Lightlike and ideal tetrahedra

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Finally, we compute the volumes of both types of tetrahedra as functions of their edge lengths or dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume formula of ideal hyperbolic tetrahedra.


Introduction
Ideal hyperbolic tetrahedra Hyperbolic ideal tetrahedra are fundamental building blocks in 3d hyperbolic geometry. They are geodesic tetrahedra in H 3 with vertices in the ideal boundary ∂ ∞ H 3 ∼ = CP 1 . As they are determined by their vertices, they are parametrized, up to isometries, by a single complex parameter z ∈ C\{0, 1}, its shape parameter or cross-ratio.
The general approach to the construction of 3d hyperbolic structures via hyperbolic ideal tetrahedra was introduced by Thurston in [Th80]. Starting with a topological 3-manifold M with a topological ideal triangulation, one chooses hyperbolic structures on the tetrahedra that glue smoothly into a hyperbolic structure on M . The consistency conditions for the gluing determine a system of algebraic equations on the set of shape parameters. Under a few additional assumptions, solutions to these gluing equations define a smooth hyperbolic structure on M .
This construction is a powerful tool in 3d hyperbolic geometry. Given a hyperbolic 3-manifold M with a geodesic ideal triangulation and solutions of Thurston's gluing equations, one can in principle compute many invariants of M . In particular, the hyperbolic volume of M can be computed as the sum of volumes of each ideal tetrahedron [Th80], see also [NZ85], which is a well-know function of the shape parameter [Mi82].

Generalized ideal tetrahedra This description of hyperbolic 3-manifolds in terms of
ideal hyperbolic tetrahedra can be generalized to other geometries. In [Da11,Da14] Danciger introduced a generalized notion of ideal tetrahedra in 3d anti-de Sitter and 3d half-pipe spaces and studied a generalized version of Thurston's gluing equations.
Denoting by Y Λ the 3d hyperbolic space for Λ > 0, the 3d anti-de Sitter space for Λ < 0 and the 3d half-pipe space for Λ = 0, one can describe these generalized ideal tetrahedra as geodesic tetrahedra in Y Λ with vertices at the ideal boundary ∂ ∞ Y Λ and with spacelike edges. The additional condition that the edges are spacelike imposes restrictions on the relative position of the vertices at the asymptotic boundary. Nonetheless, generalized ideal tetrahedra are also parametrized, up to isometries, by a single shape parameter, now taking values in the ring of generalized complex numbers C Λ . See also [Lu15], for a general discussion of gluing equations over commutative rings.
Generalized ideal tetrahedra share many properties with their hyperbolic counterparts and thus offer the prospect to generalize results and constructions from hyperbolic geometry to 3d anti-de Sitter and half-pipe geometry. In particular, they were applied by Danciger in [Da11,Da14] to construct geometric transitions between hyperbolic and anti-de Sitter structures, going through half-pipe structures, and were also used as building blocks for the study of more general polyhedra in [DMS14].
A particularly interesting quantity in this respect is the hyperbolic volume. The volume of a generalized ideal tetrahedron can be defined as the integral of a 3-form invariant under the action of the isometry group, which is unique up to global rescaling. However, so far there is no anti-de Sitter or half-pipe analogue of the Milnor-Lobachevsky formula for the volume in this setting. This raises Question 1: Is there a simple formula for the volume of a generalized ideal tetrahedron in Y Λ as a function of its shape parameter and the parameter Λ that controls the geometric transitions? 3d Lorentzian geometry Another strong motivation to investigate generalized ideal tetrahedra is the close relation between structures from 2d and 3d hyperbolic geometry and 3d Einstein geometry in Lorentzian signature. Every 3d Lorentzian Einstein manifold M is locally isometric to a homogeneous and isotropic Lorentzian 3d manifold X Λ of constant curvature Λ, namely the 3d de Sitter space for Λ > 0, the 3d anti-de Sitter space for Λ < 0 and the 3d Minkowski space for Λ = 0. The geometry of M can then be described by geometric structures modeled on X Λ and with structure group G Λ = Isom 0 (X Λ ), that is, by an atlas of coordinate charts valued in X Λ with isometric transition functions.
Under additional assumptions on causality, namely maximal global hyperbolicity and the completeness of a Cauchy surface S, there is a full classification result [Ms07,Sc99,Ba05,BB09], which characterizes the 3d Einstein manifolds in terms of structures from 2d and 3d hyperbolic geometry. More specifically, it identifies the moduli space GH Λ (M ) of maximal globally hyperbolic Einstein metrics, modulo isotopy, on a 3-manifold M = R × S with the bundle ML(S) of bounded measured geodesic laminations over the Teichmüller space T (S) of the Cauchy surface.
For each value of Λ, this identification is given by a Lorentzian counterpart of the grafting construction from 3d hyperbolic geometry. Moreover, the Lorentzian grafting construction is directly related to hyperbolic grafting via the Wick-rotation and rescaling theory developed by Benedetti and Bonsante [BB09]. It was also shown by the first author in [Me07] that these constructions admit a unified description via the ring of generalized complex numbers C Λ .

Symplectic structures and mapping class group actions The moduli spaces
GH Λ (M ) admit a symplectic structure induced by Goldman's symplectic structure [Go84,Go86] on the spaces of holonomies Hom(π 1 (S), G Λ )/G Λ . This is a natural Lorentzian generalization of (the imaginary part of) Goldman's symplectic structure on the moduli space of quasi-Fuchsian hyperbolic 3-manifolds or, more generally, the moduli space of hyperbolic end 3-manifolds. In fact, these structures are closely related via Wick-rotation and rescaling theory. More precisely, it was shown by the second author in joint work with Schlenker [SS18], that Wick rotations induce symplectic diffeomorphisms between the moduli spaces GH Λ (M ) and the moduli space of hyperbolic end 3-manifolds for all values of Λ.
In [MSc16] we showed that these symplectic structures can be given a unified description in terms of C Λ -valued shear coordinates associated with ideal triangulations of a punctured Cauchy surface. This description generalizes the Weil-Petersson symplectic structure on Teichmüller space T (S), and leads to a simple description of the mapping class group action in terms of 2d Whitehead moves. Interestingly, they involve C Λ -analytic continuations of classical dilogarithms, which suggests a close relation to the volumes of ideal hyperbolic tetrahedra.
Generalized ideal tetrahedra and their duals The role of hyperbolic structures in 3d Lorentzian geometry suggests that there should be a distinguished class of tetrahedra in 3d de Sitter, Minkowski and anti-de Sitter space with structural similarities to ideal tetrahedra, such as a simple description in terms of shape parameters.
Question 2: Are there analogues of generalized ideal tetrahedra in the spaces X Λ with similar geometric properties?
If the answer to this question is yes, one may generalize Question 1 to these tetrahedra and ask whether the geometry of these tetrahedra is simple enough to admit a volume formula in terms of simple quantities such as shape parameters and similar to the Milnor-Lobachevsky formula.
Question 3: Is there a simple volume formula for these tetrahedra in X Λ ?
In this article, we show that the answers to these three questions are positive. More specifically, we show that the analogues of generalized ideal tetrahedra in the Lorentzian spaces X Λ are the geodesic tetrahedra whose faces lie in lightlike geodesic planes.
We also find that they are related to Danciger's generalized ideal tetrahedra from [Da14] via the projective duality between the spaces X Λ and Y Λ (Theorem 4.19). This duality pairs points in one space with (totally) geodesic spacelike planes in the other. It admits a natural extension to the ideal boundary, which assigns points in ∂ ∞ Y Λ to lightlike geodesic planes in X Λ , and hence pairs generalized ideal tetrahedra in Y Λ and lightlike tetrahedra in X Λ .
We achieve this via a unified description of the spaces X Λ and Y Λ in terms of 2 × 2-matrices with entries in C Λ . This description leads to simple expressions for the geodesics, geodesic planes, metrics and isometry group actions on both spaces, and also for the ideal boundary of Y Λ . It allows us to parametrize both lightlike and ideal tetrahedra, to investigate their geometry in detail and to explicitly relate them.
In particular, we show in Proposition 4.2 that lightlike tetrahedra are also parameterized by pair of real parameters α, β ∈ R or, equivalently, by a generalized complex number z ∈ C Λ . These parameters have simple geometric interpretations, analogous to the ones for ideal tetrahedra. For example, the parameters |α|, |β|, |α + β| represent edge lengths of the lightlike tetrahedron, with opposite edges having equal length. Under duality, these lengths correspond to the dihedral angles of the dual ideal tetrahedron.

