Einstein tori and crooked surfaces
Jean-Philippe Burelle Virginie Charette Dominik Francoeur William Goldman









In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.

HAWC Observations Strongly Favor Pulsar Interpretations of the Cosmic-Ray Positron Excess
    and Ke Fang^f,g
    
=========================================================================================





§ INTRODUCTION

Polyhedra bounded by crooked surfaces form fundamental domains in the Einstein Universe for Lorentzian Kleinian groups (<cit.>, <cit.>).  Crooked surfaces are assembled from pieces of certain hypersurfaces, namely light cones and Einstein tori.  This motivates our study of  these hypersurfaces, and how they intersect.


The theory of crooked planes, in the context of Minkowski space, has been very successful in understanding and classifying discrete groups of affine transformations acting properly on ℝ^3 (<cit.>,<cit.>,<cit.> and <cit.>). Crooked planes are piecewise linear surfaces in Minkowski 3-space which bound fundamental domains for proper affine actions. In 2003, Frances <cit.> studied the boundary at infinity of these quotients of Minkowski space by introducing the conformal compactification of a crooked plane. In this paper, we call conformally compactified crooked planes crooked surfaces.

Recently, Danciger-Guéritaud-Kassel <cit.> have adapted crooked planes to the negatively curved anti-de Sitter space. In a note shortly following the DGK paper, Goldman <cit.> unified crooked planes and anti-de Sitter crooked planes. More precisely, Minkowski space and anti-de Sitter space can be conformally embedded in the Einstein universe in such a way that crooked planes in both contexts are subsets of a crooked surface.

A crooked surface is constructed using three pieces : two wings, and a stem. The wings are parts of light cones, and the stem is part of an Einstein torus. In order to understand the intersection of crooked surfaces, we first focus on Einstein tori. Our first result classifies their intersections.



Let T_1,T_2⊂ be Einstein tori.
Suppose that T_1≠ T_2.
Then T_1∩ T_2 is nonempty, and exactly one of the following possibilities
occurs:


  * T_1∩ T_2 is a union of two photons which intersect in exactly one point.

  * T_1∩ T_2 is a spacelike circle and the intersection is transverse.

  * T_1∩ T_2 is a timelike circle and the intersection is transverse.

A single geometric invariant η(T_1,T_2), related to the Maslov index, distinguishes the three cases.

  
  * η(T_1,T_2)=1 if and only if  T_1∩ T_2 is a union of two photons which intersect in exactly one point,
  
  * η(T_1,T_2) > 1 if and only if T_1∩ T_2 is spacelike, and
  
  * η(T_1,T_2) < 1 if and only if T_1∩ T_2 is timelike.



We next show how to further interpret this result in the three-dimensional case. The Lagrangian Grassmannian in dimension 4 is a model of the 3-dimensional Einstein universe. The relationship between the two models was studied extensively in <cit.>. We develop the theory of Einstein tori in the space of Lagrangians and characterize η as the determinant of a linear map.

A simple consequence of Theorem <ref> is

Let T_1,T_2 be a pair
 Einstein tori. Then, T_1∩ T_2 is non-contractible as a subset of T_1 or T_2.


We use this corollary to prove a complete disjointness criterion for crooked surfaces,
generalizing the construction  in Charette-Francoeur-Lareau-Dussault <cit.>
and the criterion for disjointness of anti-de Sitter crooked planes in Danciger-Guéritaud-Kassel <cit.> :


  
  Two crooked surfaces C,C' are disjoint if and only if the four photons on the boundary of the stem of C are disjoint from C', and the four photons on the boundary of the stem of C' are disjoint from C.


The Lagrangian model of the Einstein universe allows us to express the condition in this theorem explicitly in terms of symplectic products. In that model, a pair of simple inequalities guarantee that a photon does not intersect a crooked surface.

Finally, we show that the criterion in theorem <ref> reduces to the criterion for disjointness of anti-de Sitter crooked planes from <cit.> when specializing to crooked surfaces adapted to an anti-de Sitter patch.



§ NOTATIONS AND TERMINOLOGY


If V is a vector space, denote the associated
projective space (V), defined as the space
of all 1-dimensional linear subspaces of V.
If v̌∈ V is a nonzero vector in a vector space V,
then denote the corresponding point (projective equivalence class) in the projective space (V) by [v̌]∈(V). We call a real vector space endowed with a nondegenerate bilinear form a bilinear form space.
If v̌∈ V is a nonzero vector in a bilinear form space (V,),
then

    v̌^⊥ :=  {w̌∈ V |v̌w̌ = 0 }

is a linear hyperplane in V.
When v̌ is non-null, then v̌^⊥ is nondegenerate and
defines an orthogonal decomposition

    V = v̌⊕v̌^⊥.

More generally, if S⊂ V is a subset, then define

    S^⊥ :=  {w̌∈ V |v̌w̌ = 0,   ∀v̌∈ S}.






















§ EINSTEIN GEOMETRY

This section briefly summarizes the basics of the geometry of .
For more details, see <cit.>.



 §.§ The bilinear form space ^n,2


Let  be a (n+2)-dimensional real vector space endowed with a signature
(n,2) symmetric bilinear form

    ×   ⟶
    
    (ǔ,v̌)    ↦ǔv̌.

Define the null cone:

    () := {v̌∈|v̌v̌ = 0}.

The Einstein universe is the projectivization of () :

    :=(()).

 carries a natural conformal Lorentzian structure coming from the product on . More precisely,
smooth cross-sections of  the quotient map ()⟶ determine Lorentzian structures on .
Furthermore these Lorentzian structures are  conformally equivalent
to each other.

The orthogonal group Ø(n,2) of  acts conformally and transitively on
. In fact, the group of conformal automorphisms of  is exactly Ø(n,2).



 §.§ Photons and light cones

A photon is the projectivization (P) of a totally isotropic 2-plane
P⊂. It corresponds to a lightlike geodesic in the conformal Lorentzian metric of .
A spacelike circle (respectively timelike circle)
is the projectived null cone
((S)) of a subspace S⊂ which has signature
(2,1) (respectively signature (1,2)).

A light cone is the projectivized null cone ((H))
of a degenerate hyperplane H⊂.
Such a degenerate hyperplane H = ň^⊥ for some null vector
ň∈(). In terms of the synthetic geometry of ,
the light cone defined by p = [ň]∈ equals the union of all
photons containing p. We will denote it by (p).

One can consider a different homogeneous space, the space of photons of , denoted n. It admits a natural contact structure (see <cit.>) in which the photons in a lightcone form a Legendrian submanifold. The contact geometry of photon space is intimately related to the conformal Lorentzian geometry of the Einstein universe. This relation stems from the incidence relation between the two spaces. We say that a point p∈ is incident to a photon ϕ∈n whenever p∈ϕ. By extension, two points p,q∈ are called incident when they are incident to a common photon, and two photons ϕ,ψ∈n are called incident when they intersect in a common point.



 §.§ Minkowski patches


The complement in  of a light cone is a Minkowski patch.
Its natural structure is Minkowski space ,
an affine space with a parallel Lorentzian metric.
Any geodesically complete simply-connected flat Lorentzian manifold
is isometric to .
As such it is the model space for flat Lorentzian geometry.

Following <cit.>, for  we use quadratic form of signature (n,2) given by

    v̌v̌ := v_1^2 + v_2^2 + … + v_n-1^2 - v_n^2 - v_n+1v_n+2
 and work in the embedding of Minkowski space

    ⟶n
    v̌
     v^n    ⟼v̌
     v^n 
    ‖v̌‖^2 - (v^n)^2  
     1.

