SymplecticForm.isLagrangianMap_continuousLinearMap_iff).
Main declarations #
TauCeti.SmoothTwoForm.IsNondegenerate.IsIsotropicMap,IsCoisotropicMap,IsLagrangianMap: the three tangent-image predicates.TauCeti.SmoothTwoForm.IsNondegenerate.isIsotropicMap_iff_mpullback_eq_zero: isotropy is the vanishing of the pulled-back form.TauCeti.SmoothTwoForm.IsNondegenerate.isLagrangianMap_comp_diffeomorph_iff: independence of the parametrization.TauCeti.SmoothTwoForm.IsNondegenerate.isLagrangianMap_diffeomorph_comp_iff: transport by symplectomorphisms.TauCeti.SmoothTwoForm.IsNondegenerate.IsLagrangianMap.two_mul_finrank: the dimension count.TauCeti.SmoothTwoForm.IsNondegenerate.IsLagrangianImmersionandTauCeti.SmoothTwoForm.IsNondegenerate.isLagrangianImmersion_iff: Lagrangian immersions.
References #
- A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001, Section 3.
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Sections 2.3 and 3.4.
The three predicates #
A map f : N → M is isotropic for a nondegenerate two-form σ when every tangent image
df_x(T_x N) is an isotropic subspace of (T_(f x) M, σ_(f x)).
The tangent image is taken with mfderiv, which is 0 at points where f is not
differentiable, so there the condition holds trivially: a nowhere-differentiable map counts as
isotropic. Isotropy is meaningful for differentiable maps, and the statements relating it to the
pulled-back form assume f smooth.
Instances For
A map f : N → M is coisotropic for a nondegenerate two-form σ when every tangent
image df_x(T_x N) is a coisotropic subspace of (T_(f x) M, σ_(f x)).
Instances For
A map f : N → M is Lagrangian for a nondegenerate two-form σ when every tangent image
df_x(T_x N) is a Lagrangian subspace of (T_(f x) M, σ_(f x)).
Instances For
Isotropy of a map is pointwise isotropy of its tangent images.
Coisotropy of a map is pointwise coisotropy of its tangent images.
The Lagrangian condition on a map is the pointwise condition on its tangent images.
Isotropy of a map is the vanishing of σ on pairs of pushed-forward tangent vectors.
A Lagrangian map is both isotropic and coisotropic, and conversely.
A Lagrangian map is isotropic.
A Lagrangian map is coisotropic.
Isotropy is the vanishing of the pullback. A smooth map is isotropic exactly when it pulls the two-form back to zero.
Reparametrization #
Precomposing with a diffeomorphism of the source does not change isotropy.
Precomposing with a diffeomorphism of the source does not change coisotropy.
Independence of the parametrization. Precomposing with a diffeomorphism of the source does not change whether a map is Lagrangian.
Transport by symplectomorphisms #
A symplectomorphism preserves isotropic maps: if ψ pulls σ₂ back to σ, then ψ ∘ f is
isotropic for σ₂ exactly when f is isotropic for σ.
A symplectomorphism preserves coisotropic maps: if ψ pulls σ₂ back to σ, then ψ ∘ f
is coisotropic for σ₂ exactly when f is coisotropic for σ.
Transport by symplectomorphisms. If ψ pulls σ₂ back to σ, then ψ ∘ f is Lagrangian
for σ₂ exactly when f is Lagrangian for σ.
Dimension counts #
An isotropic map with injective differential at some point comes from a manifold of at most
half the dimension: 2 dim N ≤ dim M.
A Lagrangian map with injective differential at some point comes from a manifold of exactly
half the dimension: 2 dim N = dim M.
A coisotropic map from a nonempty manifold, witnessed by a point x, comes from a manifold of
at least half the dimension: dim M ≤ 2 dim N. No injectivity of the differential is needed.
An isotropic map with injective differentials from a manifold of half the dimension is Lagrangian.
Lagrangian immersions #
A Lagrangian immersion for a nondegenerate two-form σ on a finite-dimensional manifold
M is a smooth immersion f : N → M with isotropic tangent images from a manifold of half the
dimension of M. Its image may have self-intersections.
- isImmersion : Manifold.IsImmersion J I (↑⊤) f
A Lagrangian immersion is a smooth immersion.
- isIsotropicMap : IsIsotropicMap J hσ f
A Lagrangian immersion has isotropic tangent images.
A Lagrangian immersion comes from a manifold of half the dimension.
Instances For
A Lagrangian immersion is a Lagrangian map.
An immersion from a nonempty manifold is a Lagrangian immersion exactly when it is a Lagrangian map.
A smooth immersion is Lagrangian exactly when its pulled-back two-form vanishes and its source has half the ambient dimension. This includes empty sources.
Linear Lagrangian subspaces #
In a finite-dimensional symplectic vector space with its constant symplectic form, a continuous linear map is a Lagrangian map exactly when its range is a Lagrangian subspace.