Building Three-Dimensional Differentiable Manifolds Numerically

In this first step, the multicube structure consisting of a collection of cubic regions, $\mathcal{B}_{A}$, is enhanced by defining a set of domains, called the star-shaped domains, $\mathcal{S}_{I}$, that overlap the boundaries between the primary cubic regions. One star-shaped domain surrounds each distinct vertex of the multicube structure. It is constructed from (copies of) all the cubic regions that intersect at that vertex point. (A particular cubic region $\mathcal{B}_{A}$ may be included more than once in a star-shaped domain if two or more of its vertices are identified with each other.) Each of the star-shaped domains, $\mathcal{S}_{I}$, has the topology of an open ball in $\mathbb{R}^{3}$. The index $A$ is used to label the cubes $\mathcal{B}_{A}$ in the multicube structure, while the index $I$ labels the star-shaped domains $\mathcal{S}_{I}$, or equivalently the distinct vertices in the multicube structure. The structures of the individual star-shaped domains depend on the global properties of the multicube structure, in particular on how many cube vertices intersect in the manifold at the center of each $\mathcal{S}_{I}$. Figure 1 illustrates several examples of star-shaped domains having different numbers of cubic regions intersecting at their central vertex points.

A code designed to use multicube structures can be enhanced to assemble the $\mathcal{S}_{I}$ in a fairly straightforward way: Any multicube structure code must include the cube face identification maps. Starting at one vertex of one cubic region, the identities of the three cubes whose faces are identified with the faces of $\mathcal{B}_{A}$ adjacent to this vertex are determined from the multicube maps. This can be done, for example, by following the interface identification maps for points near this vertex on each of the three faces that meet at that point. Copies of the three cubes identified as neighbors in this way are added to $\mathcal{S}_{I}$. This identification step is repeated for the adjacent faces of each of the additional cubes, and then iterated until (copies of) all the cube vertices that intersect the original vertex point are included in $\mathcal{S}_{I}$. Once a star-shaped domain $\mathcal{S}_{I}$ is complete, if some cube vertices in the full multicube structure remain un-assigned to the center of some star-shaped domain, then a new star-shaped domain $\mathcal{S}_{I+1}$ is constructed around this vertex using the same procedure. The process terminates when all the cube vertices have been included at the center of some star-shaped domain. There are a finite number of cube vertices in any multicube structure (that can be used for practical numerical work), so in practice this process always terminates after a finite number of steps.

The second step in the procedure to construct global $C^{\,0}$ reference metrics builds a flat metric in each of the star-shaped domains, $\mathcal{S}_{I}$, introduced in Sec. 2.1. Each $\mathcal{S}_{I}$ consists of a cluster of cubes that intersect at its central point. If these cubes are appropriately distorted into parallelograms (by adjusting the dihedral angles between the cube faces), they can be fitted together (without overlapping and without leaving gaps between them) to form an isometric subset of $\mathbb{R}^{3}$, and thus inherit a natural flat metric. Figure 1 illustrates several simple examples of star-shaped domains isometrically embedded in $\mathbb{R}^{3}$.

To understand whether the cubic regions of a multicube structure can always be fitted together in this way, consider a small two-sphere surrounding the central vertex of $\mathcal{S}_{I}$. This sphere intersects all the cubic regions that meet at this central point. The intersections of this sphere with the faces and edges of each cube form triangles on this sphere. Figure 2 illustrates the spherical triangles that result from these intersections. The intersection of one of these cubes, $\mathcal{B}_{A}$, is displayed as the spherical triangle with solid (red) line edges. The intersections of other nearby cubes in $\mathcal{S}_{I}$ are displayed with dash-dot (green) line edges. Together the intersections from all the cubic regions in $\mathcal{S}_{I}$ form a triangulation of this two-sphere.