In the expression above,

    v̌ :=  v^1 
    ⋮
     v^n-1 ∈

is a vector in Euclidean space with Euclidean norm
‖v̌‖, and the Lorentzian norm in   is:

    (v̌, v^n) ⟼‖v̌‖^2 - (v^n)^2.


The complement of this embedding of  is a light cone, and we will denote its vertex by p_∞. This vertex is called the improper point in <cit.>, and its coordinates in a basis as above are:

    p_∞⟷0̌
     0 
     1 
     0 .

The closure in  of every non-null geodesic γ in  contains p_∞
and the union γ∪{p_∞} is a spacelike circle
or a timelike circle according to the nature of γ.
Conversely every timelike or spacelike circle which
contains p_∞ is the closure of a timelike or spacelike geodesic in .

The light cone of a point which is not p_∞, but belongs to its light cone, intersects the
Minkowski patch  in an affine hyperplane
upon which the Lorentzian structure on  restricts to a field of degenerate
quadratic forms,  that is,  a null hyperplane.



If we choose an origin p_0 for a Minkowski patch, then we get an identification
of the patch with a Lorentzian vector space. The trichotomy of vectors into timelike,
spacelike and lightlike has an intrinsic interpretation with respect to p_0 and p_∞:

A point is :


  * timelike if it lies on some timelike circle through p_0 and p_∞,

  * spacelike if it lies on some spacelike circle through p_0 and p_∞, and

  * lightlike if it lies on a photon through p_0.


One and only one of these three happens for every point in the Minkowski patch.



 §.§ Einstein hyperplanes


An Einstein hyperplane  H corresponds to a linear hyperplane ℓ^⊥⊂ orthogonal to a spacelike line ℓ⊂.
A linear hyperplane ℓ^⊥ is conveniently described by a
normal vector š⊂ℓ,  which we may assume satisfies
šš = 1. In that case š is determined up to multiplication by
± 1.

The hyperplane š^⊥ is a bilinear form space isomorphic to
^n-1,2 and its projectivized null cone is a model for n-1.
In dimension n=3, an Einstein hyperplane is homeomorphic to a 2-torus
S^1× S^1 so we will call it an Einstein torus.
Under the embedding (<ref>),
an Einstein hyperplane which passes through the point p_∞
meets the Minkowski patch  in an affine hyperplane upon which the Lorentzian structure on  restricts to a Lorentzian metric, that is, a timelike hyperplane.

Since an Einstein torus is a totally geodesic embedded copy of 2, it has a pair of natural foliations by photons. This is because the light cone of a point in 2 is a pair
of photons through that point.
As described in Goldman <cit.> for n=3,
the complement of an Einstein hyperplane has the natural structure of the
double covering of anti-de Sitter space. This identification is presented in more detail
in section <ref>.

































§ PAIRS OF EINSTEIN HYPERPLANES

The purpose of this section is to define the invariant η≥ 0 characterizing
pairs of hyperplanes in  and to prove theorem <ref>. We describe the moduli space of equivalence
classes of pairs, and reduce to the case n=3.
Then <ref> reinterprets  in terms of symplectic geometry
using the local isomorphism ⟶Ø(3,2).




 §.§ Pairs of positive vectors

A linearly independent pair of two unit-spacelike vectors š_1, š_2
spans a 2-plane ⟨š_1,š_2⟩⊂ which is:


  * Positive definite ⟺ |š_1š_2| < 1;

  * Degenerate ⟺ |š_1š_2| = 1;

  * Indefinite ⟺ |š_1š_2| > 1.

The positive definite and indefinite cases respectively determine orthogonal splittings :

    ≅^n,2 = ^2,0⊕^n-2,2
       ≅^n,2 = ^1,1⊕^n-1,1.

In the degenerate case, the null space is spanned by
š_1 ±š_2, where

    š_1š_2 = ∓ 1.

By replacing š_2 by -š_2 if necessary, we may assume
that š_1š_2 = 1.
Then š_1 - š_2 is null.
Since  itself is nondegenerate, there exists v̌_3∈ such
that

    (š_1 -š_2)v̌_3 = 1.
 Then  š_1,š_2,v̌_3 span
a nondegenerate 3-plane of signature (2,1).

In all three cases, there is a 5-dimensional subspace of signature (3,2) containing š_1 and š_2.
For that reason, the discussion of pairs of Einstein hyperplanes can be
reduced to the case of n=3.

The absolute value of the product

    η(H_1, H_2) := |š_1·š_2 |

is a nonnegative real number, depending only on the pair
of Einstein hyperplanes H_1 and H_2.
Specifying the above discussion to the case n=3 we have proved Theorem <ref> :



  * If the span of š_1,š_2 is positive definite (η(H_1,H_2)<1), then the intersection of the corresponding Einstein
tori is the projectivised null cone of a signature (1,2) subspace, which is a timelike circle.


  * If the span of š_1,š_2 is indefinite (η(H_1,H_2)>1), then the intersection is the projectivised null cone
of a signature (2,1) subspace, which is a spacelike circle.


  * Finally, if the span of š_1,š_2 is degenerate (η(H_1,H_2)=1), the span š_1 + š_2 is a degenerate 2-plane in š_1 + š_2 + v̌_3 ≅^2,1. The orthogonal complement of this ^2,1 is of signature (1,1) and is contained in (š_1 + š_2)^⊥ = š_1^⊥∩š_2^⊥. Since this last subspace is of dimension 3 and must also contain the degenerate direction of š_1 + š_2, it is a degenerate subspace with signature (+,-,0). Its null cone is exactly the union of two isotropic planes intersecting in the degenerate direction, so when projectivising we get a pair of photons intersecting in a point.




The intersection of two Einstein tori is non-contractible in each of the two tori.

An Einstein torus is a copy of the 2-dimensional Einstein universe. Explicitly, we can write it as ℙ() where  is the null cone in ℝ^2,2. A computation shows that all timelike circles are homotopic, all spacelike circles are homotopic and these two homotopy classes together generate the fundamental group of the torus. Similarly, photons are homotopic to the sum of these generators.










 §.§ Involutions in Einstein tori

Orthogonal reflection in š defines an involution of  which fixes the corresponding
hyperplane H=š^⊥.

The orthogonal reflection in a positive vector š is defined by:

    R_š(v̌) = v̌ - 2v̌š/ššš.


We compute the eigenvalues of the composition R_šR_š'̌, where š,š'̌ are unit spacelike vectors, and relate this to the invariant η.

The orthogonal subspace to the plane spanned by š and š'̌ is fixed pointwise by this composition. Therefore, 1 is an eigenvalue of multiplicity n. In order to determine the remaining eigenvalues, we compute the restriction of R_šR_š'̌ to the subspace š + š'̌.


    R_šR_š'̌(š)    = R_š(š - 2(šš'̌)š'̌)
       =-š - 2(šš'̌)(š'̌-2(š'̌š)š)
       =(4 (š'̌š)^2-1)š -2(š'̌š)š'̌.
    
    
    R_šR_š'̌(š'̌)    = R_š(-š'̌)
       = -š'̌ + 2(šš'̌)š.

The matrix representation of R_šR_š'̌ in the basis š,š'̌ is therefore:

    [ 4 (š'̌š)^2-1      2(šš'̌);     -2(š'̌š)           -1 ].

The eigenvalues of this matrix are:

    2(šš'̌)^2 - 1 ± 2 (šš'̌)√((šš'̌)^2-1).

