I Minkowski’s theorem for curved thetrahedra

Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps


We present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron’s faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. A concrete realization of this is provided by the relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in 3+1 dimensional covariant loop quantum gravity in the presence of a cosmological constant.

1]Hal M. Haggard, 2]Muxin Han, 3]and Aldo Riello \affiliation[1]Physics Program, Bard College, Annandale-on-Hudson, NY 12504, USA \affiliation[2]Institut für Quantengravitation, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstraße 7/B2, 91058
Erlangen, Germany \affiliation[3]Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario Canada N2L 2Y5 \emailAddhhaggard(AT)bard(DOT)edu \emailAddmuxin.han(AT)gravity(DOT)fau(DOT)de \emailAddariello(AT)perimeterinstitute(DOT)ca \keywordsDiscrete Geometry, Curved Geometry, Polyhedra, Minkowski Theorem, Flat Connections, Chern Simons, Cosmological Constant \arxivnumber

1 Introduction

In 1897, Hermann Minkowski proved a reconstruction theorem stating that to each non-planar polygon with edges , one can associate a unique convex polyhedron in Euclidean three-space with faces. The area and outward pointing normal of its -th face are and , respectively (40); (5). One hundred years later, in 1996, Michael Kapovich and John J. Millson showed how the space of polygons with fixed edge lengths admits a natural phase space structure (35). The combination of these results is remarkable: it points out that discrete geometries are a natural arena for dynamics. One may then wonder whether this arena is related to the theory of dynamic geometry par excellence, general relativity. The answer turns out to be positive, though not in a trivial way. In fact, the Kapovich-Millson phase space can be quantized via geometric quantization techniques (18) and this quantized space gives a compelling interpretation of the Hilbert space of loop quantum gravity (45); (49) (restricted to a single graph) in terms of discrete quantum geometries (11); (29). The main notions behind this are the following. In loop quantum gravity, the fundamental phase-space variables are fluxes (momenta) and holonomies (coordinates) carried by the Faraday-Wilson lines of the gravitational field. These lines cross at nodes, where gauge invariance is imposed as a momentum conservation equation, often referred to as the Gauß or closure constraint


here labels the Faraday-Wilson lines at a node (all supposed outgoing), and is the right invariant vector field on the -th copy of , i.e. the flux operator along the -th Faraday-Wilson line. Since the norm of the flux of the gravitational field carried by one of these lines is associated to the area it carries (46), it is physically meaningful to reinterpret this equation using Minkowski’s theorem. In this way, it can be read as the definition of a quantum convex polyhedron at each intersection of gravitational Faraday-Wilson lines. How these polyhedra are glued to one another and how they encode the extrinsic geometry of the three-space they span is more complicated and we refer to the cited literature for more details. Nevertheless, the crucial point here is that to each kinematical state of loop quantum gravity one can associate a discrete piecewise-flat quantum geometry thanks to Minkowski’s theorem.

In this paper we move toward the generalization of this construction to the case where the model space for the discrete geometry is curved instead of flat. In other words, we generalize Minkowski’s theorem to tetrahedra whose faces are flatly embedded in the three-sphere and hyperbolic three-space , and conjecture that a similar construction may hold for general curved polyhedra. From a purely mathematical point of view the generalization of Minkowski’s theorem is interesting in its own right, and requires new inputs in order to replace “the notion of face direction by some notion not relying on parallelism in the Euclidean sense” ((5), p. 346), or, in other words, to deal with the parallel transport of the face normals to a single base point. Moreover, the question arises whether the space of curved tetrahedra also admits a natural phase space structure, and eventually how close it is to the Kapovich-Millson one. We will show that a natural phase space structure exists, and it coincides with the one studied by Thomas Treloar in (50).

Surprisingly, this phase space structure is the same in both the spherical and hyperbolic case, which have a unified description in our framework, and it is exactly the generalization of the Kapovich-Millson phase space to geodesic polygons embedded in . (This is not the manifold in which the curved tetrahedron is embedded and, again, underlies both the positively- and negatively-curved cases.) Beside pure mathematics, this generalization is relevant to physics as well. Indeed, this construction is thought to bear strong relations to quantum gravity in the presence of a cosmological constant. On the one hand, this is apparent through the requirement that the simplicial decomposition of the bulk geometry be a solution of Einstein’s field equations with the cosmological term within each building block (8). On the other, curved tetrahedra made an appearance already in the semiclassical limit of the Turaev-Viro state sum (51); (41); (47); (48), which in turn, is known to be related to Edward Witten’s Chern-Simons quantization of three-dimensional gravity with cosmological constant (52). The relations with quantum gravity in (Anti-)de Sitter space constitute our main motivation (31).

Several works, with close connections to ours, have focused on three spacetime dimensions. Notably, in (21); (15); (14); (22); (17) and (42); (43), the precise connection was investigated between the Chern-Simons quantization of three dimensional gravity and the spinfoam or loop-theoretic polymer quantizations, respectively. With this in mind, we should emphasize that the present work studies three dimensional discrete geometries as boundaries of four dimensional spacetimes; this is in contrast to the research cited above, which focused on the description of geometries in two-plus-one dimensions. This difference in dimensionality implies a mismatch in the geometrical quantities encoded in the Faraday-Wilson lines: in four and three spacetime dimensions these carry units of area and length respectively. Unsurprisingly, the geometrical reconstruction theorems are completely different in the two cases.

A second interesting divergence of the two approaches is the fact that the sign of the geometric curvature must be decided a priori in the two-plus-one case (in particular, Girelli el al. restrict their analysis to the hyperbolic case), while it is determined at the level of each solution in our case. Again, this is because our formalism automatically allows for—in fact, requires—both positively and negatively curved geometries. It is intriguing to attribute this difference to the lack of a local curvature degree of freedom in three-dimensional gravity (since these are purely kinematical constructions, one should take this statement cum granu salis). In spite of these differences, there is an important feature the two constructions share: in both cases one is naturally lead to consider phase-space structures and symmetries that are deformed with respect to the standard ones of quantum gravity. In particular the momentum space of the geometry is curved and the symmetries are distorted to (quasi-)Poisson Lie symmetries, which are the classical analogues of quantum-group symmetries. We leave the discussion of the quantization of our phase space and its symmetries for a future publication.