Any triangulation on a two-sphere can be realized geometrically in an infinite number of ways. Given any one realization, an infinite number of others can be created simply by moving the vertices of the triangles around on the sphere by small amounts (i.e., much smaller than the sizes of the triangles), and then replacing the edges with geodesics (great circles) between vertices. Each spherical triangle with geodesic edges represents the intersection of a parallelogram (whose dihedral angles match the angles of the triangle) with the two-sphere. Thus there are an infinite number of ways to construct distorted parallelograms that fit together in the correct way to represent $\mathcal{S}_{I}$ as an isometric subset of $\mathbb{R}^{3}$.

An algorithm designed to compute a flat metric on $\mathcal{S}_{I}$ must choose from among the infinite possibilities in some way. Making and implementing that choice is expected to be a complicated optimization problem that we plan to analyze fully in a future study. For the purposes of the present study, however, we have chosen to adopt a simple pragmatic approach: choosing the dihedral angles to have uniform sizes around each edge. This simple approach limits the class of multicube structures to which it can be applied. However, it is general enough that we have been able to construct compliant examples (see Sec. 4) from most of the Thurston geometrization classes, plus examples of several well known manifolds like the Poincaré dodecahedral space [11], Seifert-Weber space [12], and all six compact orientable three-manifolds that admit flat metrics [13, 14], including the Hantzsche-Wendt space [15] (E6).

Before proceeding with the details of constructing flat metrics on the $\mathcal{S}_{I}$ in these simple multicube structures, it will be helpful to establish some basic notation. The notation $\partial_{\alpha}\mathcal{B}_{A}$ (or more compactly $A\{\alpha\}$) is used to refer to the $\alpha$ face of cubic region $\mathcal{B}_{A}$. The index $\alpha$ can have the values $\{-x$, $+x$, $-y$, $+y$, $-z$, $+z\}$. The edge of region $\mathcal{B}_{A}$ formed by the intersection of the $A\{\alpha\}$ and $A\{\beta\}$ faces is referred to as $A\{\alpha\beta\}$, and the vertex formed by the intersections of the $A\{\alpha\}$, $A\{\beta\}$, and $A\{\gamma\}$ faces is referred to as $A\{\alpha\beta\gamma\}$. The dihedral angle between the $A\{\alpha\}$ and $A\{\beta\}$ faces is denoted $\psi_{A\{\alpha\beta\}}$, while the angle between the axes at the edges of the $A\{\alpha\}$ face is denoted $\theta_{A\{\alpha\}}$. The $\psi_{A\{\alpha\beta\}}$ are also equal to the angles of the spherical triangle created by the intersection of cube $\mathcal{B}_{A}$ with a small sphere (see Fig. 2), and the $\theta_{A\{\alpha\}}$ are also equal to the arc lengths of the edges of this triangle.

The uniform dihedral angle spacing assumption adopted here requires the dihedral angles of all the cubic regions that intersect along an edge to be the same. In addition to being reasonably simple to impose, it has the advantage of imposing a rigid uniformity that prevents any cubic region from being more distorted than its neighbors. To prevent conical singularities along the cube edges, the sum of the dihedral angles around each edge must be exactly $2\pi$. The uniform dihedral angle assumption therefore implies that the dihedral angle at the $A\{\alpha\beta\}$ edge must be given by

where $K_{A\{\alpha\beta\}}$ is the number of cubic regions that intersect along this edge.

The uniform dihedral angle assumption also implies that the triangulations of the two-sphere at the center of a star-shaped domain, $\mathcal{S}_{I}$, must have a special local reflection symmetry. Figure 3 illustrates two neighboring triangles in one of these triangulations. If the uniform dihedral angle assumption has been imposed then $\psi_{1}=\bar{\psi}_{1}=2\pi/K_{1}$ and $\psi_{2}=\bar{\psi}_{2}=2\pi/K_{2}$. The spherical geometry analog of the angle-side-angle congruence theorem from Euclidean geometry then implies that $\psi_{3}=\bar{\psi}_{3}$. With the uniform dihedral angle assumption, this means that $K_{3}=\bar{K}_{\bar{3}}$, i.e. the number of edges that meet at vertex $3$ in this triangulation must be the same as the number that meet at vertex, $\bar{3}$, of the neighboring triangle. This symmetry must apply to every edge of every triangle in the triangulations of the two-spheres at the centers of each star-shaped domain $\mathcal{S}_{I}$. Therefore, this simple assumption is quite limiting, and is not satisfied by most two-sphere triangulations and consequently most multicube structures.