We observe that they only depend on the invariant η=|šš'̌|. The composition of involutions has real distinct eigenvalues when the
intersection is spacelike, complex eigenvalues when the intersection is timelike, and a double real eigenvalue when the
intersection is a pair of photons.

The case when š_1·š_2 = 0 is special:
in that case the two involutions commute and we will say that the Einstein hyperplanes are orthogonal. As observed at the end of section <ref>, the complement of an Einstein torus in  is
a model for the double covering space of
anti-de Sitter space ^3, which has a complete Lorentzian metric of constant curvature -1.
In this conformal model of ^3 (see <cit.>),
indefinite totally geodesic 2-planes are represented by tori which are orthogonal to ∂^3.




































§ THE SYMPLECTIC MODEL

We describe a model for Einstein 3-space in terms of 4-dimensional symplectic algebra,
an alternative approach which is simpler for some calculations.

Let (,ω) be a 4-dimensional real symplectic vector space,
that is,  is a real vector space of dimension 4 and
× is a nondegenerate skew-symmetric
bilinear form. Let 𝗏𝗈𝗅∈Λ^4(V) be the element defined by the equation (ω∧ω)(𝗏𝗈𝗅)=-2. The second exterior power Λ^2() admits a nondegenerate symmetric bilinear form ·
of signature (3,3) defined by


    (ǔ∧v̌)∧(ǔ'∧v̌')=(ǔ∧v̌)·(ǔ'∧v̌')𝗏𝗈𝗅.


The kernel

    := (ω)⊂Λ^2()

inherits a symmetric bilinear form which has signature (3,2).

Define the vector ω^*∈Λ^2 to be dual to ω by the equation

    ω^*·(ǔ∧v̌) = ω(ǔ,v̌),

for all ǔ,v̌∈.
Because of our previous choice of 𝗏𝗈𝗅, we have
ω^*ω^*=-2.
The bilinear form , together with the vector ω^* define a reflection

    𝖱_ω^*:Λ^2()   →Λ^2()
    α   ↦α + (αω^*)ω^*.

The fixed set of this reflection is exactly the vector subspace  orthogonal to ω^*.

The Plücker embedding ι : 𝖦𝗋(2,) →(Λ^2()) maps 2-planes in  to lines in Λ^2(). We say that a plane in  is Lagrangian if the form ω vanishes identically on pairs of vectors in that plane. If we restrict ι to Lagrangian planes, then the image is exactly the set of null lines in .

The form ω yields a relation of (symplectic) orthogonality on 2-planes in . Lagrangian planes are orthogonal to themselves, and non-Lagrangian planes have a unique orthogonal complement which is also non-Lagrangian. The following proposition relates orthogonality in  with an operation on Λ^2().

  
  A pair of 2-dimensional subspaces S,T⊂ are orthogonal with respect to ω if and only if [𝖱_ω^*(ι(S))] = [ι(T)].
  
    First, assume S is Lagrangian. This means that S=S^⊥, and that ι(S)∈ω^*⊥. Hence,
    
    𝖱_ω^*(ι(S)) = ι(S)=ι(S^⊥).


    Next, if S is not Lagrangian, then we can find bases (ǔ,v̌) of S and (ǔ',v̌') of S^⊥ satisfying ω(ǔ,v̌)=ω(ǔ',v̌')=1 and all other products between these four are zero. Then,
    
    𝗏𝗈𝗅=-ǔ∧v̌∧ǔ'∧v̌'

    and
    
    ω^*=-ǔ∧v̌ -ǔ'∧v̌'.

    Consequently,
    
    [𝖱_ω^*(ι(S))] = [ǔ∧v̌ + ω(ǔ,v̌)ω^*] = [-ǔ'∧v̌'] = [ι(S^⊥)].

  




 §.§ Symplectic interpretation of Einstein space and photon space









The natural incidence relation between  and 3 is
described in the two algebraic models ( and ) as follows.
A point p∈ and a photon ϕ∈3 are
incident if and only if (p,ϕ) satisfies one
of the two equivalent conditions:


  * The null line in  corresponding to p lies in
the isotropic 2-plane in  corresponding to ϕ.

  * The Lagrangian 2-plane in  corresponding to p contains the line in  corresponding to ϕ.

These two are equivalent because of the following proposition :

  
  Let P,Q⊂ be two-dimensional subspaces. Then, P∩ Q= 0 if and only if ι(P)ι(Q)≠0.
  
    Choose bases ǔ,v̌ of P and ǔ',v̌' of Q. Then,
    
    ǔ∧v̌∧ǔ'∧v̌' ≠ 0

    if and only if ǔ,v̌,ǔ',v̌' span  which is equivalent to P and Q being transverse.
  


The light cone (p) of a point p∈ is the union of
all photons containing p.
It corresponds to the orthogonal hyperplane [p]^⊥⊂ of the null line corresponding to p.
In photon space (), the photons containing
p form the projective space (L) of the Lagrangian
2-plane L corresponding to p.






 §.§ Timelike or spacelike triples and the Maslov index

Fixing a pair of non-incident points in the Einstein universe induces a trichotomy on points, as explained in section <ref>.
The corresponding data in the Lagrangian model is related to the Maslov index of a triple of Lagrangians.

Two non-incident points correspond to a pair of transverse Lagrangians L,L'. This induces a splitting =L⊕ L'.
Together with the symplectic form ω, this splitting defines a quadratic form defined by

    q_L,L'(v̌) := ω(π_L(v̌),π_L'(v̌)).


The Maslov index of a triple of pairwise transverse Lagrangians L,P,L' is the integer m(L,P,L')=sign(q_L,L'|_P), where sign(q)  is the difference between the number of positive and negative eigenvalues of q. Transversality
implies that q_L,L' restricted to P is nondegenerate. This index classifies orbits of triples of pairwise transverse Lagrangians <cit.>.

Lagrangians which are nontransverse to L correspond to lightlike points, Lagrangians P with |m(L,P,L')|=2
correspond to timelike points, and Lagrangians P with m(L,P,L')=0 correspond to spacelike points.






 §.§ Nondegenerate planes and symplectic splittings


We describe the algebraic structures equivalent to an
Einstein torus in . As a reminder, these are
hyperplanes of signature (2,2) inside ≅^3,2,
and describe surfaces in 3 homeomorphic to a 2-torus.

In symplectic terms, an Einstein torus corresponds to a splitting
of  as a symplectic direct sum of two nondegenerate
2-planes. Let us detail this correspondence.

Define a 2-dimensional subspace S⊂ to be nondegenerate if and only if
the restriction ω|_S is nondegenerate.
A nondegenerate 2-plane S⊂ determines a splitting as follows. The plane


    S^⊥ := {v̌∈|ω(v̌,S) = 0}

is also nondegenerate, and defines a symplectic complement to S.
In other words,  splits as an (internal) symplectic direct sum:

    = S ⊕ S^⊥.

The corresponding Einstein torus is then the set of Lagrangians which are non-transverse
to S (and therefore also to S^⊥).

The lines in S determine a projective line in 3 which is
not Legendrian. Conversely, non-Legendrian
projective lines in 3 correspond to nondegenerate 2-planes. This non-Legendrian line in 3, as a set of photons, corresponds to one of the two rulings of the Einstein torus.
The other ruling corresponds to the line (S^⊥).


In order to make explicit the relationship between the descriptions of Einstein tori in the two models, define a map μ as follows:

    μ : 𝖦𝗋(2,)    →()
    
                           S    ↦[ι(S) + 1/2ω(ι(S))ω^*],

where ι(S) is any representative in  of the projective class ι(S).

The map μ is the composition of the Plücker embedding ι with the orthogonal projection onto .

