Encoding Curved Tetrahedra in Face Holonomies by Haggard, Han and Riello

This week I have been studying ‘Encoding Curved Tetrahedra in Face Holonomies’. This paper is closely related to the posts:

In this paper the authors  present a generalization of Minkowski’s classic theorem on the reconstruction of tetrahedra which will apply 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.

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.

An example of this is  the relation with the spin-network states of loop quantum gravity. This paper provides a 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.

Minkowski’s theorem for curved tetrahedra

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


a tetrahedron in E³, whose faces have area al  and outward normals nl.

The paper’s  main result is that Minkowski’s theorem and the closure equation, admit a natural generalization to curved tetrahedra in S³ and H³. The curved closure equation is


where e denotes the identity in SO(3). This equation encodes
the geometry of curved tetrahedra.

As in the flat case, the variables appearing in the closure equation are associated to the faces of the tetrahedron. The {Ol} will be interpreted as the holonomies of the Levi-Civita connection around
each of the four faces of the tetrahedron.

The holonomy Ol around the l-th face of the spherical tetrahedron, calculated at the base point P contained in the face itself, is given by


where {J} are the three generators of so(3), al is the area of the face, and n(P) is the direction normal to the face in the local frame at which the holonomy is calculated.

The vertices of the geometrical tetrahedron are labelled as shown below:


This numbering induces a topological orientation on the tetrahedron, which must be consistent with the geometrical orientation of the paths around the faces.



The holonomies along the simple paths, {Ol}, can be expressed more explicitly by introducing the edge holonomies {oml}, encoding the parallel transport from vertex l to vertex m along the edge connecting them.


The holonomies {Ol } and the normals appearing in their exponents are defined at vertex 4, which is shared by all three faces. Therefore, indicating with θlm the external dihedral angle between faces l and m,



and similarly,


The Gram Matrix

For a given a tetrahedron, flatly embedded in a space of constant positive, negative or null curvature, the Gram matrix is defined  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:


Curved Minkowski Theorem for tetrahedra

Four SO(3) group elements Ol , l= 1,..,4 satisfying the closure equation  can be used to reconstruct a unique generalized  constantly curved convex tetrahedron, provided:

  1. the {Ol } are interpreted as the Levi-Civita holonomies around the faces of the tetrahedron
  2.  the path followed around the faces is of the so-called simple type , and has been uniquely fixed by the choice of one of the two couples of faces (24) or (13),
  3.  the orientation of the tetrahedron is fixed and agrees with that of the paths used to calculate the holonomies,
  4.  the non-degeneracy condition det Gram(Ol )  ≠ 0 is satisfied.

Minkowski’s theorem establishes a one-to-one correspondence between closed non-planar polygons in E³ and convex polyhedra, via the interpretation of the vectors defining the sides of the polygon as area vectors for the polygon. This theorem can be extended to curved polyhedra.

In this paper the authors have given a generalization of Minkowski’s theorem for curved tetrahedra. This establishes a correspondence between non-planar, geodesic quadrilaterals in S³, encoded in four SO(3) group elements {Ol} whose product is the group identity, and flatly embedded tetrahedra in either S³ or H³.

This can be used to  reinterpret the Kapovich-Millson-Treloar symplectic structure of closed polygons on a homogeneous space as the symplectic structure on the space of shapes of curved
tetrahedra with fixed face areas.

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The geometry of the tetrahedron and asymptotics of the 6j-symbol

This week I have been studying a useful PhD thesis ‘Asymptotics of quantum spin networks‘ by van der Veen. In this post I’ll look a the section dealing with the geometry of the tetrahedron.

Spin networks and their evaluations

In the special case of the tetrahedron graph the author presents a complete solution to the rationality property of the generating series of all evaluations with a fixed underlying graph using the combinatorics of the chromatic evaluation of a spin network, and a complete study of the asymptotics of the  6j-symbols in all cases: Euclidean, Plane or Minkowskian ,using the theory of Borel transform.


 The three realizations of the dual metric tetrahedron depending on the sign of the Cayley-Menger determinant det(C). Also shown are the two exponential growth factors Λ ± of the 6j-symbol.