Another piece of recent work in the loop gravity literature that interestingly shares some features with our construction is by Bianca Dittrich and Marc Geiller (20); (19); (9). While constructing a new representation (a new “vacuum”) for loop quantum gravity, adapted to describe states of constant curvatures (and no metric), they are lead to deal with exponentiated fluxes as the meaningful operators. As a consequence, areas in their formulation are also associated to —instead of —elements. Nonetheless, the parallel seems limited, since it appears that they are not forced to deform their phase space and symmetry structures.

Finally, Kapovich and Millson, and later Treloar, recognized that the phase space structure of polygons corresponds to William Goldman’s symplectic structure on the moduli space of flat connections on an times punctured two-sphere. Our result provides this correspondence with a more direct and physical interpretation. In fact, the punctures on the two-sphere can be understood as arising from the gravitational Faraday-Wilson lines piercing an ideal two-sphere surrounding one of their intersection points. Exactly as in the flat case, these lines characterize the face areas and define a polyhedron. This picture can be used to extend the covariant loop quantum gravity framework in four spacetime dimensions, the spinfoam formalism, to the case with a non-vanishing cosmological constant. This is the goal of our companion paper (31), which proposes a generalization of the spinfoam models constructed by John Barrett and Louis Crane (10); (7), and which eventually developed into the Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK) models (24); (23); (26).

In the companion paper (31), we analyze Chern-Simons theory with a specific Wilson graph insertion. The model can be viewed as a deformation of the EPRL/FK model aimed at introducing the cosmological constant in the covariant loop quantum gravity framework. The semiclassical analysis of the model’s quantum amplitude is given by the four-dimensional Einstein-Regge gravity, augmented by a cosmological term, and discretized on homogeneously curved four simplices. Interestingly, the key equation studied in this paper, i.e. the generalization of the Gauß (or closure) constraint to curved geometry, arises in that context simply as one of the equations of motion.

The present paper is divided in two parts. In the first we introduce the generalization of Minkowski’s theorem to curved tetrahedra. First we discuss the strategy underlying the theorem in the spherical case (2). We then gradually extend our analysis to more general settings eventually including both signs of the curvature (3 to 5). The first part concludes with the statement and proof of the theorem in its general form (6). The second part is dedicated to the description of the phase space of shapes of discrete tetrahedra. We first introduce the subject and its relations with curved polygons and moduli spaces of flat connections (7). Then, we review the quasi-Poisson structure one can endow with (8), which serves as a preliminary step to the actual description of the phase space of shapes (9). We conclude this part with a brief account of the equivalent quasi-Hamiltonian approach (10). The paper closes with some physical considerations and an outlook towards future developments of the work (11 and 12).

Part I Minkowski’s theorem for curved thetrahedra

2 General strategy and the spherical case

In the flat case, Minkowski’s theorem associates to any solution of the so-called “closure equation”


a tetrahedron in , whose faces have area and outward normals . We call the vectors area vectors.

Our generalization begins with the most symmetrical curved spaces, the three-sphere and hyperbolic three-space , which have positive and negative curvature, respectively. In these ambient spaces, define a curved polyhedron as the convex region enclosed by a set of flatly embedded surfaces (the faces) intersecting only at their boundaries. A flatly embedded surface is a surface with vanishing extrinsic curvature and the intersection of two such surfaces is necessarily a geodesic arc of the ambient space. In the spherical case, the flatly embedded surfaces and the geodesic arcs are hence portions of great two-spheres and of great circles, respectively. Note that in the hyperbolic case this definition includes curved polyhedra extending to infinity. This and other properties specific to the hyperbolic case are discussed beginning in the next section.

Our main result is that Minkowski’s theorem and the closure equation, Eq. (2), admit a natural generalization to curved tetrahedra in and . The curved closure equation is


where denotes the identity in . In the remainder of this section, we explain how this equation encodes the geometry of curved tetrahedra. Before going into this, we want to stress the essential non-commutativity of this equation, which mirrors the fact that the model spaces are curved, and therefore is the crucial feature of our approach. Indeed, non-commutativity has far-reaching consequences that are particularly apparent in the last section of the paper where the curved closure equation is used as a moment map. This non-commutativity will also be the source of an ambiguity in the reconstruction that is unique to the curved case.

As in the flat case, the variables appearing in the closure equation are associated to the faces of the tetrahedron. Indeed, the shall be interpreted as the holonomies of the Levi-Civita connection around each of the four faces of the tetrahedron. Since the faces of the tetrahedron are by definition flatly embedded surfaces in , any path contained within them parallel transports the local normal to the face at its starting point into the local normal to the face at its endpoint. Therefore, choosing at every point of the face a frame in which the local normal is parallel to , one can reduce via a pullback the connection to an one without losing any information. (Note that there always exists a unique chart covering an open neighborhood of the whole face.) In this two-dimensional setting, it is a standard result that a vector parallel transported around a closed (non self-intersecting) loop within the unit sphere gets rotated by an angle equal to the area enclosed by the loop. Therefore, the holonomy around the -th face of the spherical tetrahedron, calculated at the base point contained in the face itself, is given by


where are the three generators of , is the area of the face, and is the direction normal to the face in the local frame at which the holonomy is calculated. Let us for a second ignore the issues related to curvature and non-commutativity, and discuss what happens in the flat Abelian limit where the radius of curvature of the three-sphere goes to infinity and the ambient space becomes nearly flat. To make this explicit, introduce the sphere radius into the previous expression:


In the limit , the curved closure equations reduces, at the leading order, to the flat one:


Importantly, the geometrical meaning of the variables is exactly the same as in Eq. (2). Thus, our formulation subsumes the flat one as a limiting case.