  
  For S a nondegenerate plane, the image of μ is always a spacelike line, and μ(S)=μ(S^⊥).
  
    For the first part, let š be any vector representative of the line ι(S). Then,
    
    (š + 1/2ω(š)ω^*) ∧(š +  1/2ω(š)ω^*) = 1/2ω(š)^2 𝗏𝗈𝗅,

    and therefore μ(S) is spacelike.

    The second part is a consequence of the correspondence between orthogonal complements and reflection in ω^* (Proposition <ref>) and the fact that a vector and its reflected copy have the same orthogonal projection to the hyperplane of reflection.
  




  The map μ induces a bijection between spacelike lines in  and symplectic splittings of . Under the Plücker embedding ι, the Einstein torus defined by the symplectic splitting S⊕ S^⊥ is sent to the Einstein torus defined by the spacelike vector μ(S)∈.

Let ǔ∈ be a spacelike vector normalized so that ǔ·ǔ= 2.
Then, both vectors ǔ±ω^* are null.
By the fact that null vectors in Λ^2()
are decomposable,
each ǔ±ω^* corresponds to a 2-plane in .
These 2-planes are nondegenerate since

    (ǔ±ω^*) ∧ω^* =
    -ω(ǔ±ω^*)𝗏𝗈𝗅 = 2≠ 0.

The two planes ǔ±ω^* are orthogonal since they are the images of each other by the reflection 𝖱_ω^*, and so they are the summands for a symplectic splitting of .

This map is inverse to the projection μ defined above.

To prove the last statement in the proposition, we apply proposition <ref>. The Einstein torus defined by the splitting S,S^⊥ is the set of Lagrangian planes which intersect S (and S^⊥) in a nonzero subspace. Let P be such a plane. Then, ι(S)ι(P)=0, which means that

    (ι(S) + 1/2 (ι(S)ω^*)ω^*)ι(P)=0,

so ι(P) is in the Einstein torus defined by the orthogonal projection μ(S). Similarly, if ι(P) is orthogonal to μ(S) then P intersects S in a nonzero subspace.









 §.§ Graphs of linear maps

Now we describe pairs of Einstein tori in terms of symplectic splittings of
(,ω) more explicitly.

Let A,B be vector spaces of dimension 2 and A⊕ B their
direct sum. If AB is a linear map,
then the graph of f is the linear subspace
(f)⊂ A⊕ B consisting of all
ǎ⊕ f(ǎ), where ǎ∈ A.
Every 2-dimensional linear subspace  L ⊂ A⊕ B which is transverse to
B = 0 ⊕ B⊂ A⊕ B equals (f) for a unique  f.
Furthermore, L = (f) is transverse to A = A ⊕ 0 if and only if
f is invertible, in which case L = (f^-1) for the inverse map
B  A.

Now, suppose A and B are endowed with nondegenerate alternating bilinear forms ω_A,
ω_B, respectively.
Let A  B be a linear map.
Its adjugate is the linear map

    B  A

defined as the composition

    B  B^*
     A^*
     A

where  ω_A^#, ω_B^# are isomorphisms induced by
ω_A, ω_B respectively, and f^† is the transpose of f.
If ǎ_1,ǎ_2 and b̌_1,b̌_2 are bases of A and B respectively
with

    ω_A(ǎ_1,ǎ_2)    = 1 
    ω_B(b̌_1,b̌_2)    = 1,

then the matrices representing f and (f) in these bases are related by:

    f_11    f_12
     f_21    f_22 =  f_22    -f_12
     -f_21    f_11.

In particular, if f is invertible, then

    (f)  =  (f) f^-1

where (f) is defined by
f^*(ω_B) = (f) ω_A.


Let = S ⊕ S^⊥.
Let S  S^⊥ be a linear map
and let P = (f)⊂ be the corresponding
2-plane in  which is transverse to S^⊥.


  * P is nondegenerate if and only if (f) ≠ -1.

  * If P is nondegenerate, then its complement P^⊥
is transverse to S, and equals the graph

    P^⊥ = (-(f) ),

of the negative of the adjugate map to f

    S^⊥S.




Choose a basis ǎ, b̌ for S.
Then ǎ⊕ f(ǎ) and b̌⊕ f(b̌)
define a basis for P, and

    ω(ǎ⊕ f(ǎ),b̌⊕ f(b̌))     =
    ω(ǎ,b̌) + ω( f(ǎ),f(b̌))
    
        = (1 + (f)) ω(ǎ,b̌),

since, by definition,

    ω( f(ǎ),f(b̌)) = (f)
    ω(ǎ,b̌).

Thus P is nondegenerate if and only if 1 + (f)≠ 0, as desired.

For the second assertion, suppose that P is nondegenerate.
Since P,P^⊥, S, S^⊥⊂ are each 2-dimensional,
the following conditions are equivalent:


  * P is transverse to S^⊥;

  * P∩ S^⊥ = 0;

  * P^⊥ + S =;

  * P^⊥ is transverse to S.

Thus P^⊥ = (g) for a linear map S^⊥ S.

We express the condition that ω(P,P^⊥) = 0 in terms of f and g:
For š∈ S and ť∈ S^⊥, the symplectic product is zero if anly only if

    ω(š + f(š), t + g(ť) ) =
    ω(š,  g(ť) ) + ω(f(š), ť)

vanishes.
This condition easily implies that g = -(f) as claimed.



The following proposition relates the invariant η defined for a pair of spacelike vectors with the invariant  associated to a pair of symplectic splittings.

  Let S⊕ S^⊥ be a symplectic splitting and f:S→ S^⊥ be a linear map with (f)≠ -1. Let T=(f) be the symplectic plane defined by f. Then,
  
    η(μ(S),μ(T))=
      
       |1-(f)|/|1+(f)|.

  
    Let ǔ,v̌ be a basis for S such that ω(ǔ,v̌)=1. Then, ǔ+f(ǔ),v̌+f(v̌) is a basis for T. Moreover,
    
    ǔ∧v̌∧(ǔ + f(ǔ))∧(v̌ + f(v̌)) = ǔ∧v̌∧ f(ǔ) ∧ f(v̌).

    We can compute which multiple of 𝗏𝗈𝗅 this last expression represents by using the normalization (ω∧ω)(𝗏𝗈𝗅)=-2  and the computation
    
    (ω∧ω)(ǔ∧v̌∧ f(ǔ)∧ f(v̌)) =  2(f).

    We deduce that
    
    ǔ∧v̌∧ f(ǔ) ∧ f(v̌)=-(f)𝗏𝗈𝗅.


    Using the product formula from lemma <ref>, we find that
    
    √(2)ǔ∧v̌ + ω^*/√(2)

    is a unit spacelike representative of μ(S), and
    
    √(2)(ǔ + f(ǔ))∧(v̌ + f(v̌))/1+(f) + ω^*/√(2)

    is a unit spacelike representative of μ(T). Their product is
     
    -2(f)/1+(f) + 1 = 1-(f)/1+(f),

     proving the proposition.

    
    
    
    
    
    
    
    
  





























































§ EXAMPLES OF PAIRS OF TORI

The purpose of this section is to describe a few basic examples
illustrating our general theory.
These examples include indefinite affine planes
(Einstein tori containing the distant point)
and one-sheeted hyperboloids.



 §.§ The inner product space associated to (,ω)

Let e_1,e_2,e_3,e_4∈ be a basis for which the
symplectic form ω is defined by ω(e_1,e_3) = ω(e_2,e_4) = 1 and other products zero.
We define the inner product on = (ω) ⊂Λ^2() which has signature (3,2).