Volumes of generalized ideal tetrahedra and their duals
We also apply the explicit parametrization of lightlike and ideal tetrahedra to derive a unified formula for their volumes as a function of the parameters α, β. For a generalized ideal tetrahedron I ⊂ Y Λ the resulting formula in Theorem 5.1 is a generalization of the Milnor-Lobachevsky volume formula for ideal hyperbolic tetrahedra, involving Λ as a deformation parameter Here, Cl Λ is a generalized Clausen function. It coincides with the usual Clausen function for Λ > 0, the hyperbolic Clausen function for Λ < 0 and the integral of a logarithmic function for Λ = 0. The volume computation for a lightlike tetrahedron L ⊂ X Λ is more involved and is achieved in Theorem 5.2. The result is again a very simple expression involving Λ as a deformation parameter This can also be seen as the quotient of the hyperboloid of unit timelike vectors in R 2,0,2 by the antipodal map and thus inherits a Lorentzian metric of constant sectional curvature −1.
The group of orientation preserving isometries of AdS 3 is PO 0 (2, 2) ∼ = PSL(2, R) × PSL(2, R). It acts transitively on AdS 3 . The full group of isometries of AdS 3 is the group PO(2, 2) ⊂ PGL(2, R) × PGL(2, R). It is a double cover of PO 0 (2, 2) and is generated by PO 0 (2, 2) together with the isometry [( de Sitter space The Klein model of 3d de Sitter space can be defined similarly as the space of spacelike lines through the origin in R 1,0,3 It is the quotient of the hyperboloid of unit spacelike vectors in R 1,0,3 by the antipodal map and thus inherits a Lorentzian metric with sectional curvature +1. Note that with this definition dS 3 is orientable, but not time orientable. The group of orientation preserving isometries is PO 0 (1, 3) ∼ = PGL(2, C). It acts transitively on dS 3 . The full isometry group is the group PO(1, 3), generated by PO 0 (1, 3) and

Minkowski space
We also consider a Klein model of 3d Minkowski space. This is defined as the space of lines through the origin in R 1,1,2 transversal to the hyperplane x 2 = 0 As Mink 3 can be identified with the hyperplane H = {x ∈ R 1,1,2 | x 2 = 1}, it inherits a Lorentzian metric of sectional curvature 0. The group of orientation preserving isometries of Mink 3 is the Poincaré group in 3 dimensions PO 0 (1, 1, 2) = PO 0 (1, 2) ⋉ R 1,2 ∼ = PSL(2, R) ⋉ sl(2, R). It acts transitively on Mink 3 . The full isometry group of Mink 3 is the group PO(1, 1, 2) = PO(1, 2) ⋉ R 1,2 ∼ = PGL(2, R) ⋉ sl(2, R). It is generated by PO 0 (1, 1, 2) and the isometry In the following, we denote these three projective quadrics in RP 3 by X Λ , where Λ ∈ {−1, 0, 1} is the sectional curvature of the quadric Dual models The projective quadrics X Λ ⊂ RP 3 can also be characterized by their duality to three other projective quadrics Y Λ ⊂ RP 3 for Λ = −1, 0, 1. The latter are defined as the spaces of timelike lines through the origin in R 4 (2.5) with respect to the symmetric bilinear form y, y Λ = −y 2 1 + Λy 2 2 + y 2 3 + y 2 4 . (2.6) As Y Λ is the quotient of the set of timelike unit vectors for ·, · Λ by the antipodal map, it also inherits a constant curvature metric. For Λ = −1, this is again a Lorentzian metric of sectional curvature −1, and Y −1 is identical to X −1 = AdS 3 . For Λ = 1 one obtains a Riemannian metric of sectional curvature −1, and Y 1 is the Klein model of 3d hyperbolic space H 3 . For Λ = 0 one has a degenerate metric of signature (0, 0, 2), and Y 0 = H 2 × R is the product of 2d hyperbolic space with the real line, the so called co-Minkowski or half-pipe space, see for instance [Da11,Da13,BF18,FS16]. Thus, For each value of Λ, the isometry group of Y Λ agrees with the isometry group of X Λ . The isotropy groups, however, are different.

Projective duality
Geodesics lines and geodesic planes in X Λ and Y Λ are obtained as the intersections of X Λ and Y Λ with projective lines and with projective planes in RP 3 . The latter are the projections of 2d and 3d linear subspaces of R 4 to RP 3 . As usual, a geodesic in X Λ or Y Λ is called timelike, lightlike or spacelike if its tangent vectors are timelike, lightlike or spacelike. A geodesic plane in X Λ or Y Λ is called timelike, if it contains a timelike geodesic, spacelike if all of its geodesics are spacelike, and lightlike, if it contains a lightlike but no timelike geodesics.
The projective duality between X Λ and Y Λ is a bijection between points in one space and (totally) geodesic spacelike planes in the other. For Λ = 0, it is induced by orthogonality with respect to the ambient bilinear form ·, · Λ on R 4 from (2.6). To a point [x] ∈ X Λ it assigns the spacelike plane x * ⊂ Y Λ and to a point [y] ∈ Y Λ the spacelike plane y * ⊂ X Λ with where [x], [y] ∈ RP 3 denote the equivalence classes of x, y ∈ R 4 in RP 3 . This duality also induces a bijection between spacelike geodesics in X Λ and in Y Λ . It assigns to a spacelike geodesic g the intersection p * ∩ q * for any two points [p], [q] ∈ g. This intersection is a spacelike geodesic and independent of the choice of [p], [q] in g.
For Λ = 0 the ambient bilinear form ·, · Λ becomes degenerate and the duality cannot be directly interpreted in terms of orthogonality. One can, however, understand the duality for Λ = 0 as a limit of the other two cases via certain blow-up procedures, see [FS16]. The duality between points and geodesic planes in X 0 and Y 0 is then given by The geometric interpretation of the duality is the following. Half-pipe space Y 0 = H 2 × R can be identified with the set of spacelike affine planes in Minkowski space, whose normal vector is given by a point in H 2 and whose offset in the direction of the normal vector by a real parameter. The duality sends a point in Y 0 to the associated spacelike affine plane in Mink 3 . Conversely, a point x ∈ Mink 3 is dual to the graph of the map f : H 2 → R, n → x, n 1,1,2 , which defines a spacelike geodesic plane in half-pipe space.
The duality between points and geodesic planes extends to more general convex subsets X Λ and Y Λ . A set in RP 3 is called convex if it is the projection of a convex cone in R 4 that contains no non-trivial linear subspace. The projective dual of a convex set is then defined as the projection of the corresponding dual cone.
Convex sets in X Λ and Y Λ can then be defined as the restriction of convex set in RP 3 to each of these projective quadrics. The projective duality can thus be defined with respect to the duality between of convex cones in R 4 . We refer the reader to [FS16] for more details. Geometrically, the dual of a convex set can also be characterized as the set of spacelike geodesic planes which do not intersect the convex set.

Ideal points and lightlike planes
The spaces Y Λ admit a natural compactification in the projective quadric model. Namely, we can consider the closure of Y Λ in RP 3 , given by Its boundary in RP 3 is the projective lightcone This can be viewed as the asymptotic ideal boundary of Y Λ . It generalizes the description of the boundary ∂H 3 as the set of lightlike rays in R 1,0,3 . We will see in Section 3.6 that the ideal boundary ∂ ∞ Y Λ can be identified with RP 1 × RP 1 for Λ = −1, with CP 1 for Λ = 1 and with RP 1 × R for Λ = 0.
The projective duality (2.7) between points and spacelike planes in X Λ and Y Λ admits a natural extension to a duality between points [y] ∈ ∂ ∞ Y Λ and lightlike planes y * ⊂ X Λ , given again by (2.7).