In the curved setting, it is crucial to keep track of the holonomy base point; for within a curved geometry, only quantities defined at, or parallel transported to, a single point can be compared and composed with one another. Therefore, all four of the holonomies appearing in the curved closure equation must have the same base point. In spite of this, there is no point shared by all four faces of the tetrahedron at which one can naturally base the holonomies, and therefore at least one of them must be parallel transported away from its own face before being multiplied with the other three. Actually, the curved closure equation itself has no information about the base points of the holonomies or about which paths they have been parallel transported along to arrive at a common frame. This must be an extra piece of information that needs to be fed into the reconstruction algorithm. Analogous interpretational choices—though for clear reasons less numerous—have to be made in the flat case. Here, we provide a standard set of paths on an abstract tetrahedron embedded in along which the are assumed to be calculated. Such a choice of standard paths must also account for the presence of the identity element on the right hand side of the curved closure equation. This comes from the fact that the chosen standard paths compose to form a homotopically trivial loop. Interestingly, the curved closure equation can also be related to an integrated version of the Bianchi identities for the three dimensional Riemann tensor (see e.g. (27)).

Label the vertices of the geometrical tetrahedron as in Figure 1. This numbering induces a topological orientation on the tetrahedron, which must be consistent with the geometrical orientation of the paths around the faces. Faces are labeled via their opposite vertex (e.g., face 4 is the one at the bottom of Figure 1), while edges are labeled by the two vertices they connect.

Figure 1: A standard numbering of the vertices of the tetrahedron, which also induces a particular topological orientation. The BurntOrangep character of this tetrahedron is simply a device for underlining its spherical nature. However, similar pictures result from stereographic projection of a spherical tetrahedron in onto . This projection sends great two-spheres of into spheres of , though possibly with different radii. Note that the convex or concave aspect of the stereographically projected spherical tetrahedron has no intrinsic meaning.

Each face is traversed in a counterclockwise sense when seen from the outside of the tetrahedron. This is consistent with the tetrahedron’s topological orientation. The normals appearing in the holonomies, Eq. (4), are hence the outward pointing normals to the face whenever the base point of the holonomy lies on that face (right-handed convention). There is no natural common base point for all four faces. However, any three faces do share a point. Pick faces , which share vertex 4, and base the holonomies at this vertex:


Then, in the case of holonomies , the vectors are outward normals to their respective faces in the frame of vertex 4. Clearly, this is not the case for the normal . Thus, we must specify the path used to define the holonomy around face 4 and its transport to vertex 4. By now, this path is completely fixed by the curved closure equation. It consists of defining in an analogous way to the and then parallel transporting it to vertex 4 through the edge . The set of relevant paths is shown in Figure 2. Up to the choice of the base point, this is manifestly the simplest (and shortest) set of paths going around each face in the order required by the closure equation and composing to the trivial loop. For this reason, we will call these simple paths.

Figure 2: The set of simple paths used to interpret the holonomies .

The holonomies along the simple paths, , can be expressed more explicitly by introducing the edge holonomies , encoding the parallel transport from vertex to vertex along the edge connecting them (we use leftward composition of holonomies). Thus, , and


Let us stress once more that, since the closure equation is preserved by a cyclic permutation of the holonomies, the assignment to a specific holonomy of the label “4” is indeed an extra input needed by the reconstruction. We call this vertex the special vertex.

Another important symmetry of the closure equation is its invariance under conjugation of the four holonomies by a common element of :


This maps the areas into themselves, and the normals into . (Here and in the rest of the paper, bold-face symbols stand for matrices in the fundamental representation; e.g. in the previous equation, is the matrix corresponding to .) This symmetry can be interpreted either as a change of reference frame at the base point 4, or as the effect of a further parallel transportation of the along another piece of path from vertex 4 to some other base point. The latter interpretation is particularly compelling when : the result of this transformation is an exchange of the rôle of vertices (and therefore faces) 4 and 2. We conclude that picking vertex 4 or 2 as special, are gauge equivalent choices. So, it is more appropriate to refer to the edge as the special edge rather than referring to 2 or 4 as special vertices.

A set of holonomies that close modulo simultaneous conjugation, is naturally interpreted as the moduli space of flat connections on a sphere with four punctures. Indeed, since the holonomies of a flat connection can only depend on the homotopy class of the (closed) path along which they are calculated, these connections are maps from the fundamental group of the punctured sphere to . Therefore, the moduli space of flat connections is this space of maps modulo conjugation:


Conjugation is the residual gauge freedom left at the arbitrarily chosen base point of the holonomies. The connection with the tetrahedron’s geometry arises from the observation that the fundamental group of the 4-punctured sphere is isomorphic to that of the tetrahedron’s one-skeleton. However, this isomorphism is not canonical, and constitutes the extra piece of information that is needed to run the reconstruction, i.e. the knowledge of the precise paths associated to the .

The choice of a special edge, e.g. (24), breaks permutation symmetry. However, conjugation is a true symmetry of the problem, and is the analogue of rotational invariance for the flat case. Therefore, any quantity with an intrinsic geometrical meaning must be obtained through conjugation invariant combinations of the . The normals are not gauge invariant observables, but their scalar and triple products are.

Scalar products between the normals have a clear meaning: they encode the dihedral angles between the faces of the tetrahedron. Because the faces of the tetrahedron are flatly embedded, these dot products are invariant along the edge shared by two faces and hence the dihedral angles are well defined. For faces 1, 2, and 3, the situation is simple. The holonomies and the normals appearing in their exponents are defined at vertex 4, which is shared by all three faces. Therefore, indicating with the (external) dihedral angle between faces and , see Figure 3,

Figure 3: The dihedral angle spans the arc from outward normal to . Here we illustrate the case .