Let e^1,e^2,e^3,e^4∈^* be the basis dual to
e_1,e_2,e_3,e_4; in this basis

    ω = e^1 ∧ e^3 + e^2 ∧ e^4.

and the corresponding volume form is:

    ω∧ω = -2 e^1∧ e^2 ∧ e^3 ∧ e^4.

Denote the element ω^*∈Λ^2()
dual to ω by:

    ω^* := e_1∧ e_3 + e_2∧ e_4/2

so that ω(ω^*)=1.
For α∈Λ^2(),

    α∧ω^*
    = ω(α)
    ( - e_1 ∧ e_2 ∧ e_3 ∧ e_4/2).

For α,β∈Λ^2(), define their
inner product α·β by:

    α∧β =:α·β e_1 ∧ e_2 ∧ e_3 ∧ e_4/2.

so that
ω^*·ω^* = -1 and
ω(α) = - ω^*·α.

Projection onto 
    =(ω) = (ω^*)^⊥

is:

    Λ^2()    ⟶
    α   ⟼α - ω(α)ω^*

The innner product (<ref>) on
Λ^2() has signature (3,3).
Since

    ω^*·ω^* = -1,

the induced bilinear form on its orthogonal complement
 is nondegenerate with signature (3,2).



 §.§ Symplectic splittings from positive vectors


If v∈ is a positive vector, say with v· v= 1,
then the both vectors v ±ω^* are null
(since ω^*·ω^* = -1 and  v·ω^*=0).
By the well-known   
fact that null vectors in Λ^2()
are decomposable,
each v ±ω^* corresponds to a 2-plane in .
This 2-plane is nondegenerate since

    (v±ω^*) ∧ω^* =
    -ω(v±ω^*)( e_1 ∧ e_2 ∧ e_3 ∧ e_4/2) ≠ 0.

These two planes are the summands for a
symplectic splitting of .





 §.§ Some specific examples

The reference <cit.> uses an inner product space
(denoted ^3,2) which has coordinates (X,Y,Z,U,V)
where the inner product has associated quadratic form
equals X^2 + Y^2 - Z^2 - UV;
compare (<ref>).
(Reference <cit.> uses the standard diagonal form
X^2 + Y^2 + T^2 - Z^2 - W^2,
where the coordinates (X,Y,T,Z,W) correspond to:

    T    : = (U-V)/2, 
    
    W    := (U+V)/2.



The coordinate basis
of Λ^2() projects to the following  basis for :

    {
    e_1∧ e_2,e_1∧ e_3 - e_2 ∧ e_4,e_1∧ e_4, e_2∧ e_3,  e_3∧ e_4}.

and 
(X,Y,Z,U,V)∈ corresponds to:

    1/2(
    X  (e_1∧ e_3 - e_2 ∧ e_4)    +
    Y  (e_1∧ e_2 + e_3 ∧ e_4)  
     +
    Z  (e_1∧ e_2 - e_3 ∧ e_4)      -
    U  e_1 ∧ e_4 +
    V  e_2 ∧ e_3
    ).

The inner product is then

    (X_1,Y_1,Z_1,U_1,V_1)    ·
    (X_2,Y_2,Z_2,U_2,V_2)  
    
    =      X_1X_2 + Y_1Y_2 -   Z_1Z_2  -
    
    (U_1V_2 + V_1U_2)/2

as desired.


























 §.§ Planes parallel to the yz-plane


Let x_0∈.
The plane defined by x=x_0 in  completes to
an Einstein torus denoted .
In , it corresponds to the
hyperplane in  defined by X = x_0 V,
or, equivalently,

    (X,Y,Z,U,V) ·ξ_x_0 = 0

where

    ξ_x_0 = (1,0,0,2x_0,0) = 1/2
    (e_1∧ e_3 - e_2 ∧ e_4 - 2 x_0  e_1 ∧ e_4)

is unit positive.

Now we apply the construction from <ref>:
Since

    ξ_x_0 + ω^*     =
    e_1∧ e_3 - x_0 e_1∧ e_4 =
    e_1∧ (e_3 - x_0 e_4)  
    ξ_x_0 - ω^*     =
    e_2∧ e_4 - x_0 e_1∧ e_4 =
    (e_2 - x_0  e_1) ∧ e_4,

the corresponding symplectic splitting
= S_x_0⊕ S_x_0^⊥ is defined by:

    S_x_0   := ⟨ e_1, e_3 - x_0 e_4 ⟩
    
    S_x_0^⊥     := ⟨ e_2 + x_0 e_1, e_4 ⟩.

These nondegenerate 2-planes are graphs of linear maps
S_0 → S_0^⊥ as follows:
S_x_0 = (f_x_0) where:

    e_1    f_x_0 0 
    
    e_3    ⟼ -x_0  e_4

However, S_x_0^⊥ is not transverse to S_0
and cannot be represented as such a graph.
On the other hand, S_x_0^⊥ is transverse to S_0 and

    S_x_0^⊥ = (S_0^⊥ S_0)
 
where

    e_2    f_x_0^⊥ -x_0 e_1 
    
    e_4    ⟼ 0

Similarly  S_x_0 is not transverse to S_0 and cannot be
represented as a graph of a map S_0→ S_0^⊥.
In all of these cases (f) = 0.

For x_0≠ 0, the tori  and 𝒳(0) intersect in a pair of ideal
photons intersecting at the distant point: these photons correspond to
the isotropic planes:

    {(0,Y,± Y, U, 0) | Y,U∈}⊂

































 §.§ The xz-plane

Denote the Einstein corresponding to the xz-plane
y=0 by .
This corresponds to the hyperplane
Y = 0 in , the orthogonal complement
of

    ξ := 1/2(
    e_1∧ e_2 + e_3∧ e_4
    ).

Since

    ξ + ω^*     =1/2 (e_1-e_4) ∧ (e_2 + e_3) 
    ξ - ω^*     = 1/2 (e_1+e_4) ∧ (-e_2 + e_3),

the corresponding symplectic splitting is:

    = ⟨ e_1 - e_4 , e_2  + e_3  ⟩⊕⟨ e_1 + e_4 , -e_2 + e_3 ⟩.

Let = S ⊕ S^⊥ be the symplectic
splitting corresponding to 𝒳(0).
Give S and S^⊥ the ordered bases
(e_1,e_3), (e_2,e_4), respectively.
In terms of these bases S and S^⊥,
the summands corresponding to  are the graphs of

    0     -1 
     1     0 ,
     0     1 
     -1     0 ,

respectively.
𝒳(0) meets this torus in the z-axis
defined by x = y = 0, a timelike geodesic (particle) in .



 §.§ The one-sheeted hyperboloid


Another interesting torus is the one-sheeted hyperboloid
ℋ defined by
x^2 + y^2 - z^2 = 1 in . This corresponds to
the hyperplane U = V in . The unit positive vector normal to this hyperplane is:

    ξ() :=   0 
     0 
     0 
     -1 
     1 ⟷
    e_1 ∧ e_4 + e_2 ∧ e_3 ∈.

The corresponding symplectic splitting is:
= H ⊕ H^⊥, where

    h_1     := e_1 + e_2  
    
    h_2     := e_3 + e_4

and

    h_1^⊥    := e_1 - e_2  
    
    h_2^⊥    := e_3 - e_4

are bases of H and H^⊥ respectively.
Since

    h_1 + h_1^⊥   ∈ S_x_0, 
    
    h_2 + 1-x_0/1+x_0  h_2^⊥   ∈ S_x_0,

S_x_0 = (f) where

    f =  1     0 
     0    
    (1+x_0)/(1-x_0)
     .