3d geometries via generalized complex numbers
In this section, we give a unified description of the projective quadrics X Λ and Y Λ in terms of 2 × 2-matrices with entries in a commutative real algebra C Λ , whose multiplication depends on Λ. For the spaces Y Λ , this description was introduced in [Da11, Da13,Da14]. For the spaces X Λ similar descriptions were considered by the first author in [Me07,MS08] and by both authors in [MSc16]. In Sections 3.1 to 3.3 we summarize the results from [Da11, Da13,Da14] and [Me07,MS08,MSc16] and combine both descriptions in a common framework. In Section 3.4 we derive simple parametrizations of geodesics and geodesic planes in these spaces, which are applied in Section 3.5 to investigate the geometry of lightlike geodesic planes in X Λ . Section 3.6 summarizes Danciger's description of the ideal boundary from [Da11, Da13,Da14] and interprets his results in terms of Lorentzian geometry by duality with the spaces X Λ .

Generalized complex numbers
For any Λ ∈ R we define the ring of generalized complex numbers C Λ as the quotient of the polynomial ring in one variable ℓ by the ideal generated by ℓ 2 + Λ Elements in C Λ can thus be parametrized uniquely as z = x + ℓy, with real x, y and ℓ 2 = −Λ.
We write x = Re(z) and y = Im(z) and refer to x and y as the real and imaginary parts of z ∈ C Λ . We also define generalized complex conjugates by z = x − ℓy and the modulus |z| 2 = zz.
Note that, up to isomorphisms, C Λ only depends on the sign of Λ. We therefore restrict attention to Λ = 1, 0, −1. For Λ = 1, this yields the field C of complex numbers, and for Λ = 0, −1 the dual numbers and hyperbolic numbers, respectively. Note that for Λ = 0, −1 the ring C Λ is not a field, as there are nontrivial zero divisors. These are real multiples of ℓ for Λ = 0 and real multiples of 1 ± ℓ for Λ = −1. The group of units in C Λ is The real algebra C Λ becomes a 2d Banach algebra for all values of Λ when equipped with an appropriate norm. This allows one to consider power series and analytic functions on C Λ and on the algebras Mat(n, C Λ ) of n × n matrices with entries in C Λ . In particular, any real analytic function f : I → R on an open interval I ⊂ R can be extended to a unique analytic function F : Ω → C Λ on an appropriate open set I ⊂ Ω ⊂ C Λ , via The analytic continuation F satisfies a generalization of the Cauchy-Riemann equations on Ω Using the exponential map, we define generalized trigonometric functions c Λ , s Λ : R → R by They satisfy the following generalized trigonometric identities and their derivatives are given bẏ We also introduce the generalized tangent and cotangent functions and denote by t −1 Λ and ct −1 Λ their inverse functions with t −1 Λ (r) ∈ (− π 2 , π 2 ) and ct −1 Λ (r) ∈ (0, π) if Λ = 1.

A unified description of X Λ and Y Λ
To obtain a unified description of the quadrics X Λ and Y Λ , we consider the ring Mat(2, C Λ ) of 2 × 2-matrices with entries in C Λ . This allows one to identify the orientation preserving isometry groups of the projective quadrics X Λ and Y Λ with the projective linear group over C Λ , see [Da11] PGL + (2, More explicitly, the group isomorphisms between PGL + (2, C Λ ) and the orientation preserving isometry groups of X Λ and Y Λ are given by The description of the projective quadrics X Λ and Y Λ in terms of matrices with entries in C Λ is obtained from a pair of involutions •, † : Mat(2, C Λ ) → Mat(2, C Λ ), given by The sets of fixed points under these involutions are four-dimensional real vector spaces. The spaces X Λ and Y Λ can then be realized as their subsets of positive determinant matrices modulo rescaling Explicitly, the identification of the quadrics X Λ from (2.2), (2.3) and (2.4) with (3.5) is given by the linear map and the identification of the quadrics Y Λ from (2.5) with (3.6) by These maps identify R 4 with the set of matrices A, B ∈ Mat(2, C Λ ) satisfying A = A • and B = B † , respectively. With these identifications, the action of the group PGL + (2, C Λ ) on X Λ and Y Λ takes the form The full isometry group of X Λ and Y Λ is generated by PGL + (2, C Λ ) together with generalized complex conjugation.
The fact that PGL + (2, C Λ ) acts transitively on the spaces X Λ and Y Λ can then be seen as a consequence of the following lemma.

They can be chosen to satisfy A • = A and
Then the matrix and thus define an element in PGL + (2, C Λ ) with the desired properties. The proof for y ∈ Y Λ is analogous.

Tangent vectors
The tangent spaces T x X Λ and T y Y Λ can also be given a simple matrix description [Da11,MS08]. With Lemma 3.1, points in X Λ and Y Λ can be parametrized as x = A ⊲ 1 and y = B ⊲ 1, with A • = A and B † = B. The tangent spaces T x X Λ and T y Y Λ can then be parametrized by The induced actions of Stab(1, X Λ ) on x Λ and of Stab(1, Y Λ ) on y Λ are given by Note that x Λ and y Λ are endowed with invariant bilinear forms These are unique up to real rescaling and are transported to the tangent spaces at x = A⊲1 ∈ X Λ and at y = B ⊲ 1 ∈ Y Λ via the PGL + (2, C Λ )-action. More precisely, for X ∈ x Λ , Y ∈ y Λ and A, B ∈ PGL + (2, C Λ ) the metrics on the tangent spaces at A ⊲ 1 and B ⊲ 1 are defined by Note also that x Λ = ℓ sl(2, R) = ℓ Lie PSL(2, R) and that the bilinear form ·, · x Λ is proportional to the Killing form on sl(2, R). This shows that the tangent space T x X Λ with the metric from (3.11) and (3.12) is isometric to 3d Minkowski space for all values of Λ. We therefore call a matrix X ∈ x Λ timelike, lightlike or spacelike, if X, X < 0, X, X = 0 or X, X > 0, respectively. This is equivalent to the statement that the matrix exp(Im X) ∈ PSL(2, R) is elliptic, parabolic or hyperbolic, respectively.
These conventions allow one to refine Lemma 3.1 and to parametrize points x ∈ X Λ and y ∈ Y Λ in terms of exponentials of unit tangent vectors.

Lemma 3.2.
Any point x ∈ X Λ or y ∈ Y Λ can be expressed as By rescaling A and B we can achieve det(A) = det(B) = 1 and tr(A), tr(B) ≥ 0. Using the parametrizations (3.7), (3.8) and (3.10), we can express them as with a, b, c, d ≥ 0 and unit matrices X ∈ x Λ and Y ∈ y Λ . The condition det(A) = det(B) = 1 then read a 2 + Λσ(X)b 2 = 1 and c 2 + σ(Y )d 2 = 1. We can thus parametrize with θ ≥ 0 and θ < 2π for Λσ(X) < 0 or σ(Y ) < 0. A direct matrix computation using the definition of x Λ and y Λ in (3.10) then shows that these expressions for A, B coincide with exp( θ 2 X) and exp( θ 2 Y ).

Proposition 3.3. The subgroups of
for some a, b ∈ R and X U ∈ x Λ with a 2 −σ(X U )b 2 = 0. Furthermore, the condition U XU −1 = X for a spacelike or timelike vector X ∈ x Λ implies X U = X, up to rescaling. The proof for Y Λ is analogous.

Geodesics and geodesic planes
The description of the spaces X Λ and Y Λ in terms of generalized complex matrices allows one to parametrize their geodesics in terms of the matrix exponential. As the isometry group PGL + (2, C Λ ) acts transitively on these spaces, all geodesics are obtained from geodesics through 1 via the action of the isometry group. Geodesics through 1 are obtained by exponentiating matrices in x Λ and y Λ .