Recall that is first defined at vertex 2 and then parallel transported to vertex 4 along the edge . Because of the gauge equivalence of 2 and 4 as special vertices, and because vertex 2 is shared by faces 1, 3, and 4, calculating the dihedral angles between these face is as simple as before:


To see this in a more direct way, note that, for example, , which is exactly the result of the previous equation.

The remaining dihedral angle, between the opposite, special faces 2 and 4, is more delicate. This is because neither of the vertices 2 or 4 is shared by the faces 2 or 4. To calculate , we use the normals at vertex 3:

The paths used for transporting the normals from vertex 3 to vertex 4 are not accidental; they lie within their own face up to the point where the face holonomy is based, and then move on, when necessary, to vertex 4 through the special edge (24). All paths lying within a single face are equivalent because of the flat embedding and so we use the most convenient choice.
Had we chosen to define at vertex 1 instead of 3, the result would have been

A quick check shows that these two results are equivalent, thanks to the closure equation and the relation . Summarizing,


Notice that in order to have a convex tetrahedron, and although this condition would be redundant for a tetrahedron in flat space, one could use the sphere’s non-trivial topology to build non-convex spherical tetrahedra.1 We are not interested in reconstructing such objects. Moreover, the previous condition implies that we can invert Eq. (14) to obtain the values of the themselves. These formulas require only data entering the curved closure equation, and not the edge holonomies , as expected from considerations of gauge invariance.

There is still a subtle point to clarify. How can the directions of the outward normals be extracted from the ? The face areas of the tetrahedron are positive real numbers lying in the interval due to the tetrahedron’s convexity (see footnote 2). However, the holonomy cannot distinguish between two triangles lying on the same great two-sphere in that have areas and , respectively, and corresponding normals and . In formulas:


This is a consequence of the fact that both the trivial loop and a great circle have trivial -holonomy. To resolve this ambiguity, it is enough to appeal to convexity by checking the signs of the triple products among the normals. Indeed, the triple products are naturally associated to the vertices of the tetrahedron, and their signs relate to its convexity (as well as to our choice of its topological ordering, and to the outward pointing property of the normals ), see Figure 4.

Figure 4: The three vectors involved in the triple product at vertex 4. Given the topological orientation of the tetrahedron, its convexity, and supposing all normals are outward pointing, one finds that .

Concretely, this translates into the following requirements for the normals:


After parallel transporting to the common base point, vertex 4, these conditions read


A moment of reflection shows that these conditions are exactly what is needed to solve the ambiguity expressed in equation (15). In fact, among the possible redefinitions of the normals by change of signs , one and only one of them satisfies Eq. (17).

It is interesting to express the intrinsic geometrical quantities of the tetrahedron, such as areas, dihedral angles, and triple products, directly in terms of the holonomies . The simplest conjugation invariant set of observables are traces of products of the . These turn out to be quite involved. A convenient alternative is given by the same invariants for the lifts of the to . Call these lifts , and their matrices in the fundamental representation . The twofold ambiguity associated with the lift reflects the geometric ambiguity of Eq. (15), which is already present at the level of . It is tempting to conjecture that considering closures solves this ambiguity, and that the holonomies can be automatically associated to the spin connection of the homogeneously curved space. Unfortunately, this is not the case, since by multiplying the geometrical values of any two (or four) of the holonomies by , one obtains another sensible closure equation that looses its geometrical interpretation.2 Hence, we are lead to allow any consistent lift with the closure


and eventually correct for the sign of (an even number of) the holonomies in such a way that all the inequalities of Eq. (17) are satisfied. A slightly different way of stating this, with closure only holding up to a sign, is that what we are really considering are closures, and only these are in one to one correspondence with curved tetrahedra. In the following we will mostly deal with holonomies, to which we associate geometries in an almost one-to-one way.

The convenience of using the comes from the simple identity:


where are the Pauli matrices, and . Define the connected part of the half-trace of the product of holonomies, :


It is then straightforward to check that the geometrical quantities of interest are normalized versions of these quantities:


with the appropriate generalization for and the missing triple products. The signs can be thought of as representing the branches of the respective square roots, and are uniquely fixed by imposing the positivity of the triple products, i.e. the convexity of the tetrahedron. They are eventually related to the areas via , which is another way of stating the ambiguity .

At this point we are left with the simple exercise of reconstructing a spherical tetrahedron from its known dihedral angles . Notice that the areas are not needed. Their consistency with respect to the reconstructed geometry will be proved in complete generality in 6. The key equation in the reconstruction is the spherical law of cosines, relating the edge lengths of a spherical triangle to its face angles. With the notation of Figure 5, this law and its inverse read

Figure 5: A spherical triangle illustrating the notation for Eqs. (22a) and (22b), the spherical cosines laws.

where is the arclength (on the unit sphere) between the vertices and , and is the angle between the arcs and at point . By putting an infinitesimal sphere around the vertex of the spherical tetrahedron, and looking at the spherical triangle defined by the intersections of this sphere with the edges stemming from vertex , one can use Eq. (22a) to deduce the three face angles at the vertex from the tetrahedron’s dihedral angles. Once all the face angles are known, Eq. (22b) yields the edge lengths for each face of the tetrahedron. Therefore, by using just one formula and its inverse, it is possible to deduce the full geometry of the spherical tetrahedron from its dihedral angles. This is possible, in the curved case, because the radius of curvature provides a natural scale to translate angles into arclengths. In this respect, the flat case is a degenerate limit in which scale invariance appears. In the flat closure, Eq. (2), the areas can all be rescaled by a common factor without altering the normals. No analogously simple symmetry is present in the curved case.

3 A first look at the hyperbolic case

In the case of hyperbolic tetrahedra, the reconstruction theorem proceeds in essentially the same way as above. Once again, the faces of the curved tetrahedron are required to be flatly embedded, which implies the holonomies around them have a form completely analogous to those in the spherical case (Eq. (4)):


where all the symbols are interpreted in the same manner, and the minus sign is due to the negative sign of the curvature. A crucial fact about this formula is that the holonomies are again in , and not in some other group with different signature. The simple reason for this is that is the group of symmetries of the tangent space (at a point) of both and .