Any open subset of $\mathbb{R}^{3}$ inherits the flat Euclidean metric of $\mathbb{R}^{3}$. Thus any $\mathcal{S}_{I}$ constructed from parallelograms whose dihedral angles satisfy the uniform dihedral angle assumption will naturally inherit a flat metric. The illustration in Fig. 3 shows the relationship between the dihedral angle $\psi_{\{\alpha\beta\}}$ and the angle between unit normals to the cube faces, $\vec{n}_{A\{\alpha\}}=c_{\{\alpha\}}\vec{\nabla}x$ for $\alpha=\pm x$, $\vec{n}_{A\{\alpha\}}=c_{\{\alpha\}}\vec{\nabla}y$ for $\alpha=\pm y$, and $\vec{n}_{A\{\alpha\}}=c_{\{\alpha\}}\vec{\nabla}z$ for $\alpha=\pm z$. The constants $c_{\{\alpha\}}$ are chosen to ensure that the $\vec{n}_{A\{\alpha\}}$ are the outgoing unit normals. The inner products of the outgoing unit normals are related to the dihedral angles by $\vec{n}_{A\{\alpha\}}\cdot\vec{n}_{A\{\beta\}}=\cos\bigl{(}\pi-\psi_{A\{\alpha\beta\}}\bigr{)}=-\cos{\psi_{A\{\alpha\beta\}}}$. The inner products of the coordinate gradients also determine the coordinate components of the inverse metric: $e^{\,ab}=\vec{\nabla}x^{a}\cdot\vec{\nabla}x^{b}$. Therefore the flat inverse metric, $e^{\,ab}_{A\{\alpha\beta\gamma\}}$, associated with the vertex $A\{\alpha\beta\gamma\}$, expressed in terms of the local Cartesian coordinates of region $\mathcal{B}_{A}$, is given by

where the constants $c_{\{+x\}}=-c_{\{-x\}}=c_{\{+y\}}=-c_{\{-y\}}=c_{\{+z\}}=-c_{\{-z\}}=1$ ensure the unit normals are outgoing. In Eq. (2) $\alpha=\pm x$, $\beta=\pm y$ and $\gamma=\pm z$. These metrics have the correct dihedral angles between coordinate faces to allow them to fit together smoothly with the metrics in neighboring regions. Since the derivatives of these metrics vanish throughout each region, they are all flat.

The final step in constructing a flat metric on the $\mathcal{S}_{I}$ domain is to show that the intrinsic parts of the metrics constructed in Eq. (2) are continuous across the interface boundaries between the cubic regions in $\mathcal{S}_{I}$. Equation (2) shows that these metrics depend only on the dihedral angles of the edges of the cubic region. The simple uniform dihedral angle assumption adopted for this study implies the local reflection symmetry of the triangulations described above. This symmetry guarantees that the dihedral angles of each cubic region $\mathcal{B}_{A}$ are the same as those of the neighboring cubic regions. The metrics in two neighboring regions will therefore be related to one another by the local reflection symmetry across the interface boundary between them. It follows that the intrinsic parts of the metrics must be continuous across the interface boundary. In general the gauge components of the metric will not be continuous when expressed in the Cartesian coordinates of the multicube structure.

The next step in our procedure for constructing a reference metric is to build a partition of unity that can be used to combine the flat metrics from the various overlapping domains into a global non-singular metric that is smooth within each cubic region, and whose intrinsic parts are continuous across the interfaces with each neighboring region. The needed partition of unity function $u_{A\{\alpha\beta\gamma\}}(\vec{x})\geq 0$ has the value $1$ at the $A\{\alpha\beta\gamma\}$ vertex of domain $\mathcal{B}_{A}$, and falls smoothly to zero on the faces of $\mathcal{B}_{A}$ that do not intersect this vertex. The $u_{A\{\alpha\beta\gamma\}}(\vec{x})$ are positive within the star-shaped domain $\mathcal{S}_{I}$ centered on the $A\{\alpha\beta\gamma\}$ vertex, and vanish on its outer boundary. They are used as weight functions to compute averages of the flat inverse metrics $e_{A\{\alpha\beta\gamma\}}^{\,ab}$ defined on the $\mathcal{S}_{I}$ domains in Eq. (2). The inverse of the resulting average, $\hat{g}_{ab}$, is the global $C^{\,0}$ reference metric.

