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.

Summary
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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