We deduce the dihedral angles of the tetrahedron following similar reasoning to that of the previous section. The extra minus sign of Eq. (23) has consequences only for the formulas that express the triple products of the normals in terms of connected traces (Eqs. (16) and (21c)); the right-hand sides of these equations should be multiplied by -1. The formulas for the dihedral angles, which involve two normals, are only sensitive to the overall agreement in sign of the triple products, which is granted in both the spherical and hyperbolic cases. This latter fact will be crucial in the following.

Once the dihedral angles have been calculated, the tetrahedron can be straightforwardly reconstructed using the hyperbolic law of cosines:


Note the extra minus sign in the first equation. These formulas conclude the list of ingredients needed for the reconstruction in the hyperbolic case.

In section 5, however, we shall see that these ingredients are not quite enough to cover all the possible hyperbolic cases naturally arising from the closure equation. A new generalization of hyperbolic geometry has to be introduced.

4 Spherical or hyperbolic? The Gram matrix criterion

Up to now we have described two possible reconstruction procedures, one for spherical and one for hyperbolic tetrahedra. Nonetheless, the starting point we are proposing for the reconstruction theorem is the same closure equation, Eq. (3). The natural question arises, whether there is an a priori criterion to decide which type of tetrahedron one should reconstruct when given only the four closing holonomies (and the choice of a special edge). There is such a criterion. The key is the unambiguousness character of the dihedral angles discussed above: once a sign (either for the moment) of the four triple products has been fixed, the dihedral angles of the curved tetrahedron are uniquely determined, irregardless of the curvature. But also, the dihedral angles encode all the necessary information to reconstruct the full tetrahedron, including its curvature. In this section we briefly review how the curvature can be deduced from the dihedral angles alone. While part of this is standard, it allows us to introduce concepts and notation useful in the following section.

To begin, we reverse the logic, and suppose we are actually given a tetrahedron, flatly embedded in a space of constant positive, negative or null curvature. Then, define its Gram matrix, as the matrix of cosines of its (external) dihedral angles:


One of the main properties of the Gram matrix is that the sign of its determinant reflects the spherical, hyperbolic, or flat nature of the tetrahedron:


A straightforward way to understand this result is by embedding in .

Let us start from the flat case, the simplest one. In this case, the parallel transport is trivial and . By introducing the four 4-vectors , one can write the matrix in terms of the matrix whose components are , we denote the component index with greek letters ranging from 1 to 4, and:


The last equality follows from the obvious fact that the are not linearly independent, since they are just four 3-vectors in disguise. Nonetheless, these 4-vectors have a useful geometric interpretation; imagine the tetrahedron as embedded in the model space orthogonal to the 4-vector . Then, each is the 4-normal to another hyper-plane in that picks out a face of the tetrahedron when it intersects orthogonally. This is depicted in one lower dimension in Figure 6, where it is also clear that the (cosine of the) hyper-dihedral angle is equal to the (cosine of the) tetrahedron’s dihedral angle . Note that in this case, using the Euclidean () or Lorentzian metric () does not make any difference, since . This reflects the fact that the flat case is a degenerate version of both the spherical and hyperbolic geometries.

Figure 6: A one-dimension lower representation of a flat, a spherical, and a hyperbolic tetrahedron as embedded in , , and , respectively. The lower picture shows a section of the hyperbolic case to highlight the Lorentzian representation used for the hyperboloid.

In the spherical case, one embeds the tetrahedron into the unit sphere . The -th face of the curved tetrahedron will then lie on a great 2-sphere of identified by the intersection of the unit sphere with a hyper-plane passing through the origin of and orthogonal to the 4-vector (we use the same symbol as in the flat case). Once more, the (cosine of the) hyper-dihedral angle is equal to the (cosine of the) dihedral angle between the faces and of the tetrahedron (provided orientations are chosen consistently). This fact gives the relation


from which it follows


where the zero value has been excluded because it would correspond to a degenerate tetrahedron, which we will not treat here.

The easiest way to understand the hyperbolic case (see Figure 6), is in terms of a “Wick rotation” of the spherical one. One obtains


from which it follows


since . Here too, the zero value has been excluded because it corresponds to degenerate cases. The new metric is needed because the Euclidean normals to the planes that intersect the unit hyperboloid are not tangent to the hyperboloid at the points of contact, and therefore the Euclidean scalar product between these normals does not reproduce the tetrahedron’s Gram matrix. Related to this, there is the fact that the hyperboloid of Figure 6 has negative curvature only when calculated within the Lorentzian metric (time direction pointing upwards in the figure).

Interestingly, there is a direct way to calculate the sign of the determinant of the Gram matrix just in terms of the holonomies and the choice of a special edge. We are free to choose the special vertex 4 of the curved tetrahedron to be located at the north pôle of (or of , respectively), in which case for , where the are determined up to a global sign by the procedure discussed in the previous section. The last three components of are then completely determined by the equations


where is given by Eq. (14) and can be either or . Explicitly:


Hence, by using the condition that must be of unit norm, in either the Euclidean or the Lorentzian metric, it is easy to realize that the sign of the determinant of the Gram matrix is given by


Now that we have been able to determine a priori the nature of the curved tetrahedron, we can run the correct form of the reconstruction according to whether the holonomies turn out to be associated with a non-degenerate spherical () or hyperbolic () geometry. If , our equations should be interpreted as some sort of degenerate spherical or hyperbolic geometry, which we do not attempt to reconstruct. Indeed, they cannot correspond to a flat tetrahedron, because in that case all holonomies should be trivial, irregardless of the shape of the tetrahedron!