First introduce a set of non-negative weight functions, $w_{A\{\alpha\beta\gamma\}}(\vec{x})\geq 0$, whose support is centered on the vertex $A\{\alpha\beta\gamma\}$. In the two-dimensional case [7] simple separable functions of the global Cartesian coordinates were used successfully for these weight functions. The three-dimensional analogs of those two-dimensional functions are

where the index $A{\{\alpha\beta\gamma\}}$ refers to the vertex of the cubic region $\mathcal{B}_{A}$, and $L$ is the coordinate size of the regions. The vectors $\vec{\Delta x}_{A{\{\alpha\beta\gamma\}}}=\left(\Delta x_{A\{\alpha\beta\gamma\}},\Delta y_{A\{\alpha\beta\gamma\}},\Delta z_{A\{\alpha\beta\gamma\}}\right)$ are defined by

where $\vec{x}$  are the global Cartesian coordinates of the multicube structure that are aligned with the cube faces, and where $\vec{c}_{A}+\vec{\nu}_{\{\alpha\beta\gamma\}}$ are the coordinates of the vertex $A{\{\alpha\beta\gamma\}}$. The values of $\vec{c}_{A}$, the locations of the centers of regions $\mathcal{B}_{A}$, are specified as part of the definition of the multicube structure; and the values of $\vec{\nu}_{\{\alpha\beta\gamma\}}$, the positions of the vertices relative to the centers are given in Table 1. They are the same for all the regions.

The functions $h(s)$ used in Eq. (3) are chosen to have the values $h(0)=1$ and $h(\pm 1)=0$. With the arguments specified in Eq. (3) this corresponds to setting $w_{A{\{\alpha\beta\gamma\}}}=1$ at the vertex point $A\{\alpha\beta\gamma\}$, and $w_{A{\{\alpha\beta\gamma\}}}=0$ on the boundary faces of $\mathcal{B}_{A}$ that do not intersect this vertex. Each of the functions $w_{A{\{\alpha\beta\gamma\}}}(\vec{x})$ is also continuous across the $A\{\alpha\}$, $A\{\beta\}$, and $A\{\gamma\}$ interfaces with the corresponding functions in the neighboring domains centered on this same vertex. We find that the functions

with integers $k>0$ and $\ell>0$, work quite well in practice. Some of these functions are illustrated in Fig. 4, with integer values in the range that worked best in our numerical tests.

The final task in constructing special partitions of unity for the region $\mathcal{B}_{A}$ is to construct the normalizing functions $W_{A}(\vec{x})$:

where $w_{A{\{\alpha\beta\gamma\}}}(\vec{x})$ is defined in Eq. (3). These $W_{A}(\vec{x})$ are strictly positive, so they can be used to define the partition of unity functions:

This normalization ensures that these functions satisfy the inequalities $0\leq u_{A{\{\alpha\beta\gamma\}}}(\vec{x})\leq 1$, and also the usual partition of unity normalization condition

for each $\vec{x}$  in each $\mathcal{B}_{A}$.

A global reference inverse metric $\hat{g}^{\,ab}(\vec{x})$ for $\vec{x}$  in region $\mathcal{B}_{A}$ can now be constructed by combining the flat metrics $e_{A{\{\alpha\beta\gamma\}}}^{\,ab}$ defined in Eq. (2) with the partition of unity functions defined in that region by Eq. (7):

The sum is over the eight vertices of region $\mathcal{B}_{A}$. This inverse metric is positive definite since it is a linear combination of positive definite inverse metrics, $e_{A{\{\alpha\beta\gamma\}}}^{\,ab}$, using the non-negative weight functions $u_{A{\{\alpha\beta\gamma\}}}(\vec{x})$. A global continuous reference metric, $\hat{g}_{ab}(\vec{x})$, is then obtained by inverting $\hat{g}^{\,ab}(\vec{x})$ at each point $\vec{x}$. The metric $\hat{g}_{ab}$ has continuous intrinsic parts across all of the multicube interface boundaries because it is constructed from flat metrics and partition of unity functions that are each appropriately continuous across those interfaces.