. Then for any unit tangent vector
and for any unit (3.14) Proof. As the expressions for x = A ⊲ 1 and y = B ⊲ 1 are obtained from the ones for x = 1 and y = 1 via the action of the isometry group, it is sufficient to consider the cases A = B = 1.
Geodesics in X Λ or Y Λ are obtained by projecting planes in R 4 . The identifications (3.7) and (3.8) of R 4 with the sets of hermitian matrices for • and † then shows that their image is contained in constant. The first two conditions follow directly from (3.13) and (3.14), the last conditions from the identitiesẋ which are obtained using (3.2) and (3.3).
Note that a geodesic x : R → X Λ or y : R → Y Λ is timelike, lightlike or spacelike, respectively, if the vectors X ∈ x Λ or Y ∈ y Λ from Proposition 3.4 are timelike, lightlike or spacelike. Equation (3.14) implies that a geodesic in Y Λ is closed if and only if it is timelike, which is possible only for Λ = −1. By equation (3.13) a geodesic in X Λ is closed if and only if it is spacelike and Λ = 1 or timelike and Λ = −1.
The parameter t ∈ R in (3.13) and (3.14) can be readily identified as the arc length parameter of a spacelike or timelike geodesic. By an abuse of notation, we write d(x, x ′ ) and d(y, y ′ ) for the arc length of a geodesic segment with endpoints x, x ′ ∈ X Λ or y, y ′ ∈ Y Λ . This segment is of course non-unique whenever there is a closed geodesic containing x, x ′ or y, y ′ . In this case, any identity stated for d(x, x ′ ) and d(y, y ′ ) is understood to hold for all such choices.
For Λ = 0 one also has Proof. Let x : R → X Λ be a spacelike or timelike geodesic parametrized as in (3.13) with The proof for points y, y ′ ∈ Y Λ is analogous.
For Λ = 0 and x, x ′ ∈ X Λ the geodesic with x(0) = A ⊲ 1 =x and x(t) =x ′ is given by where we used that X is a unit vector and that The explicit description of geodesics in Proposition 3.4 also allows one to compute their stabilizer groups.
Proposition 3.6. For a spacelike or timelike geodesic x : R → X Λ , parametrized as in (3.13), the subgroup of PGL + (2, C Λ ) stabilizing x(R) and preserving its orientation is given by Similarly, for a spacelike or timelike geodesic y : R → Y Λ , parametrized as in (3.14), the subgroup of PGL + (2, C Λ ) stabilizing y(R) and preserving its orientation, is given by Proof. As all geodesics are obtained from geodesics through 1 by the action of the isometry groups, we can assume Due to invariance of the bilinear form (3.11) on x Λ , because x is spacelike or timelike and because T preserves the orientation of x, we have U ⊲ X = X, and the claim follows from Proposition 3.6. The proof for geodesics in Y Λ is analogous.
The parameter θ in Proposition 3.6 describes a translation along the geodesics x : R → X Λ and y : R → Y Λ , which corresponds to a shift t → t+θ in the parametrization in Proposition 3.4. It is the arc length of the geodesic segment between a point on x or y and its image. The parameters a and b that define the elements U = a1 + b Im X ∈ Stab(X) and V = a1 + ℓbY ∈ Stab(Y ) via Proposition 3.3 describe generalized angles between geodesic planes through x and y. More precisely, these angles are given by for geodesics x : R → X Λ and y : R → Y Λ , respectively. In the first case, the parameter ϕ is the rapidity of a Lorentzian boost or the angle of a rotation around the geodesic x : R → X Λ . In hyperbolic geometry, which corresponds to Y Λ for Λ = 1, the parameter ϕ describes the angle between a plane containing the geodesic y and its image. We will use the nomenclature derived from hyperbolic geometry and call θ and φ the shearing and bending parameters along x and y, respectively.
The parametrization of geodesics in terms of the matrix exponential in Proposition 3.4 also gives rise to a parametrization of the geodesic planes in X Λ . As the isometry group PGL + (2, C Λ ) acts transitively on X Λ , the geodesic planes through x = A ⊲ 1 are obtained from the geodesic planes containing 1 by the action of isometries. Using the parametrization of the geodesics in Proposition 3.4 and the non-degenerate bilinear form on x Λ from (3.12), one then obtains for any linearly independent pair X 1 , X 2 ∈ N ⊥ . We call X a normal vector to P based at x.

Lightlike geodesic planes in X Λ
In this section, we derive some elementary properties of lightlike planes in X Λ that will be identified as the duals of certain statements about the ideal boundary ∂ ∞ Y Λ in the next section.
Recall that a geodesic plane in X Λ is called lightlike, if it contains a lightlike geodesic, but no timelike geodesics. This is equivalent to its normal vector from Proposition 3.7 being lightlike.
It follows directly that two distinct lightlike planes in X Λ that intersect always intersect in a spacelike geodesic. Conversely, for any spacelike geodesic in X Λ , there is a unique pair of lightlike planes that intersect in this geodesic. In the following we often need an explicit parametrization of his intersection geodesic.
Lemma 3.8. If two distinct lightlike planes P 1 , P 2 in X Λ intersect, then for any point x ∈ P 1 ∩P 2 , there is an isometry that sends x to 1, their intersection to the spacelike geodesic and their normal vectors in x to Proof. By applying isometries, we can assume x = 1. The action of PSL(2, R) Λ ⊂ Stab(1, X Λ ) on x Λ = ℓsl(2, R) then coincides with the action of PSL(2, R) on Minkowski space, and the action on the normal vectors of these planes with the PSL(2, R)-action on the set of lightlike rays in 3d Minkowski space. This can be identified with the PSL(2, R)-action on ∂H 2 , which is known to be 3-transitive. Hence, there is an isometry in PSL(2, R) that sends the normal vectors of the planes to (3.16). Then we have N ⊥ 1 ∩ N ⊥ 2 = RX with X unique up to real rescaling and given by (3.15).
An analogous parametrization exists for triples of lightlike planes that intersect in a common point. In this case, the 3-transitivity of the PSL(2, R)-action on ∂H 2 implies uniqueness up to permutations.
Lemma 3.9. If three distinct lightlike planes in X Λ intersect in a common point x, then there is an isometry that sends x to 1 and their normal vectors in x to This isometry is unique up to isometries permuting the three planes.
Proof. By applying isometries, we can assume that the intersection point of these planes is 1.
The action of PSL(2, R) Λ ⊂ Stab(1, X Λ ) on x Λ = ℓsl(2, R) then coincides with the action of PSL(2, R) on Minkowski space, and the action on the normal vectors of these planes on the PSL(2, R)-action on the set of lightlike rays in 3d Minkowski space. This can be identified with the PSL(2, R)-action on ∂H 2 , which is known to be 3-transitive.