The intersection ℋ∩
will be a pair of photons if x_0 = ± 1, a spacelike
circle if -1 < x_0 < 1 and a timelike circle if |x_0|>1 (see fig. <ref>).
The determinant (f) of f is never -1 (since all the
2-planes are nondegenerate).
The ruled torus corresponds to one of two the summands,
and the torus itself corresponds to the splitting, that
is, to the pair (f, -f/(f)).



 §.§ Vertical translates

Translating  by the vertical vector (0,0,z_0)
yields

    (z_0) := { x^2 + y^2 - (z-z_0)^2 = -1 },

which is defined by the unit positive vector

    ξ((z_0)) :=  0 
     0 
     z_0 
     -(1+z_0^2) 
     1 .

These vectors have inner products:

    ξ((z_1)) ·ξ((z_2 ))  = 1 + 1/2 (z_1 - z_2)^2.

In particular

    (z_0) ∩(-z_0) =  { x^2 + y^2 = 1,  z = 0},

the horizontal circle of radius √(1+z_0^2) and
(f) = -(1 + z_0^2).

Here is a slight variant, where the intersection is the horizontal unit circle
x^2 + y^2 - 1 = z = 0.  Take ξ(z_0) to be the unit positive vector:

    ξ_± :=
     0 
     0 
      z_0 
     -√(1+z_0^2)
    √(1+z_0^2)

The corresponding tori are the hyperboloids (z_0) defined by :

    x^2 + y^2 - (z - z_0/√(1+z_0^2))^2  =  1/1+z_0^2

The intersection (z_0)∩(-z_0) equals the horizontal unit circle. The inner product

    ξ_+ ·ξ_- = -(1 + 2z_0^2) < -1.

The corresponding determinant is d = -z_0^2/(1 + z_0^2) for which
-1<d<0.



































































































§ DISJOINT CROOKED SURFACES


In this section we apply the techniques developed above in order to prove a full disjointness criterion for pairs
of crooked surfaces.

We work in the symplectic framework of section <ref> with the symplectic vector space (,ω).

Let ǔ_+,ǔ_-,v̌_+,v̌_- be four vectors in  such that 
    ω(ǔ_+,v̌_-)=ω(ǔ_-,v̌_+)=1
 and all other
products between these four vanish. This means that we have Lagrangians

    P_0    := v̌_+ + v̌_-,
    
      P_∞   := ǔ_+ + ǔ_-, and
    
      P_±   :=v̌_± + ǔ_±

representing the points of intersection of the photons associated to [ǔ_+],[ǔ_-],[v̌_+],[v̌_-]. We call this configuration of four points and four photons a lightlike quadrilateral.

The crooked surface C determined by this configuration is a subset of  consisting of three pieces : two wings and a stem (see Figure <ref>).
The two wings are foliated by photons, and we will denote by 𝒲_+, 𝒲_- the sets of photons covering the wings. Each wing is a subset of the light cone of P_+ and P_-, respectively. Identifying points in () with the photons they represent, the foliations are as follows:

    𝒲_+ = {[t ǔ_+ + s v̌_+]   |   t s ≥ 0},


    𝒲_- = {[t ǔ_- + s v̌_-]   |   t s ≤ 0}.

We will sometimes abuse notation and use the symbol 𝒲_± to denote the collection of points in the Einstein universe which is the union of these collections of photons.

The stem 𝒮 is the subset of the Einstein torus determined by the splitting
S_1 ⊕ S_2 := (ǔ_+ + v̌_-) ⊕ (ǔ_- + v̌_+)
consisting of timelike points with respect to P_0,P_∞ :

    𝒮 = {L = w̌ + w̌'  | w̌∈ S_1, w̌'̌∈ S_2, |m(P_0,L,P_∞)| = 2}.

Note that this definition gives only the interior of the stem as defined in <cit.>. A crooked surface is the closure in  of a crooked plane in the
Minkowski patch defined by the complement of the light cone of P_∞
(see <cit.>).




Let C_1, C_2 be two crooked surfaces such that their stems intersect. Then, the stem of C_1 intersects a wing of C_2 or vice versa. That is, crooked surfaces cannot intersect in their stems only.

The stem consists of two disjoint, contractible pieces. To see this, note that this set is contained in the Minkowski patch defined by P_∞. There, the Einstein torus containing the stem is a timelike plane through the origin, and the timelike points in this plane form two disjoint quadrants. Let K be the intersection of the two Einstein tori containing the stems of C_1 and C_2. Then, K is non-contractible in either torus (Corollary <ref>), so it can't be contained in the interior of the stem. Therefore, K must intersect the boundary of the stem which is part of the wings.





Let p_0,p_∞,p∈ be three points in the Einstein universe. The point p is timelike with respect to p_0,p_∞ if and only if the intersection of the three light cones of p,p_0,p_∞ is empty.

We work in the model of  given by lightlike lines in a vector space of signature (3,2). If p is timelike with respect to p_0,p_∞, then it lies on a timelike curve which means that the subspace generated by p,p_0,p_∞ has signature (1,2). Therefore, its orthogonal complement is positive-definite and contains no lightlike vectors, so the intersection of the light cones is empty. The converse is similar.




A photon represented by a vector p̌∈ is disjoint from the crooked surface C if and only if the following two inequalities are satisfied:

    ω(p̌,v̌_+)ω(p̌,ǔ_+)>0


    ω(p̌,v̌_-)ω(p̌,ǔ_-)<0.


Write p̌ in the basis ǔ_+,ǔ_-,v̌_+,v̌_- :

    p̌ = a ǔ_+ + b ǔ_- + c v̌_+ + d v̌_-.

Then,

    a=ω(p̌,v̌_-)    b=ω(p̌,v̌_+)


    c=-ω(p̌,ǔ_-)    d=-ω(p̌,ǔ_+).


The photon p̌ is disjoint from 𝒲_+ if and only if the following equation has no solutions with ts≥ 0:

    ω(p̌, t ǔ_+ + s v̌_+) = 0.

This happens exactly when b d < 0.
Similarly, p̌ is disjoint from 𝒲_- if and only if a c >0. These two equations are equivalent to the ones in the statement of the Lemma, therefore it remains only to show that under these conditions, p̌ is disjoint from the stem.

The Lagrangian plane P representing the intersection of p̌ with the Einstein torus containing the stem is generated by p̌ and a ǔ_+ + d v̌_-. This is because a ǔ_+ + dv̌_- represents the unique photon in one of the foliations of the Einstein torus which intersects the photon p̌, and hence the span p̌ + (a ǔ_+ + d v̌_-) is their intersection point, the Lagrangian P. We want to show that P is not timelike with respect to P_0,P_∞. By Lemma <ref>, this is equivalent to showing that the triple intersection of the lightcones of P_0, P, and P_∞ is non-empty.

The intersection of the light cones of P_0 and P_∞ consists of planes of the form: (s ǔ_+ + t ǔ_-) + ( s' v̌_+ + t' v̌_-) where st' + ts'=0. We want to show that no point represented by such a plane is incident to P. Two Lagrangian planes are incident when their intersection is a non-zero subspace. Equivalently, they are incident if they do not span . We have :

    (p̌,a ǔ_+ + d v̌_-, s ǔ_+ + t ǔ_-, s'v̌_+ + t' v̌_-)
     = (-bdss' + catt')(ǔ_+,ǔ_-,v̌_+,v̌_-)
    
     = k(bds^2 + act^2)(ǔ_+,ǔ_-,v̌_+,v̌_-),

where t'=kt, s'=-ks, k≠ 0. There exist t,s making this determinant vanish because bd,ac have different signs. This means that the point where p̌ intersects the Einstein torus containing the stem is not timelike and therefore outside the stem.




