Building Three-Dimensional Di ﬀ erentiable Manifolds Numerically II: Limitations

show that only a small fraction, (23 + 80) / 744 ≈ 0 . 138, of the multicube structures constructed from eight or fewer triangulations in the Regina [3] catalog allow dihedral angles that satisfy all the constraints. These include only three manifolds constructed from eight tetrahedra, and an even smaller fraction is expected for the manifolds based on triangulations with more tetrahedra. These results reveal that the methods developed in Ref. [1], including the generalizations presented here, are unfortunately quite limited in their ability to construct reference metrics on all the manifolds based on the triangulations in the Regina [3] catalog.


Summary
Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations.This note generalizes those methods by expanding the class of suitable triangulations, significantly increasing the number of manifolds to which these methods apply.These new results show that this expanded class of triangulations is still a small subset of all possible triangulations.This demonstrates that fundamental changes to these methods are needed to further expand the collection of manifolds on which differentiable structures can be constructed numerically.

Fixing the Dihedral Angles
The method for constructing reference metrics in Ref. [1] begins with the construction of a flat metric in the neighborhood of each vertex of a multicube structure, which can be obtained from a triangulation of that manifold.These flat metrics are then combined using partition of unity functions to produce a global C 0 metric, and then smoothed to C 1 by a sequence of additional steps described in Ref. [1].These flat metrics are constructed by fixing the dihedral angles of each cube edge.The simple method used in Ref. [1] fixes those dihedral angles to be 2π/K, where K is the number of cube edges that intersect along a particular edge.This choice ensures the sum of the dihedral angles around each edge is 2π, the condition needed to avoid a conical singularity there.This uniform dihedral angle condition severely limits the class of multicube structures on which it can be applied.This simple condition is replaced here with more complicated but less restrictive conditions.
The sum of the dihedral angles, ψ A{αβ} , from the cubes that intersect along an edge must equal 2π to avoid a conical singularity along that edge.This constraint can be written explicitly: where the sum is over all the edges that intersect along edge A{αβ}.Many of these 12N cubes constraints are redundant, but for simplicity all are enforced in the numerical analysis here.
Another set of important angles are the vertex angles between the edges of the cube, see Fig. 1.The notation θ A{γ}{αβ} is used for these angles, where the A index labels the cube, {γ} one of the cube faces, and {αβ} the edge that intersects {γ} at the {αβγ} vertex.These θ A{γ}{αβ} are the angles between vectors tangent to the {αγ} and the {βγ} edges.There are three vertex angles θ A{γ}{αβ} associated with each vertex, so 24 for each cube and 24N cubes total for the multicube structure.The law of cosines from spherical trigonometry gives the relationship between a vertex angle θ A{γ}{αβ} and the dihedral angles ψ A{αβ} associated with the cube edges that intersect at that vertex: The θ A{γ}{αβ} can therefore be considered functions of the ψ A{αβ} .between the corner of a cubic region and a small sphere centered on one of the vertices of that cube.This sphere is depicted as the dashed (blue) curve; the intersection of this cubic region with the sphere is a spherical triangle shown as solid (red) curves; the solid (black) straight lines are the edges of the cube.The dihedral angles ψ {αβ} between the cube faces are also the angles of this spherical triangle.The vertex angles, θ {α}{βγ} , are the angles between the edges of the cube, and are also the arc lengths of the sides of this spherical triangle.
Consider two cube faces, A{α} and A ′ {α ′ } that are identified in the multicube structure.The intrinsic metrics associated with these cube faces can only be continuous across the interface between cubes if the vertex angles θ A{α}{βγ} are the same as the corresponding angles θ A ′ {α ′ }{β ′ γ ′ } on the identified face.These additional constraints on the dihedral angles ψ A{αβ} can be written: Any interface with C A{α}{βγ} 0 has a metric discontinuity and consequently a curvature singularity at that interface.Half of these vertex angle constrains are redundant, but for simplicity all are enforced in the numerical analysis here.
In any multicube structure the 12N cubes constraints in Eq. ( 1) and the 24N cubes constraints in Eq. ( 3) must be satisfied by the 12N cubes dihedral angle parameters.Since there are more constraints than freely specifiable parameters, many (most) multicube structures will not admit solutions to all the constraints.In Ref. [1] a relatively small collection of multicube structures were found that admit uniform dihedral angle solutions to these constraints.Solutions that do not satisfy the uniform dihedral angle condition are found here for a wider class of manifolds.
The vertex angle constraints, Eq. ( 3), are very nonlinear, and general analytic solutions are not known.More general solutions can be found numerically, however, by finding the minima of the combined constraint norm, ||C ||, defined by where the sums are over the 12N cubes edge and the 24N cubes vertex angle constraints defined in Eqs. ( 1) and ( 3).This norm is a function of the dihedral angles, ||C || 2 = ||C (ψ A{αβ} )|| 2 that is bounded below by zero, so a minimum always exists.If min ||C || 2 = 0 then all the constraints are satisfied.If min ||C || 2 0 then the constraints are not satisfied and it is not possible to build a non-singular C 0 metric on that multicube structure in this way.
The numerical search for a minimum of ||C || 2 was started by setting initial guesses for ψ A{αβ} to their uniform dihedral angle values: ψ A{αβ} = 2π/K A{αβ} , where K A{αβ} is the number of cube edges that intersect edge A{αβ}.The numerical search for a minimum was carried out using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [2, p. 136] in the Python library scipy.optimize.Any numerical minimum with ||C || ≤ 10 −12 was considered to be a good numerical solution to all the constraints, while any minimum with ||C || > 10 −12 was rejected.
Constraint-satisfying dihedral angles ψ A{αβ} were searched for numerically on the 744 multicube structures constructed from the triangulations having eight or fewer tetrahedra included in the Regina [3] catalog of compact orientable three-dimensional manifolds.Table 1 lists the 23 manifolds from this search that satisfy all the constraints as well as the uniform dihedral angle condition.Table 2 lists 80 additional manifolds that admit non-uniform dihedral angle solutions with ||C || ≤ 10 −12 .The manifold names used in these tables are those from the Regina [3] catalog.These tables also list the number of multicube regions, N cubes , and the maximum number of edges, K max = max K A{αβ} , that overlap in each multicube structure.Table 2 also includes two parameters, min A A{αβγ} and min det g −1  A{αβγ} , that measure how distorted the constraint-satisfying dihedral angles make each multicube region.The quantity A A{αβγ} is the solid angle subtended by the cube at the A{αβγ} vertex (i.e. the area of the spherical triangle in Fig. 1).This solid angle would equal π 2 in an un-distorted cube, so 2 π min A A{αβγ} is a good measure of the maximum distortion in a multicube structure.The quantity det g −1 A{αβγ} represents the determinant of the inverse C 0 metric constructed in Ref. [1] from the dihedral angles, evaluated at the vertex A{αβγ}.This determinant would equal one in an un-distorted cube, so min det g −1 A{αβγ} is another good measure of the maximum distortion in a multicube structure.Multicube structures with 1 π min A A{αβγ} < 10 −6 or min det g −1 A{αβγ} < 10 −6 were excluded from the list in Table 2.These results show that only a small fraction, (23 + 80)/744 ≈ 0.138, of the multicube structures constructed from eight or fewer triangulations in the Regina [3] catalog allow dihedral angles that satisfy all the constraints.These include only three manifolds constructed from eight tetrahedra, and an even smaller fraction is expected for the manifolds based on triangulations with more tetrahedra.These results reveal that the methods developed in Ref. [1], including the generalizations presented here, are unfortunately quite limited in their ability to construct reference metrics on all the manifolds based on the triangulations in the Regina [3] catalog.