The author also computes the asymptotic expansions for all possible colorings,  including the degenerate and non-physical cases. They find that  quantities in the asymptotic expansion can be expressed as geometric properties of the dual tetrahedron graph interpreted as a metric polyhedron.

letting ( Γ,γ ) denote the tetrahedral spin network colored by an admissible coloring . Consider the planar dual graph which is also a tetrahedron with the same labelling on the edges. Regarding the labels of as edge lengths one may ask whether ( Γ,γ ) can be realized as a metric Euclidean tetrahedron with these edge lengths. The admissibility of ( Γ,γ ) implies that all faces of  the dual tetrahedron satisfy the triangle inequality. A well-known theorem of metric geometry implies that ( Γ,γ ) can be realized in exactly one of three flat geometries

(a) Euclidean 3-dimensional space R³
(b) Minkowskian space R²’¹
(c) Plane Euclidean R².

Which of the above applies  is decided by the sign of the Cayley-Menger determinant det(C) which is a degree six polynomial in the six edge labels.

Its assumed that ( Γ,γ ) is nondegenerate in the sense that all faces of are two dimensional. In the degenerate case, the evaluation of 6-symbol is a ratio of factorials, whose asymptotics are easily obtained from Stirling’s formula.

The geometry of the tetrahedron

The asymptotics of the 6j-symbols are related to the geometry of the planar dual tetrahedron. The 6j-symbol is a tetrahedral spin network ( Γ,γ ) admissibly labeled as shown below:


with  γ= (a, b, c, d, e, f). Its dual tetrahedron  ( Γ,γ ) is also labelled by  γ. The tetrahedron and its planar dual, together with an ordering of the vertices and a colouring of the edges of the dual is depicted below. When a more systematic notation for the edge labels is needed  they will be denoted them by dij it follows that

(a, b, c, d, e, f) = (d12, d23, d14, d34, d13, d24)


We can interpret the labels of the dual tetrahedron as edge lengths in a suitable flat geometry. A condition that allows one to realize ( Γ,γ )  in a flat metric space such that the edge lengths equal the edge labels. Labeling the vertices of ( Γ,γ ). We can formulate such a condition in terms of the Cayley-Menger determinant. This is
a homogeneous polynomial of degree 3 in the six variables a2, . . . , f2. A  definition of the determinant is:

Given numbers dij we can define the  Cayley-Menger matrix by

Cij = 1−d²ij/2 for i, j ≥1


Cij = sgn(i−j) for when i = 0 or j = 0.

In terms of the coloring = (a, b, c, d, e, f) of a tetrahedron, we have:


The sign of the Cayley-Menger determinant determines in what space the tetrahedron can be realized such that the edge labels equal the edge lengths

  • If det(C) > 0 then the tetrahedron is realized in Euclidean space R³.
  •  If det(C) = 0 then the tetrahedron is realized in the Euclidean plane R².
  • If det(C) < 0 then the tetrahedron is realized in Minkowski space R²’¹.

In each case the volume of the tetrahedron is given by


The 6 dimensional space of non-degenerate tetrahedra consists of regions of Minkowskian and regions of Euclidean tetrahedra. It turns out to be a cone that is made up from one connected component of three dimensional Euclidean tetrahedra and two connected components of Minkowskian tetrahedra. The three dimensional Euclidean and Minkowskian tetrahedra are separated by Plane tetrahedra. The Plane tetrahedra also form two connected components, representatives of which are depicted below:


The tetrahedra in the Plane component that look like a triangle with an interior point are called triangular and the Plane tetrahedra from that other component that look like a quadrangle together with its diagonals are called quadrangular.  The same names are used for the corresponding Minkowskian components.

An integer representative of the triangular Plane tetrahedra is not easy to find as the smallest example is (37, 37, 13, 13, 24, 30).

Let’s look at the dihedral angles of a tetrahedron realized in either of the three above spaces. The cosine and sine of these angles can be expressed in terms of certain minors of the Cayley-Menger
matrix. Define the adjugate matrix ad(C) whose ij entry is (−1)i+j times the determinant of the matrix obtained from C by deleting the i-th row and the j-th column. Define Ciijj to be the matrix obtained from C by deleting both the i-th row and column and the j-th row and column.

The Law of Sines and the Law of Cosines are well-known formulae for a triangle in the Euclidean plane. let’s look at the Law of Sines and the Law of Cosines for a tetrahedron in all three flat geometries.

If we let θkl be the exterior dihedral angle at the opposite edge ij. The following formula is valid for all non-degenerate tetrahedra:


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