Two crooked surfaces C,C' given respectively by the configurations ǔ_+,ǔ_-,v̌_+,v̌_- and ǔ'_+,ǔ'_-,v̌'_+,v̌'_- are disjoint if and only if the four photons ǔ'_+,ǔ'_-,v̌'_+,v̌'_- do not intersect C and the four photons ǔ_+,ǔ_-,v̌_+,v̌_- do not intersect C'.


Since the four photons on the boundary of the stem are part of the crooked surface,  the forward implication is clear.

We now show the reverse implication. Assume that the four photons ǔ'_+,ǔ'_-,v̌'_+,v̌'_- do not intersect C and the four photons ǔ_+,ǔ_-,v̌_+,v̌_- do not intersect C'. Let us first show that the wing 𝒲_+ of C does not intersect C'. By lemma <ref>, it suffices to show that

    ω(tǔ_+ + sv̌_+, v̌'_+)ω(tǔ_+ + sv̌_+, ǔ'_+) > 0

and

    ω(tǔ_+ + sv̌_+, v̌'_-)ω(tǔ_+ + sv̌_+, ǔ'_-) < 0

for all s,t∈ such that st≥ 0 (with s and t not both zero).

We have

    ω(tǔ_+ + sv̌_+, v̌'_+)ω(tǔ_+ + sv̌_+, ǔ'_+) 
       =t^2ω(ǔ_+,v̌'_+)ω(ǔ_+,ǔ'_+) + stω(ǔ_+,v̌'_+)ω(v̌_+,ǔ'_+)
            + stω(v̌_+,v̌'_+)ω(ǔ_+,ǔ'_+) + s^2ω(v̌_+,v̌'_+)ω(v̌_+,ǔ'_+).

By hypothesis, neither ǔ_+, v̌_+ intersect C', and neither ǔ'_+, v̌'_+ intersect C. Therefore, using again lemma <ref> and st≥ 0, we see that each term in this sum is non-negative and that at least one of them must be strictly positive. Therefore,

    ω(tǔ_+ + sv̌_+, v̌'_+)ω(tǔ_+ + sv̌_+, ǔ'_+) > 0.

The proof that

    ω(tǔ_+ + sv̌_+, v̌'_-)ω(tǔ_+ + sv̌_+, ǔ'_-) < 0

is similar. Therefore, 𝒲_+ does not intersect C'.

In an analogous way, one can show that 𝒲_- does not intersect C'. Therefore, the wings of the crooked surface C do not intersect C'. Hence, to show that C and C' are disjoint, it only remains to show that the stem of C does not intersect C'.

By symmetry, the wings of C' do not intersect C, which means in particular that they do not intersect the stem of C. Consequently, the stem of C can only intersect the stem of C'. However, according to theorem <ref>, if the stem of C intersects the stem of C', it must necessarily intersect its wings as well, which is not the case here. Therefore, we conclude that C and C' must be disjoint.


By lemma <ref>, this disjointness criterion can be expressed explicitly as 16 inequalities (two for each of the 8 photons defining the two crooked surfaces). There is some redundancy in these inequalities, but there does not seem to be a natural way to reduce the system.



§ ANTI-DE SITTER CROOKED PLANES



In this section, we show that the criterion for disjointness of anti-de Sitter crooked planes described in <cit.> is a special case of theorem <ref>, when embedding the double cover of anti-de Sitter space in the Einstein universe.

The 3-dimensional Anti-de Sitter space, denoted , is the manifold (2,) ≅() endowed with the bi-invariant Lorentzian metric given by the Killing form. We now recall the definition of a (right)  crooked plane.

Let ℓ be a geodesic in the hyperbolic plane . The right  crooked plane based at the identity associated to ℓ is the set of g∈(2,) such that g has a nonattracting fixed point in ℓ⊂∪∂. In other words, the isometries g∈(2,) which make up the crooked plane are :

  
  * elliptic elements centered on a point of ℓ,
  
  * parabolic elements with fixed point in ∂ℓ, and
  
  * hyperbolic elements with repelling fixed point in ∂ℓ.


A right  crooked plane based at g∈(2,) is a left-translate of one based at the identity. We will say that such a crooked plane is defined by the pair (g,ℓ).

A left  crooked plane is defined the same way, replacing nonattracting fixed point by nonrepelling fixed point. Since a right  crooked plane and a left  crooked plane always intersect, we will assume in what follows that all our  crooked planes are of the first type.


Let ℓ,ℓ' be geodesic lines of  and g∈(2,). Then, the right  crooked planes defined by (I,ℓ) and (g,ℓ') are disjoint if and only if for any endpoints ξ of ℓ and ξ' of ℓ', we have ξ≠ξ' and d(ξ,gξ')-d(ξ,ξ')<0.

In this criterion, the difference d(p,gq)-d(p,q) for p,q∈∂ is defined as follows : choose sufficiently small horocycles C,D through p,q respectively. Then, d(p,gq)-d(p,q):=d(C,GD)-d(C,D) and this quantity is independent of the choice of horocycles.



 §.§ AdS as a subspace of Ein

Let _0 be a real two dimensional symplectic vector space with symplectic form ω_0. Denote by  the four dimensional symplectic vector space =_0 ⊕_0 equipped with the symplectic form ω = ω_0 ⊕ -ω_0. This vector space  will have the same role as in section <ref>.

The Lie group (_0)=(_0) is a model for the double cover of anti-de Sitter 3-space. We will show how to embed this naturally inside the Lagrangian Grassmannian model of the Einstein Universe in three dimensions. To do this, define

    i : (_0) →Gr(2,)


    f ↦(f)


The graph of f ∈(_0) is a Lagrangian subspace of =_0⊕_0.
This means that i((_0))⊂() ≅. This map is equivariant with respect to the homomorphism:

    (_0) ×(_0) →()


    (A,B) ↦ B ⊕ A ,

where the action of (_0) ×(_0) on (_0) is by (A,B)· X = AXB^-1.


The involution of  induced by the linear map

    I ⊕ -I : _0⊕_0 ↦_0 ⊕_0,

where I denotes the identity map on _0, preserves the image of i. It corresponds to the two-fold covering (_0)→(_0). The fixed points of this involution are exactly the complement of the image of i, corresponding to the conformal boundary of AdS (see section 2 of <cit.> for details).






 §.§ Crooked surfaces and AdS crooked planes


As in <cit.>, we say that a crooked surface is adapted to an AdS patch if it is invariant under the involution I ⊕ -I. Goldman proves in <cit.> that a crooked surface is adapted to an  patch if and only if it is the closure in 3 of a left or right  crooked plane in that patch. Moreover, two  crooked planes in the same patch are disjoint if and only if their closures in 3 are disjoint.

 If a crooked surface is invariant under I ⊕ -I, then its corresponding lightlike quadrilateral is invariant. Two of the opposite vertices are fixed (they lie on the boundary of AdS) and the two others are swapped. If we denote the four photons by ǔ_-,ǔ_+,v̌_-,v̌_+, this means v̌_- = (I ⊕ -I) ǔ_- and v̌_+ = (I ⊕ -I) ǔ_+.



  §.§.§ AdS crooked planes based at the identity

For concreteness, choose a basis of  to identify it with ^4.
We will represent a plane in ^4 by a 4 × 2 matrix whose columns generate the plane, up to multiplication on the right by an invertible 2×2 matrix. For example, (f) corresponds to the matrix:

    [ I; f ].