The ideal boundary of Y Λ
Under the duality between X Λ and Y Λ from Sections 2.2 and 2.3, lightlike geodesic planes in X Λ are dual to points on the ideal boundary of Y Λ . We thus summarize the properties of the ideal boundary ∂ ∞ Y Λ from [Da11, Da13,Da14]. To make the paper self-contained, and because details will be needed in the following, we also include proofs, adapted from [Da14]. We also point out their duality with results on lightlike planes and show that in some cases this provides an additional geometric interpretation.
In the matrix parametrization of Y Λ , the ideal boundary ∂ ∞ Y Λ becomes the set of rank 1 matrices modulo real rescaling This identifies ∂ ∞ Y Λ with the generalized complex projective line (3.17) It should be mentioned, however, that the topology induced by this identification does not coincide with the one induced by RP 3 for Λ = 0.
For Λ = 1 this holds by definition. For Λ = −1 the identification is given by the map and for Λ = 0 by the map The action (3.9) of PGL + (2, C Λ ) on Y Λ extends to a PGL + (2, Under the identification of ∂ ∞ Y Λ with C Λ P 1 , this action becomes the standard action of PGL + (2, C Λ ) on C Λ P 1 via projective transformations Note that for Λ = 1 this coincides with the action of Möbius transformations on the Riemann sphere CP 1 = ∂ ∞ H 3 . In this case, the condition vv † = 0 in (3.17) simply states that v = 0. By rescaling representatives of points in CP 1 such that their second entry is 1, one obtains: (3.20) In the following, the action of PGL + (2, C Λ ) on C Λ P 1 is often described with respect to three which correspond to the points ∞, 0, 1 ∈ CP 1 = C ∪ {∞} for Λ = 1. We also write v 1 = ∞, v 2 = 0 and v 3 = 1 to denote the points v 1 , v 2 , v 3 ∈ C Λ P 1 in (3.21) for Λ = 1.
The subgroup of PGL + (2, C Λ ) that permutes v 1 , v 2 , v 3 is the group of order six generated by the classes of It permutes the points v 1 , v 2 , v 3 according to Spacelike geodesics in Y Λ have two endpoints in ∂ ∞ Y Λ , obtained from their parametrization (3.14) as the limits t → ±∞. These endpoints are the duals of the two unique lightlike planes that intersect in the dual spacelike geodesic in X Λ . The action of the isometry group PGL + (2, C Λ ) on ∂ ∞ Y Λ allows one to map these endpoints to fixed reference points, namely the points v 1 v † 1 and v 2 v † 2 for v 1 , v 2 given in (3.21). This is dual to the statement in Lemma 3.8 that by acting with isometries, one can transform the normal vectors of the lightlike planes into (3.16).
Lemma 3.10. Let y + , y − ∈ ∂ ∞ Y Λ be endpoints of a spacelike geodesic in Y Λ . Then there is an isometry B ∈ PGL + (2, C Λ ) such that Proof. Using (3.14) we can parametrize any spacelike geodesic y in Y Λ as with A ∈ PGL + (2, C Λ ) and a spacelike unit matrix Y ∈ y Λ . Any normalized spacelike matrix in y Λ can be written as The endpoints of the geodesic y are then represented by the matrices Using the identification of the boundary ∂ ∞ Y Λ with the complex projective line C Λ P 1 from (3.17), one can parametrize the endpoints as A direct computation then shows that (3.23) is satisfied for the projective matrices for a = −1 and a = 1, respectively.
It is shown in [Da14, Proposition 2], see also the remark after [Da14, Proposition 3], that this result extends to triples of points in ∂ ∞ Y Λ , provided that they are contained in a common spacelike plane. In this case one can take the three reference points v 1 = ∞, v 2 = 0, v 3 = 1 in (3.21).
Proof. By Lemma 3.10, one can assume that y 1 = v 1 v † 1 and y 2 = v 2 v † 2 . As y 3 is connected to y 1 and y 2 by spacelike geodesics, by (3.24) there are isometries A i ∈ PGL + (2, C Λ ) and vectors Y i ∈ y Λ for i = 1, 2 such that (3.25) Using the identification of ∂ ∞ Y Λ with C Λ P 1 , we can parametrize y 3 = w 3 w † 3 with w 3 ∈ C Λ P 1 . The condition | det(A i )| 2 > 0 together with (3.25) then implies that both entries of w 3 are units in C Λ , and by rescaling it, we can achieve that its second entry is 1 and its first entry is a unit z ∈ C × Λ , as in (3.20). The condition that y 1 , y 2 , y 3 lie on a common spacelike plane implies |z| 2 > 0 and that Note that for Λ = 1 Proposition 3.11 is the well-known 3-transitivity of the action of PGL(2, C) on the Riemann sphere CP 1 . However, for Λ = 0 and Λ = −1 the action of PGL + (2, C Λ ) on C Λ P 1 is in general not 3-transitive, even if one allows for permutations of the three points. In particular, the proof of Proposition 3.11 shows that an element of PGL + (2, C Λ ) that stabilizes or exchanges v 1 = ∞ and v 2 = 0 cannot map a general point v ∈ C Λ P 1 to v 3 = 1.
Proposition 3.11 can be viewed as the dual of Lemma 3.9. The dual of the spacelike plane in Y Λ containing the points y 1 , y 2 , y 3 ∈ ∂ ∞ Y Λ is a point in X Λ that lies on the dual planes to y 1 , y 2 , y 3 and hence in their intersection. The normal vectors of the lightlike planes in Lemma 3.9 are thus given by the points y 1 , y 2 , y 3 in Proposition 3.11.
Given four distinct points in ∂ ∞ Y Λ such that any three of them lie on a common spacelike plane, one can apply an isometry to send three of them to the points v 1 v † 1 , v 2 v † 2 , v 3 v † 3 , as in Proposition 3.11. As the fourth point is on a spacelike plane through v 1 v † 1 and v 2 v † 2 , it is represented by an element v 4 ∈ C Λ P 1 whose entries are units in C Λ by the proof of Proposition 3.11. Rescaling this element, one obtains Hence, up to isometries, the four points are characterized uniquely, by an element in C × Λ \ {1}, the shape parameter introduced in [Da14, Section 3.1], which can be viewed as a generalized cross-ratio.
Note that the orbit of the cross-ratio z = cr(∞, 0, 1, z) under the action of the subgroup (3.22) of PGL + (2, C Λ ) permuting v 1 , v 2 , v 3 is given by These are the familiar expressions for the transformation of a cross-ratio in CP 1 under the subgroup of Möbius transformations that permute ∞, 0, 1. Indeed, for Λ = 1, any point y ∈ CP 1 can be parametrized as in (3.26) and the cross-ratio coincides with the usual cross-ratio on CP 1 defined by This is a consequence of formula (3.20) for the PGL(2, C)-action on CP 1 and the invariance of the cross-ratio under isometries. Note, however, that for Λ = 0 and Λ = −1 the cross-ratio cannot defined globally by (3.27), since z 3 − z 2 or z 4 − z 1 need not be units in C Λ .

Lightlike and ideal tetrahedra
In this section we investigate the geometric properties of tetrahedra with lightlike faces in X Λ and their duals in Y Λ . We then show that the latter are precisely the generalized ideal tetrahedra introduced by Danciger in [Da14].
In the following, we denote by x i and y i the vertices of tetrahedra in X Λ and Y Λ , respectively, and by x ij or y ij the geodesic through the vertices x i , x j or y i , y j . In both cases, we write e ij for the edge of the tetrahedron through the vertices x i , x j or y i , y j , the geodesic segment of x ij or y ij that is part of the tetrahedron.