In conclusion notice that one can attempt to reverse the logic presented here, by taking the four hyper-planes identified by the as the primitive variables, instead of the tetrahedron. Doing so, the above construction identifies in the spherical case not one but 16 different tetrahedra on , with antipodal pairs congruent.3 The way we have defined the Gram matrix picks out only one of these tetrahedra, the one for which all four normals induced by the are outgoing. Choosing one among these 16 tetrahedra is somewhat analogous to fixing the signs of the four triple products discussed in the previous section. Also, it is interesting to note that in the flat case this multiplicity does not appear, provided the tetrahedra “opened up towards infinity” are disallowed. What about the hyperbolic case? On the (one sheeted-)hyperbolid the situation is analogous to the flat case; however, by looking at the hyperboloid as a sort of analytical continuation—we do not intend to be precise about this claim—of the sphere, one might expect to find again a remnant of the 16-fold multiplicity. A moment of reflection shows that the tetrahedra crossing the equator in the spherical case are “broken up” into two pieces, both extending to infinity, and contained in the two seperate sheets of the two-sheeted hyperboloid. In the next section we shall see why these two-sheeted tetrahedra are of interest for the curved reconstruction theorem.

5 Two-sheeted hyperbolic tetrahedra

A well-known result, easily deduced from the Gauß-Bonnet theorem, is that the area of an hyperbolic triangle cannot be larger than and is given by , where are the triangle’s internal angles. (We use throughout for triangle areas and rely on context to distinguish the geometry as spherical, hyperbolic or Euclidean.) The bound is saturated by ideal triangles, i.e. triangles with vertices “at infinity”. Nonetheless, an element representing a rotation around some fixed oriented axis is generally between 0 and , which means that the areas encoded in the holonomies generally range over these values. Spherical triangles achieve this full range of areas, but standard hyperbolic triangles do not. Is there something forcing the areas to be smaller than when the determinant of the Gram matrix, seen as a function of the four holonomies, is negative? It is not hard to find examples showing that there is not. Consequently, we need to make sense of hyperbolic tetrahedra with face “areas” in the full range . Inspired by the observations at the end of the previous section, we look to use triangles stretching across the two sheets. The aim of this section is to describe these new two-sheeted hyperbolic triangles and tetrahedra.

Figure 7: The two-sheeted hyperboloid and triangle. Left A plane passing through the origin of the embedding space (light gray) intersects the hyperboloid along two geodesics. The dark gray disk gives the Beltrami-Klein model of 2-dimensional hyperbolic space. Geodesics on the hyperboloid are mapped onto straight lines of the Beltrami-Klein disk (dashed lines). In contrast to the Poincaré disk model, the Beltrami-Klein model does not preserve angles. Center and Right In dark grey, a two-sheeted triangle. The right-most figure shows the two Beltrami-Klein disks as seen from the origin of , therefore a positively oriented triangle has a right-handed down-ward pointing orientation with respect to the plane of the page. The central, one-sheeted triangle in the lower sheet is shaded for future reference.
Figure 8: Another representation of a two-sheeted triangle. In the central image the shape of the actual hyperbolic triangle is highlighted. On the sides we represent the triangle in both the Poicaré and Beltrami-Klein models. The Poincaré model has the virtue of preserving angles, while the Beltrami-Klein models represents geodesics via straight lines and is easily recovered via the intersecting-plane construction shown in the previous picture.

We start with the 2-dimensional triangles. In Figures 8 and 8 we have illustrated what we mean by a two-sheeted triangle, and how to orient them. The key idea, is to use the planes passing through the origin of the embedding space to extend the geodesics beyond infinity to the other sheet, and to use the natural orientation of the hyperbolae provided by Lorentz boosts which is also consistent with the orientation induced by that of the planes. Figures 8 and 8 represent a two-dimensional hyperbolic geometry, and hence the geometry of the faces of a hyperbolic tetrahedron. In three dimensions, the Beltrami-Klein disk model becomes a three-ball model, in which the two-dimensional hyperboloids where the faces of the tetrahedron lie are mapped onto flat disks inscribed in the three-ball; it is one of these that is pictured.

The area of a two-sheeted triangle, however, is not just larger than , it is actually infinite. Nonetheless, what appears implicitly in the closure equation is not the area of the triangle, but the total deficit angle perceived by an observer going around it. In an homogeneously curved geometry this happens to be proportional to the area. Therefore, by defining a notion of holonomy around a two-sheeted triangle, we effectively provide a notion of “renormalized” area for these triangles. At the end of this section, we briefly comment on how far this idea can be pushed.

In order to define a holonomy around a two-sheeted triangle, it is enough to give a prescription for the parallel transport through infinity from one sheet to the other. In other words, one needs to identify the tangent spaces at the point and on the boundaries of the two Poincaré or Beltrami-Klein disks, or balls in three dimensions. However, given a geodesics and its extension to the other sheet, there is a very natural prescription for the identification of the aforementioned tangent spaces (see the left columns in each panel of Figure 9, as well as Figure 8).

Figure 9: The identification across the two sheets of boundary points along the same geodesics. The right column of each panel shows what happens when , belonging to a different geodesics, is let “collapse” onto : the point has to be identified with two different points according to which geodesics they belong to. In this sense, one is not allowed to think of “gluing” the two discs together to perform the identification. The arrows, show how a frame is parallel transported across the two disks. Arrow 2 represents the projection onto the disk of a normal to the plane of Figure 8.

This requires that: (i) the velocity vector along the a geodesic going out to infinity is identified with the incoming velocity vector on the geodesic’s continuation, (ii) the vector normal to the outgoing geodesic and pointing towards the interior of a two-sheeted triangle is identified with the only vector with both these properties at the entering point of the incoming geodesics on the other sheet. The second requirement simply preserves the notions of in and outside. In the embedded picture, it requires the vector normal to the geodesic to lie on the same side of the hyperplane defining the geodesic itself. Notice that by orienting the normals to the upper and lower sheets as future and past pointing respectively, the three-dimensional frame composed by the velocity, the normal vector to hyperplane, and the normal to the hyperboloid preserves its orientation thanks to this requirement. This construction can be generalized to the three-dimensional two-sheeted hyperboloid, by considering the normals to the flatly embedded surfaces defining the faces of the two-sheeted tetrahedron instead of the normal to the geodesic arcs defining the sides of the two sheeted triangles.