The identity element of (_0) maps to the plane

    [ I; I ]

and its image under the involution I ⊕ -I is

    [  I; -I ].

The intersection of the lightcones of the two Lagrangians (I) and (-I) consists of Lagrangians which have the form

    [  v_1  v_1;  v_2  v_2;  v_1 -v_1;  v_2 -v_2 ]

for some v_1,v_2∈ not both zero.

Therefore, the lightlike quadrilaterals containing as opposite vertices (I) and (-I) are parameterized by pairs of distinct nonzero vectors ǎ,b̌∈_0 (2× 1 column vectors), up to projective equivalence. The four vertices of the lightlike quadrilateral are then:

    [ I; I ], [  ǎ  ǎ;  ǎ -ǎ ], [  b̌  b̌;  b̌ -b̌ ], [  I; -I ].

We will say that such a lightlike quadrilateral is based at I and defined by the vectors ǎ,b̌. We choose as representatives of its lightlike edges the vectors:

    v̌_- = [ ǎ; ǎ ], ǔ_- = [ -ǎ;  ǎ ]


    v̌_+ = [ b̌; b̌ ], ǔ_+ = [  b̌; -b̌ ].


With the definition of the wings using the sign choices of section <ref>, we will see that the intersection of the associated crooked surface with the  patch is a right  crooked plane.

Indeed, the definition of the photons foliating the wing 𝒲_+ was

    𝒲_+ = {[t ǔ_+ + s v̌_+]   |   t s ≥ 0}.


Suppose that the graph Lagrangian [ I; f ] for some f∈(2,) is on such a photon, equivalently that it contains a vector of the form

    [ (t+s)b̌; (t-s)b̌ ]

with ts≥ 0. This is equivalent to

    fb̌ = (t-s/t+s)b̌.

When t s ≥ 0, the quantity |t-s/t+s| ≤ 1, hence the point [b̌]∈∂ is a nonattracting fixed point of f. By a similar calculation, we can show that 𝒲_- consists of elements (f) such that f has a nonattracting fixed point at [ǎ]∈∂.



  §.§.§ AdS crooked planes based at f

In order to get an AdS crooked plane based at a different point f∈(_0), we map the crooked plane by an element of the isometry group (_0) ×(_0) ⊂(). The easiest way is to use an element of the form :

    [ I 0; 0 f ].

This corresponds to left multiplication by f in ().

Applying f to a lightlike quadrilateral, we get a lightlike quadrilateral with vertices of the form:

    [ I; f ],  [  I; -f ] , [  ǎ -ǎ; fǎ fǎ ],
    [   b̌   b̌;  fb̌ -fb̌ ]


and edges of the form:

    [  ǎ; fǎ ], [ -ǎ; fǎ ]


    [  b̌; fb̌ ], [   b̌; -fb̌ ].






 §.§ Disjointness


The disjointness criterion for crooked surfaces in the Einstein Universe is given by 16 inequalities. Using the symmetries imposed by an AdS patch, we can reduce them to 4 inequalities.

Using the involution defining the AdS patch, we can immediately reduce the number of inequalities by half. This is because both surfaces are preserved by the involution, and their defining photons are swapped in pairs. (So for example, we only have to check that ǔ_+ and ǔ_- are disjoint from the other surface, for each surface.)

The second reduction comes from the fact that for AdS crooked planes, we only need to check that the four photons from the first crooked surface are disjoint from the second, and then the four from the second are automatically disjoint from the first.

For a crooked surface based at the identity with lightlike quadrilateral defined by the vectors ǎ,b̌∈_0 and another based at f with quadrilateral defined by ǎ',b̌'∈_0, the inequalities reduce to:


    ω_0(ǎ',b̌)^2 > ω_0(fǎ',b̌)^2


    ω_0(ǎ',ǎ)^2 > ω_0(fǎ',ǎ)^2


    ω_0(b̌',b̌)^2 > ω_0(fb̌',b̌)^2


    ω_0(b̌',ǎ)^2 > ω_0(fb̌',ǎ)^2.


What remains is to interpret these four inequalities in terms of hyperbolic geometry. We first define an equivariant map from ℙ(_0) to ∂ℍ^2. As a model of the boundary of ℍ^2, we use the projectivized null cone for the Killing form in 𝔰𝔩(_0)≅𝔰𝔩(2,). Choose a basis of _0 in which ω_0 is given by the matrix J=[  0  1; -1  0 ] and define

    η : _0    →(𝔰𝔩(2,))
    ǎ   ↦ -ǎǎ^T J,

where ǎ is a column vector representing a point in ℙ(_0). This map associates to the vector ǎ the tangent vector to at identity of the photon between I and the boundary point [  ǎ  ǎ;  ǎ -ǎ ]. Note that the image of η is contained in the upper part of the null cone.


η is equivariant with respect to the action of (_0).


    η(Aǎ) = -Aǎ(Aǎ)^TJ = -Aǎǎ^T A^T J = -A ǎǎ^T J A^-1 = A η(ǎ) A^-1.




We will denote by K the trace form on 𝔰𝔩(2,)

    K(X,Y) = (XY).

Its value is 1/8 times the Killing form.


Let ǎ,b̌∈_0. Then,
ω_0(ǎ,b̌)^2 = -K(η(ǎ),η(b̌)).


    ω_0(ǎ,b̌)^2    = -ǎ^T J b̌b̌^T J ǎ
       = ǎ^T J η(b̌) ǎ
       = ( ǎ^T J η(b̌) ǎ)
       = (ǎǎ^T J η(b̌))
       = -(η(ǎ)η(b̌))
       = -K(η(ǎ),η(b̌)).



Note that the expression ω_0(ǎ,b̌) is not projectively invariant, but the sign of ω_0(ǎ,b̌)^2 - ω_0(ǎ, fb̌)^2 is.

The following inequalities are equivalent

    ω_0(ǎ,b̌)^2 - ω_0(ǎ,fb̌)^2>0,
 
    K(η(ǎ),fη(b̌)f^-1) > K(η(ǎ),η(b̌)).




Finally, we want to show that the four inequalities (<ref>) are equivalent to the DGK criterion (Theorem <ref>).

Let A,B,A',B' denote respectively η(ǎ),η(b̌),η(ǎ'),η(b̌'). Then, A, B, A', B' represent endpoints of two geodesics g,g' in the hyperbolic plane. We want to show

    d(ξ,fξ'f^-1) - d(ξ,ξ')<0

for ξ∈{A,B} and ξ'∈{A',B'}.

We use the hyperboloid model of , {X∈𝔰𝔩(2,) | K(X,X)=-1}. Consider horocycles C_ξ(r)={X∈ |  K(X,ξ)=-r} and C_ξ'(r') = {X∈ |  K(X,ξ')=-r'} at ξ and ξ' respectively. The distance between these two horocycles is given by the formula

    d(C_ξ(r),C_ξ'(r'))=arccosh(-1/2(K(ξ,ξ')/2rr' + 2rr'/K(ξ,ξ'))).

Similarly,

    d(C_ξ(r),fC_ξ'(r')f^-1)=arccosh(-1/2(K(ξ,fξ'f^-1)/2rr'+2rr'/K(ξ,fξ'f^-1))).

We know that K(ξ,fξ'f^-1)>K(ξ,ξ'). If r,r' are sufficiently small, by increasingness of the function x↦ x+1/x for x>1 and increasingness of arccosh we conclude d(C_ξ(r),C_ξ'(r'))>d(C_ξ(r),fC_ξ'(r')), which is what we wanted.
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