Lightlike tetrahedra
We start by considering tetrahedra in X Λ whose faces are all contained in lightlike planes. We will also require that these tetrahedra are (i) convex, i. e. obtained as projections of convex cones in R 4 , (ii) non-degenerate, i. e. not contained in a single geodesic plane, and (iii) that their internal geodesics at each vertex, the geodesics that intersect the interior of the tetrahedron, are all spacelike. The last condition is relevant mainly for Λ = 1.
Definition 4.1. A lightlike tetrahedron in X Λ is a non-degenerate convex geodesic 3-simplex in X Λ with lightlike faces such that all internal geodesics starting at its vertices are spacelike.
Note that this definition implies with Lemma 3.8 that all edges of a lightlike tetrahedron are spacelike geodesic segments. The two faces containing an edge of a lightlike tetrahedron then lie on the two unique lightlike planes that intersect along this spacelike geodesic. Each vertex is the unique intersection point of the three lightlike planes containing the adjacent faces.
By applying isometries we can relate any lightlike tetrahedron to one in standard position. By this, we mean a lightlike tetrahedron with one of its vertices at x = 1 and the three lightlike normal vectors at this vertex given as in Lemma 3.9. The vertices of the lightlike tetrahedron can then characterized uniquely by its fourth lightlike normal vector, up to rescaling, and hence by a pair of real parameters.
Proof. Let A i ∈ PGL + (2, C Λ ) an isometry with A • i = A i and A i ⊲ 1 = x i , as in Lemma 3.1. Denote by A i ⊲ N ij the normal vector of the face f j at the vertex x i from Proposition 3.7. Then by Lemma 3.9 we can assume that x 4 = 1 and Denote by x ij a spacelike geodesic through x i and x j with x ij (0) = x i . Then, by Proposition 3.4, the geodesic x ij can be parametrized as where X ij ∈ x Λ is a spacelike unit vector, unique up to a sign, that is orthogonal to both N ik and N il with respect to the bilinear form (3.11) for distinct i, j, k, l ∈ {1, 2, 3, 4}. By Lemma 3.2 the remaining vertices can be expressed as where i = 1, 2, 3, α i ∈ R.
To obtain the relation between these parameters we now compute the remaining vectors X ij and N ij . For the former, note that (4.3) and (4.4) imply where t ij ∈ R is given by the condition x j = x ij (t ij ). Using this identity with expression (3.13) for the exponential and the identities Im(X 4i ) Im(X 4j ) Im(X 4i ) = −2 Im(X 4i ) − Im(X 4j ), (4.6) which follow from (4.5), one obtains for all distinct i, j ∈ {1, 2, 3}. Using again relation (4.6), expression (4.4) for the matrices A i and the parametrization (3.13) of the matrix exponential implies for all distinct i, j ∈ {1, 2, 3, 4} The matrices N ij ∈ x Λ can then be computed from the condition that N ij is orthogonal to X ik for all distinct i, j, k, and normalized such that N ij , X ij x Λ = −1. Note that this last condition is also satisfied by the matrices N 4i and X 4i from (4.2) and (4.5). A direct computation with expression (3.11) for the bilinear form on x Λ shows that X 4i , X 4i x Λ = 1 and X 4i , X 4j x Λ = −1 for distinct i, j ∈ {1, 2, 3}. Equations (4.4) and (4.8) imply X 4i = −X i4 . Together with (4.7), these identities imply that for all distinct i, j, k ∈ {1, 2, 3}. A short computation using (4.5) and (3.11) finally shows that they are all lightlike if and only if α 1 + α 2 + α 3 = 0 (modπ for Λ = 1).
By applying isometries to a lightlike tetrahedron in X Λ , we may assume that its vertices are in the standard position given in Proposition 4.2. Then, the group of isometries which fixes the vertex x 4 = 1 and permutes the lightlike planes intersecting at this vertex is precisely the subgroup of PGL + (2, C Λ ) that permutes the reference points v 1 = ∞, v 2 = 0, v 3 = 1 ∈ C Λ P 1 in (3.21). Using these symmetries we may always choose two of the parameters α, β, γ in Proposition 4.2 to be positive. For Λ = 1, due to periodicity of spacelike geodesics, we can further choose 0 < |α|, |β|, |γ| < π. The description of X Λ as a projective quadric in RP 3 then shows that the vertices in Proposition 4.2 always define a lightlike tetrahedron. It also gives rise to an explicit parametrization of lightlike tetrahedra.
Note that this imposes restrictions on the possible values of α, β, γ, but does not determine γ uniquely as a function of α, β. The condition that the internal geodesics starting at each vertex are spacelike imposes further restrictions, namely and these are satisfied for all a l as in (4.12) if and only if α 1 + α 2 + α 3 = 0.
A global parametrization of the coefficients a l in (4.12) can then be obtained via where A, B and r satisfy the conditions in (4.10) and λ ∈ R + . By comparison with (4.10) we find rX(A, B) .
Note that the formulas for Λ = 0 in (4.13) are obtained from the ones for Λ = 0 by expanding the latter as a power series in α, β and Λ. Expression (3.1) for the generalized trigonometric functions in terms of the exponential map extends to general Λ = −ℓ 2 ∈ R and defines s Λ and c Λ as power series in Λ. One can thus expand the left-and right-hand side of the equations for Λ = 0 in (4.13) as a power series in Λ. To zero-th order in Λ these equations are satisfied trivially, and at first order one obtains the equations for Λ = 0.  Remark 4.8. Corollary 4.7 shows that for Λ = −1, 0 a lightlike tetrahedron L is the intersection of the past of the geodesic containing one of the two longest edges with the future of the geodesic containing the other. For Λ = 1, the space X 1 = dS 3 is not time orientable, but it still holds that any point in L is connected to each of two longest edges by a timelike geodesic segment in L that ends on a face through the opposite edge.
Instead of using geodesics through the midpoints of its edges, we can also characterize the geometry of a lightlike tetrahedron in terms of lightlike geodesics. For this, we consider lightlike geodesics in the geodesic planes defined by its faces and through one of its vertices. The longest edges of a lightlike tetrahedron are then distinguished by the fact that such lightlike geodesics through their endpoints intersect the opposite face.
In particular, n ij intersects the tetrahedron L outside x i if and only if x ij contains one of the longest edges of L.
Proof. If n ij and x kl intersect, then x i , x k , x l lie on a common lightlike plane. Since n ij lies on the lightlike plane opposite x j , the only edge geodesic containing x j which intersects n ij is x ij , with the intersection point given by x i . Furthermore, the edge geodesics x kl opposite to x j (that is, with k, l = j) intersect n ij at a single point. This is given by x i , if k = i or l = i. For k, l = i, the intersection point can be computed solving for θ ij and t kl , where N ij and X kl are given by (4.2), (4.9), and (4.5), (4.7).
Corollary 4.9 defines canonical projections of each vertex x i on each of the geodesics x kl containing its opposite edge e kl . We will call these null projections in the following. Thus, given a vertex x i , we define the point π kl (x i ) on the geodesic x kl as the unique intersection point between x kl and the lightlike geodesic n ij , as shown in Figure 1. It should be emphasized that π kl (x i ) may lie outside of the corresponding edge e kl .
Each geodesic x kl contains exactly two such projections, namely π kl (x i ) and π kl (x j ) for the two vertices x i and x j opposite x kl . For each edge geodesic x ij , this defines two geodesic planes that intersect in x ij , the planes through x i , x j , π kl (x i ) and through x i , x j , π kl (x j ), as shown in Figure 4.1. We call them the internal planes of the lightlike tetrahedron at the edge e ij . The angles between these planes are given by the ratios of the generalized sine functions of the edge lengths.
Proposition 4.10. Let L be a lightlike tetrahedron in X Λ with vertices x i as in Proposition 4.2. Then the Lorentzian angle ϕ ij between the internal planes at the edge e ij is given by where |z ij | = |z ji | and Proof. Denote by x ij and x kl the geodesics through x i , x j and through x k , x l , parametrized as in (4.3). For any point x kl (t) on the geodesic x kl we can parametrize the plane through where X ij ∈ x Λ is a unit vector parameterizing the geodesic x ij as in (4.3) and X ij,kl (t) ∈ x Λ is the unit vector parameterizing the geodesic x ij,kl through x i and x kl (t) via x ij,kl (s) = A i ⊲ exp(sX ij,kl (t)).
These vectors can be computed directly as the normalized trace-free parts of A −1 i ⊲ x j and A −1 i ⊲ x kl (t), respectively. We can then compute the normal vector A i ⊲ N ij,kl (t) of P ij,kl (t) at x i from the conditions where X ij are the matrices from (4.5) and (4.7). This yields for all distinct i, j, k ∈ {1, 2, 3} s Λ (αj ) and N ik and N i4 given by (4.2) and (4.9). Corollary 4.9 gives the null projections of x i and x j on the opposite edge geodesic x 4k and the null projections of x 4 and x k on x ij In particular, the normal vectors at x i of the plane P ij,4k (−α i ) through x i , x j , π 4k (x i ) and of the plane P ij,4k (−α j ) through x i , x j , π 4k (x j ) are given by x kl Figure 2: Internal planes of a lightlike tetrahedron at the edge e ij .
Similarly, the normal vectors at x 4 to the planes P 4k,ij (−α i ) through x 4 , x k , π ij (x 4 ) and P ij,4k (−α j ) through x 4 , x k , π ij (x i ) are given by In both cases, we find that the Lorentzian angle between the two planes is given by The claim then follows by setting α 1 = α, α 2 = β, α 3 = γ = −α − β.
Proposition 4.10 associates to each edge of a lightlike tetrahedron L a Lorentzian angle that is given by the ratios of generalized sine functions of the edge lengths α, β, γ. Combining these with the corresponding edge lengths, we may define a generalized complex parameters z ij = z ji ∈ C × x il x jl x ik x jk where A i ∈ PGL + (2, C Λ ) with A i ⊲ 1 = x i , the tangent vector X ij of x ij is given by (4.5), (4.6), the shape parameter z ij = z ji ∈ C × Λ by (4.14) and σ ij = σ ji ∈ {±1} by Proof. This follows from the expressions (4.2), (4.5), (4.7), (4.9) for the normal and tangent vectors derived in the proof of Proposition 4.2.
By Proposition 3.6 and equation (4.3), we can parametrize T ij as The requirement that T ij maps x i to x j determines the parameter θ ij as follows. Using equation (4.4), we can rewrite this requirement as which is equivalent to By an explicit computation of the matrices R ij , one finds that this is satisfied if and only if with θ ij = θ ji . In the case Λ = 1, this holds up to multiples of π.
To investigate the action of T ij on the normal vectors of the adjacent faces, denote by A i ⊲N ik the lightlike vector at x i normal to a face f k adjacent to x ij , with the normalization N ik , X ik x Λ = −1, and with distinct i, j, k ∈ {1, 2, 3, 4}. Then T ij stabilizes the null planes intersecting along the geodesic x ij and preserves its orientation if and only if for some σ ij ∈ R × . With the condition |σ ij | = 1 this is equivalent to and, again by a direct computation, one finds with a ij = a ji , b ij = b ji and σ ij = σ ji . Factoring out −σ ij (a ij − b ij )e −ℓθij/2 and inserting α 1 = α, α 2 = β and α 3 = γ, we obtain the expressions in the proposition.