Does the identification of and and their tangent spaces obtained while moving along a given geodesic induce an identification of the boundaries of the two Beltrami-Klein disks (balls)? No. The reason is shown in the right columns of each panel of Figure 9: a point on the upper sheet is identified with different points on the lower sheet depending on the geodesics through which the point is reached. Therefore, specializing to the relevant 3-dimensional case, the parallel transport prescription we give, instead of identifying the boundaries of the two balls and , provides a 1-to-1 map between the spaces and . Here labels the space of geodesics based at a point on .

Having fixed the parallel transport prescription, finding the holonomy around a triangle is just a matter of calculation. A particularly simple way to find the holonomy in the standard case is to note that the parallel transport along a geodesic is trivial, and the only non-trivial contributions come from the “kinks” at the vertices of the triangle. Pleasantly, this remains true here because nothing happens when parallel transporting a frame across the two sheets. If the triangle lies completely within one sheet (or on the surface of a sphere) each kink contributes to the final holonomy with a rotation (around the normal to the surface) through an angle , where is the angle between the velocity vectors before and after the kink. After circuiting a triangle the total rotation amounts to , with being the internal angles of the triangle. Then, in the case of a spherical triangle we simply obtain its area (modulo ) . Similarly, for a one-sheeted hyperbolic triangle, we obtain (again modulo ) minus its area . However, if the triangle is hyperbolic and two-sheeted, we define its “renormalized” area through the parallel transport prescription we just outlined, obtaining the formula:


To avoid confusion, we will call the holonomy area of the two-sheeted triangle. This area is in the range . To show this note that the internal angles of the two-sheeted triangle are related to those of the unique (up to congruence) one-sheeted triangle identified by continuing its geodesic sides, see the rightmost, lower panel of Figure 8, where the relevant one sheeted triangle is dashed. Calling the angles of the latter triangle , , and , where is the only angle the two triangles have in common, and its area , one finds:


The same result could have been obtained by using the simple observation that the holonomy area of a (necessarily two-sheeted) hyperbolic lune of width is (to be compared to the spherical case: ). We observe that the holonomy areas only make sense modulo , and can only be calculated for regions whose boundaries are arbitrarily well approximated by piecewise geodesics lines. Consequently, it is not possible to make sense of the holonomy area of a full hyperbolic sheet, and the total area of the two sheets is zero, since it is “enclosed” by the trivial loop. Nonetheless, given that the starting point of our reconstruction theorem are the holonomies themselves, and not arbitrary regions of the two-sheeted hyperboloid, these definitions are appropriate and useful.

The generalization of this construction to higher dimensions, and in particular to two-sheeted tetrahedra, is straightforward: these tetrahedra are regions of the two-sheeted 3-hyperboloid identified by four points on it, the vertices, and delimited by the intersections of the hyperboloid with the hyperplanes generated by triplets of vertex 4-vectors. Note that to completely characterize the tetrahedron, one has to specify the orientations of the planes.

6 Curved Minkowski Theorem for tetrahedra

Now that the geometric picture has been clarified, we can state and finally prove the curved Minkowski theorem for tetrahedra. When we want to emphasize that the Gram matrix can be caclulated directly from the holonomies , e.g. using Eq. (21b) and related expressions, we write .

Theorem 1.

Four group elements , satisfying the closure equation , can be used to reconstruct a unique generalized (i.e. possibly two-sheeted in the hyperbolic case) constantly-curved convex tetrahedron, provided:

  • the are interpreted as the Levi-Civita holonomies around the faces of the tetrahedron,

  • the path followed around the faces is of the so-called “simple” type (see 2), and has been uniquely fixed by the choice of one of the two couples of faces (24) or (13),

  • the orientation of the tetrahedron is fixed and agrees with that of the paths used to calculate the holonomies,

  • the non-degeneracy condition is satisfied.

The uniqueness is understood to be modulo isometries.

In particular, condition (i) means that the ’s written in the form have the following geometrical interpretation: (1) the are the areas of the faces of the tetrahedron (possibly interpreted as holonomy areas), and (2) the are the outward pointing normals to these faces when parallel transported (along the simple path chosen) to a common reference frame. Also, it turns out that: (3) the tetrahedron has positive (negative) curvature if (, respectively); (4) the tetrahedron is double-sheeted if it has a negative curvature and the cofactors of the Gram matrix do not agree in sign. The proof is an extension of the formalism appearing at Eq. (42) below.

Observe that the four conditions to be satisfied for the theorem to hold have distinct characters: condition (i) is key to the theorem, it allows its geometric interpretation; condition (iii) is simply needed to avoid the possibility of reconstructing the parity reversed tetrahedron as well; condition (iv) is technical and, unfortunately, can be cumbersome from the point of view of the holonomies, since the Gram matrix is a nice object geometrically speaking, but not as simple algebraically; finally, condition (ii) has a somewhat strange status. Indeed, a condition of this type is certainly needed to take care of the parallel transport ambiguities present in the curved setting, but at the same time the specific form we are employing looks quite arbitrary—even if inspired by a simplicity criterion—and in principle can be modified to other choices of paths, which would, in turn, require a few somewhat obvious modifications in the reconstruction procedure. The simple-path condition naturally arises in the four-dimensional context of (31).

Before giving the proof of the main theorem we give a short proof of a useful lemma:

Lemma 1.

The principal minors of are positive, with the exception of the minor in the hyperbolic case.