Discussion
Given a set of constraint-satisfying dihedral angles ψ A{αβ} , a global C 0 reference metric can be constructed in a straightforward way using the methods developed in Ref. [1].These C 0 reference metrics determine a basic C 1 differentiable structure on those manifolds.Smoother differentiable structures are needed, however, to allow global solutions to second-order equations like Einstein's gravitational field equation.Methods for transforming the C 0 reference metrics to C 1 (or smoother via Ricci flow) are also given in Ref. [1].Those methods were used here successfully to construct C 1 reference metrics for the 23 manifolds listed in Table 1 and the 17 manifolds displayed in boldface in Table 2.The constraint-satisfying dihedral angles make the remaining 63 non-boldface manifolds in Table 2 so distorted that non-singular C 1 metrics could not be constructed in this way.Considerable effort was expended in various attempts to find improved methods that could produce non-singular C 1 metrics in those cases, but all those efforts failed.Only 23 + 17 of the 744 triangulations from the Regina [3] catalog produced C 1 reference metrics that could be used to solve Einstein's equation on those manifolds.These results demonstrate that the methods developed in Ref. [1], and generalized here, are very limited.Fundamental changes to the reference metric construction methods, or methods for transforming the more singular triangulations into nicer more compliant forms are needed.

Figure 1 :
Figure1: Figure shows the intersection between the corner of a cubic region and a small sphere centered on one of the vertices of that cube.This sphere is depicted as the dashed (blue) curve; the intersection of this cubic region with the sphere is a spherical triangle shown as solid (red) curves; the solid (black) straight lines are the edges of the cube.The dihedral angles ψ {αβ} between the cube faces are also the angles of this spherical triangle.The vertex angles, θ {α}{βγ} , are the angles between the edges of the cube, and are also the arc lengths of the sides of this spherical triangle.

Table 2 :
Multicube Structures Admitting Non-uniform Dihedral Angles With ||C || ≤ 10 −12 .The quantities 2 π min A and min det g −1 (defined in the text) measure distortion of the multicube structure, with2π min A = min det g −1 = 1 in an undistorted structure.Bold face entries also admit non-singular C 1 metrics.