Ideal tetrahedra
Corollary 4.5 shows that the edge lengths of a lightlike tetrahedron in X Λ play a similar role to the dihedral angles of an ideal hyperbolic tetrahedron: up to isometries, they determine the lightlike tetrahedron completely. Indeed, the duality between lightlike planes in X Λ and points on the ideal boundary ∂ ∞ Y Λ suggests that lightlike tetrahedra should be dual to tetrahedra in Y Λ whose vertices are points in ∂ ∞ Y Λ , pairwise connected by spacelike geodesics.
Such tetrahedra are precisely the generalized ideal tetrahedra introduced and investigated by Danciger in [Da11,Da14], up to the fact that we exclude the degenerate ones. In this section we review the results on generalized ideal tetrahedra in [Da11,Da14] that are needed in the following and relate them to the corresponding statements about lightlike tetrahedra. We then show that lightlike and ideal tetrahedra are dual under the projective duality from Sections 2.2 and 2.3.
Definition 4.12. An ideal tetrahedron in Y Λ is a non-degenerate convex geodesic 3-simplex whose vertices are points in ∂ ∞ Y Λ and whose faces lie on spacelike geodesic planes.
As all vertices of an ideal tetrahedron are contained in ∂ ∞ Y Λ and all faces lie on spacelike geodesic planes, the action of the isometry group PGL + (2, C Λ ) on ∂ ∞ Y Λ allows one to map three vertices of an ideal tetrahedron to fixed reference points in ∂ ∞ Y Λ , as in Proposition 3.11. It is shown in [Da14, Proposition 3] that the remaining vertex is then parametrized by the cross-ratio from Definition 3.12. Alternatively, this vertex is given by two real parameters α, β, which can be viewed as generalized dihedral angles.
Proof. As y 1 , y 2 , y 3 ∈ ∂ ∞ Y Λ lie on a spacelike geodesic plane, by Proposition 3.11 there is a unique isometry B ∈ PGL + (2, C Λ ) with up to a permutation of the vertices. The remaining vertex is then given by As all faces lie on spacelike geodesic planes, by the proof of Proposition 4.13 one has 1 − z ∈ C × Λ \ {1}. In particular, there exists r 1 , r 2 , β, γ ∈ R × such that Eliminating the parameters r 1 , r 2 yields and therefore Equation (4.15) relates the parameters α, β that parametrize an ideal tetrahedron in Proposition 4.13 to the generalized cross-ratio of its vertices from Definition 3.12. By considering also the images of the cross-ratio under the action of the subgroup (3.22) that permutes the vertices B ⊲ y 1 , B ⊲ y 2 and B ⊲ y 3 , one obtains all the cross-ratios of a generalized ideal tetrahedron [Da14, Section 3.1].
Corollary 4.14. The cross-ratios of vertices of the ideal tetrahedron in Proposition 4.13 are given by and their multiplicative inverses.
As for lightlike tetrahedra, using the symmetries (3.22), we may always choose two of the parameters α, β, γ in Proposition 4.13 to be positive. For Λ = 1, due to periodicity, we can further choose 0 < |α|, |β|, |γ| < π. We then obtain the following parametrization of an ideal tetrahedron that is the counterpart of Proposition 4.4.
Proof. Let y 1 , y 2 , y 3 , y 4 ∈ Y Λ be given as in Proposition 4.13, and choose lifts y ′ 1 , y ′ 2 , y ′ 3 , y ′ 4 ∈ R 4 ⊂ Mat(2, C Λ ). Up to isometries (possibly reversing orientation), we can choose We consider the convex cone in R 4 spanned by lifts of the vertices y i ∈ Y Λ ⊂ RP 3 to vectors y ′ i ∈ R 4 . This takes the form This projects to a convex tetrahedron in Y Λ if and only if y ′ , y ′ Λ < 0 for all y ′ ∈ I ′ , and this condition is satisfied for all α, β > 0 and γ = −α − β.
Any point y ∈ Y Λ that is connected to y 1 by a spacelike geodesic can be parametrized as The points on ∂ ∞ Y Λ that are connected to y 1 by a spacelike geodesic are obtained from (4.18) as the limit t → 0. Note also that for all z ∈ C Λ , the map g z : R → Y Λ , t → y(e s , z) is a spacelike geodesic in Y Λ , parametrized by arc length and with g z (∞) = y 1 . This follows because g z parametrizes the intersection of the image of a plane in R 4 under the map (3.8) with the set of matrices of unit determinant and because d(g z (s), g z (s ′ )) = |s − s ′ | by Proposition 3.5. Hence, we can view the sets for fixed t > 0 as generalized horocycles based at y 1 ∈ ∂ ∞ Y Λ . For Λ = 1, they coincide with the usual horocycles in H 3 .
The edge geodesic through y 1 and y j is obtained by setting B k = 0 for k / ∈ {1, j} in (4.17). By comparing the resulting expression with (4.18), one finds that this geodesic intersects each horocycle H t (y 1 ) in a unique point y(t, z j ) with z j given by More generally, a comparison of the parametrizations (4.17) and (4.18) shows that any geodesic through y 1 that intersects the ideal tetrahedron I intersects each horocycle H t (y 1 ) in a unique point y(t, z) with z given by The intersection point of the geodesic g r,θ : R → Y Λ , s → y(e s , z(r, θ)) with the face opposite the vertex y 1 is obtained by setting B 1 = 0 in (4.17). Parameterizing z as in (4.19) and comparing with (4.18), we find that this intersection point is given by Inserting formula (4.19) into the parametrization (4.18) then completes the proof. The parameters α, β, γ in Propositions 4.13 and 4.15 also have a geometrical interpretation, namely as generalized dihedral angles at the edges of the ideal tetrahedron. Here, our convention for the dihedral angles uses one exterior angle, namely the biggest dihedral angle α + β, and two interior angles, α and β. For Λ = 1 these are the usual dihedral angles between the faces of an ideal hyperbolic tetrahedron, up to the fact that one of them is external and given by π − θ, where θ the usual interior dihedral angle. For Λ = −1 they give a Lorentzian angle between its faces, and for Λ = 0 they are the length of the unique translation along the degenerate direction that relates adjacent faces. Using the global parametrization in Proposition 4.15, we obtain the analogue of Corollary 4.5.
Corollary 4.16. An ideal tetrahedron I is determined up to isometries by its generalized dihedral angles. If I is parametrized as in Proposition 4.15, its dihedral angles are α, β and α + β, with opposite edges having equal dihedral angles.
Proposition 4.15 and Corollary 4.16 show that the dihedral angles of an ideal tetrahedra play an analogous role to the edge lengths of lightlike tetrahedra. It is also possible to give a geometric interpretation for the ratios of their generalized sine functions as shearing distances along edges.
We define the shearing distance along an edge e ij as the signed arc length ϕ ij between the orthogonal projections of y k and y l on e ij , for all distinct i, j, k, l ∈ {1, 2, 3, 4}. The sign of ϕ ij is taken positive (resp. negative) if the orientations of e ij induced (i) by the face opposite y k and (ii) by moving from π ij (y k ) to π ij (y l ) agree (resp. disagree), see Figure 5.
Proposition 4.17. Let I ⊂ Y Λ be an ideal tetrahedron with vertices y 1 , y 2 , y 3 , y 4 parametrized as in Proposition 4.13. Then the shearing distance ϕ ij at the edge e ij is given by where |z ij | = |z ji | and Proof. Denote by B ij the unique isometry with B ij ⊲ ∞ = y i , B ij ⊲ 0 = y j and B ij ⊲ 1 = y k from Proposition 3.11, where (y i , y j , y k ) is positively ordered with respect to the orientation of I. Then the orthogonal projection of y k on e ij is given by π ij (y k ) = B −1 ij ⊲ 1 and the orthogonal projection of the remaining vertex y l by π ij (y l ) = B −1 ji ⊲ 1. Suppose B ij is normalized with | det(B ij )| = 1. Then by Proposition 3.5 the shearing distance ϕ ij satisfies As in the case of lightlike tetrahedra, the cross-ratios or shape parameters of a generalized ideal tetrahedron can also be characterized in terms of its symmetries.
The other parameters z ij are obtained by computing the isometries B −1 kl • B ij , for instance The claim then follows from the identity α + β + γ = 0.
Corollary 4.16 and Proposition 4.17 show that the cross-ratios of an ideal tetrahedron from Corollary 4.14 have a direct geometric interpretation that generalizes the one of ideal tetrahedra in H 3 . Their arguments are generalized dihedral angles between faces, and their moduli shearing distance along edges.
They are the counterparts of Corollary 4.5 and Proposition 4.10 for lightlike tetrahedra in X Λ , which state that the arguments of their shape parameters determine the edge lengths and their moduli the Lorentzian angle between the internal planes of a lightlike tetrahedron. Proposition 4.18, which characterizes the cross-ratios of an ideal tetrahedron in terms of its symmetries, is the counterpart of Proposition 4.11 for lightlike tetrahedra.
We have seen in Proposition 4.4 that given parameters α, β, γ, satisfying α + β + γ = 0, there exists a lightlike tetrahedron with edge lengths |α|, |β|, |γ|, unique up to isometries. Similarly, under the same assumptions, Proposition 4.13 proves the existence of a generalized ideal tetrahedron with generalized dihedral angles |α|, |β|, |γ|, again unique up to isometries. The following theorem gives a geometric interpretation for this correspondence between lightlike and ideal tetrahedra in terms of the projective duality of Sections 2.2 and 2.3.
Note, however, that this correspondence is not given by the duality between convex sets in X Λ and Y Λ from [FS16] discussed in Section 2.2. As explained in Section 2.2, the dual of a convex set in X Λ or Y Λ can be characterized as the set of spacelike geodesic planes that do not intersect the convex set. Here, instead, we characterize lightlike tetrahedra in X Λ as the sets of spacelike geodesic planes in Y Λ that do intersect ideal tetrahedra in two specified pairs of opposite edges. Conversely, ideal tetrahedra in Y Λ correspond to spacelike geodesic planes in X Λ that intersect a lightlike tetrahedron in all pairs of opposite edges except its longest edge pair.
Theorem 4.19. The projective duality from Section 2.2 identifies a lightlike tetrahedron in X Λ with the set of spacelike planes in Y Λ that intersect an ideal tetrahedron along two pairs of opposite edges. It identifies an ideal tetrahedron in Y Λ with the set of spacelike planes in X Λ that intersect a lightlike tetrahedron along its shortest edges.
Proof. This follows from the parameterization of lightlike tetrahedra and ideal tetrahedra as projections of the convex cones with the vertices x ′ i and y ′ j given by (4.11) and (4.16). By assumption, we have α + β + γ = 0 with α, β > 0 and α + β < π for Λ = 1. This implies x ′ i , y ′ j = 0 for i = j and (4.20) In particular, the spacelike plane in Y Λ dual to any point in the lightlike tetrahedron must intersect the ideal tetrahedron: given any x ′ ∈ L ′ there exists y ′ ∈ I ′ such that x ′ , y ′ = 0. Such spacelike planes, however, cannot not intersect the pair of edges e 12 and e 34 in I ′ : If For all other combinations of i and j, there are b i , b j ≥ 0 and b i + b j = 0 for which x ′ , y ′ = 0. By Proposition 4.4 and Corollary 4.5 the edges e 12 and e 34 are the longest edges of the lightlike tetrahedron. The proof of the second statement is analogous.
Although this correspondence is not the duality of convex sets from Section 2.2, it still identities faces and vertices of a lightlike tetrahedron with faces and vertices of a lightlike tetrahedron. Geodesics through two vertices or on two faces of a lightlike tetrahedron are identified with geodesics on the two dual faces or though the two dual vertices, respectively. In this sense, lightlike tetrahedra in X Λ and ideal tetrahedra in Y Λ are projectively dual.