The principal minors are immediate, since each is equal to 1. The minors are also easily seen to be positive since they are equal to , for the appropriate choice of indices . Finally, to show that also the minors are all positive, consider first the case of the principal minor equal to the determinant of the matrix obtained by erasing row and column 4 from :


where the unit vectors are those appearing in Eqs. (4) and (23) (with signs fixed by the triple product criterion),4 and the matrix appearing at the furthest right is the matrix which has the three 3-vectors as columns. In light of this formula is trivially positive. The same holds for . A little more effort is needed to prove that and are also positive. Explicitly:


where in the first equality we used the definition of Eq. (14), which takes into account the parallel transport of to vertex 4 along the special edge; while in the second we made use of the fact that , and therefore . Therefore, is positive. It can be shown that is positive by a very similar argument. ∎

The proof of the theorem proceeds in a completely constructive way, and without loss of generality, it is performed within the explicit choice of edge being the special one. Most of the steps necessary for the reconstruction were explained in great detail in 2, and will not be discussed again. Our attention is focused on the well-definedness and unambiguous statement of each step of the reconstruction. We will also prove the consistency of the reconstruction procedure. Therefore the theorem is subdivided into two parts: in the first, we show that the uniquely identify a Gram matrix that, in turn, is associated to a unique curved tetrahedron; in the second part we show that the Levi-Civita holonomies around the four faces of the tetrahedron are necessarily given by the themselves. Loosely speaking, in the first part we extract from the closure relation and the simple-path condition the dihedral angles of a tetrahedron which uniquely determine it, and in the second we verify that the areas of the reconstructed tetrahedron are necessarily the same as those encoded in the initial group elements .


Part one First, calculate the triple products appearing in Eq. (17) using the group elements via Eq. (21c) (properly generalized in the way discussed in the first section for or ), and fix the signs appearing there by requiring these four triple products to be positive (note that there is only one such choice). Geometrically, this completely fixes the signs of the normals by imposing the convexity of the tetrahedron.5 This allows the unambiguous specification of the entries of the (putative) Gram matrix


with the right-hand side of the first equation being calculated via Eq. (21b) and its generalization for . We stress that is a function of the ’s only. Now,


since the null case has been excluded by hypothesis. Define the matrix , to be interpreted as the metric of the four-dimensional embedding space as described in 4. Then, there exist four 4-vectors such that


or more symbolically . In particular, , and the four 4-vectors can be interpreted geometrically as the oriented unit normals to the hyperplanes passing through the origin of which, upon intersection with the the unit sphere (unit two-sheeted hyperboloid , respectively), identify the great spheres (great hyperboloid, respectively) bounding the tetrahedron itself. See the figures and discussion of 4.

The vertices of the tetrahedron are located along the intersections of the triplets of hyperplanes normal to the . Hence the matrix has columns proportional to the 4-vectors identifying the vertices of the tetrahedron (the minus sign in this formula fixes the correct sign of the vertex vectors). We define . In the spherical case, the vertex vectors completely characterize the tetrahedron; they identify four points on the unit sphere that can be connected by the shortest geodesic segments between them. However, in the hyperbolic case it is not a priori clear that the intersect the two-sheeted unit hyperboloid . Indeed, for them to do so, they must be timelike, that is they must satisfy . However, this is equivalent to the condition , which in turn must be true because of the following relations and the result of Lemma Lemma 1 (which states for all ):


where, recall, is the principal minor obtained by erasing row and column 4 from . To obtain the last equality, the fact is used that being a diagonal minor, is also equal to the -th cofactor of . Therefore, we can conclude that also in the hyperbolic case a unique generalized (i.e. possibly two-sheeted) tetrahedron can be identified. It suffices to define the “shortest” geodesic between two vertices as the generalized geodesic (i.e. possibly going through infinity) that does not pass through any point defined by the intersection of the hyperboloid and three of the four hyperplanes normals to the other than its initial and final points. This concludes the first part of the proof.

Part two The group elements and closure relation Eq. (3) specify more data than the Gram matrix alone. Thus, we have to verify the consistency of all of this data. Indeed, the construction from part one guarantees only that the dihedral angles of the reconstructed tetrahedron are compatible with the holonomy group elements , but not that the reconstructed areas also match those encoded in the . More specifically, we have claimed that the can be interpreted as holonomies of the Levi-Civita connection around the various faces of the tetrahedron, and this implies (see 2 and 5) that the rotation angles of the are the areas of the faces of the tetrahedron. We now prove this claim.

Given the reconstructed tetrahedron, one can explicitly calculate the holonomies along the specific simple path on its 1-skeleton used in the reconstruction. Call these the reconstructed holonomies, . Although they satisfy , and their 4-normals satisfy by construction, it is not yet clear whether the are necessarily equal to the (up to global conjugation, i.e. gauge). Demonstrating this is what we mean by showing consistency of the reconstruction. We once more proceed constructively, and show that both and , in the notation of Eqs. (4) and (23). We will show that the Gram matrix and the closure equation contain all the information needed to completely fix the . Because the and the have the same Gram matrix we will briefly drop the distinction and omit the tildes.

First align with by acting with a global rotation (conjugation). A second global rotation around the -axis can be used to align with ; this is always possible because . Now the system is completely gauge-fixed and there is no further freedom to rotate the vectors. The vector has a fixed angle with both and , determined by and , and there are a priori at most two vectors with this property (identified by the intersection of two cones around and , respectively). However, only one of those satisfies the additional requirement that , which was crucially used in the reconstruction.6 Similarly, is also uniquely determined. All that remains then is to show that the entries of the Gram matrix completely fix the areas.

Consider . Since , , and are all given, there exists at most two values of (in the interval ) that solve this equation (geometrically this is again the intersection of two cones). The triple product condition singles out one of these two solutions. Similarly, one fixes by using the analogous expression and . To conclude, we need to show that and are completely determined.

Consider the closure equation , where we identify , , and so on. We have completely fixed and , as well as and . The remaining unknowns are and