Volumes of lightlike and ideal tetrahedra
In this section we derive formulas for the volumes of lightlike tetrahedra in X Λ and of generalized ideal tetrahedra in Y Λ as functions of their edge lengths and dihedral angles, respectively. These formulas are obtained by direct integration of the volume forms on X Λ and on Y Λ , defined here uniquely up to global rescaling as the PGL + (2, C Λ )-invariant 3-forms on each space.

Volumes of ideal tetrahedra
We start with the computation of volumes of generalized ideal tetrahedra in Y Λ . This is technically much simpler to compute and serves as a guide for the computation of the lightlike volume below. For Λ = 1, it includes the Milnor-Lobachevsky formula [Mi82], which gives the volume of a hyperbolic ideal tetrahedron I as vol(I) = 1 2 Cl(2α) + Cl(2β) + Cl(2γ) = l(α) + l(β) + l(γ). (5.1) Here α, β and γ = π − (α + β) are the interior dihedral angles of the tetrahedron, Cl : R → R is the Clausen function of order two and l : R → R the closely related Lobachevsky function.
We will now show that the volume formulas for generalized ideal tetrahedra I ⊂ Y Λ can be computed for all values of Λ simultaneously and are simple generalizations of formula (5.1), in which Λ appears as a deformation parameter.
The standard computation of the volume for an ideal hyperbolic tetrahedron, due to Milnor [Mi82] and based on the work by Lobachevsky, proceeds by subdividing the ideal tetrahedron in three sub-tetrahedra with a higher degree of symmetry. This method can be extended to generalized ideal tetrahedra. However, for simplicity and to exhibit the analogies with the computation of the volume of lightlike tetrahedra in X Λ , we compute the volume by a different method that does not require a subdivision, namely with the parametrization from Proposition 4.15. Cl Λ (2α) + Cl Λ (2β) − Cl Λ (2(α + β)) .

Volumes of lightlike tetrahedra
We now consider the volumes of lightlike tetrahedra L ⊂ X Λ . These volumes can be computed in a similar way from the parametrization in Proposition 4.4. By a straightforward change of coordinates, this yields a parametrization in which both, the lightlike tetrahedron and its volume form become particularly simple. where ct −1 Λ is the generalized inverse cotangent given by (3.4). That the derivative of the right hand side with respect to s is indeed the integrand of the left hand side follows by a direct but lengthy computation. The derivative of the term tan(s) on the right hand side gives the second term on the left. The first term on the left is obtained from the derivatives of the inverse generalized cotangents on the right hand side with the formulas To simplify this integral further, we apply a change of variables, tan(s) = 1 − tan(t) 1 + tan(t) , to the third and fourth term to combine them with the first and second term, respectively. After some further computations involving trigonometric identities we then obtain vol(L) = 1 2ℓ 2 π 4 0 dt cos 2 (t) Note that the volume of the lightlike tetrahedron L ⊂ X Λ for Λ = 0 is also obtained from the volume formula for a 3-simplex in 3d Minkowski space. Omitting the coordinate x 2 in the identification (3.7) we can identify the vertices of L with points in R 3 . The volume is then given by the Minkowski bilinear form ·, · 1,0,2 and the Lorentzian wedge product on R 3 as vol(L) = 1 6 x 3 − x 4 , (x 1 − x 4 ) ∧ (x 2 − x 4 ) = 1 3 αβ(α + β).
It remains to clarify the relation between the volume formulas for a lightlike tetrahedron for Λ = 0 and Λ = ±1. For Λ = −ℓ 2 = 0 the division by ℓ 2 in the volume formula for Λ = ±1 is ill-defined. However, in this case we have Cl Λ (x) = −x log |2s Λ (x)| + x and hence the numerator of the volume formula for Λ = ±1 also vanishes. In fact, we can obtain the volume formula for Λ = 0 as a limit of the formula for Λ = ±1 if we extend the latter to Λ ∈ R by considering its expansion as a power series in ℓ.
Corollary 5.3. The volume of a lightlike tetrahedron L ⊂ X Λ is given as a power series in its shortest edge lengths α, β and in Λ by