This section constructs a conformal factor, $e^{\phi}$, that is used to transform the reference metric $\hat{g}_{ab}$ constructed in Sec. 2.3:

The geodesic equation for the curve $x^{a}(s)$ in the $\bar{g}_{ab}$ metric is given by

where $s$ is an arbitrary parameterization of this curve, where $\bar{\Gamma}^{a}_{bc}$ are the Christoffel symbols of the second kind for this metric, and where $A(s)$ is a parameter dependent function. The idea is to choose a conformal factor $e^{\phi}$ in Eq. (10) having two properties: a) it makes each edge of each cubic region into a geodesic of the metric $\bar{g}_{ab}$, and b) it is continuous across each cube interface.

Consider the cubic region, $\mathcal{B}_{A}$, whose Cartesian coordinates are labeled $x^{a}=\{x^{\alpha},x^{\beta},x^{\gamma}\}$, and consider the $A\{\alpha\beta\}$ edge of this region where the $A\{\alpha\}$ and $A\{\beta\}$ faces intersect. This edge is a curve with tangent vector $\mathrm{d}x^{a}/\mathrm{d}s=\{0,0,1\}$, where the parameter $s$ has been chosen to be $s=x^{\gamma}$. An equivalent form of this equation, more convenient for these purposes, is given by

where the $\bar{\Gamma}_{abc}$ are the Christoffel symbols of the first kind. The three components of this equation can be reduced to

As an interesting aside, note that Eq. (13) depends only on the intrinsic metric on face $A\{\beta\}$, and together with Eq. (15) forms the intrinsic geodesic equation on this face. Similarly Eq. (14) depends only on the intrinsic metric on face $A\{\alpha\}$, and together with Eq. (15) forms the intrinsic geodesic equation on this face. Thus the curve formed by the intersection of two surfaces is a geodesic of the full three-dimensional space if and only if it is a geodesic of the intrinsic geometry of each surface separately.

The idea now is to choose a conformal factor $\phi$ that transforms $\hat{g}_{ab}$, using Eq. (10), so the resulting $\bar{g}_{ab}$ satisfies Eqs. (13)–(15) on the edges of each cubic region. The intrinsic parts of the resulting $\bar{g}_{ab}$ will be continuous across the interfaces between regions if and only if the conformal factor $\phi$ is continuous across those interfaces. First set $\phi=0$ along the edges of the cubic region to ensure that $\bar{g}_{ab}$ is continuous there. In this case Eqs. (13)–(15) can be re-written in terms of $\hat{g}_{ab}$ and $\phi$ for points along the $A\{\alpha\beta\}$ edge:

The terms involving $\partial_{\gamma}\phi$ all vanish in these equations because $\phi=0$ along this edge. These equations place constraints on $\partial_{\alpha}\phi$ and $\partial_{\beta}\phi$. In particular, Eq. (18), determines $A(s)$ in terms of $\hat{g}_{\gamma\gamma}$, while Eqs. (16) and (17), can be re-written as boundary conditions for $\partial_{a}\phi$ along the $A\{\alpha\beta\}$ edge:

Note that these expressions imply that the conformal factor $\phi$ will not vanish everywhere on the cube faces unless $0=\hat{\Gamma}_{a\gamma\gamma}$ on the edges, in which case those edges would already be geodesics of the $\hat{g}_{ab}$ metric. Also note that the metric $\hat{g}_{ab}$ constructed in Sec. 2 rapidly approaches a constant flat metric at each vertex of the cubic region. It follows that the connection $\hat{\Gamma}_{abc}$ and the gradient $\partial_{a}\phi$ all vanish at these vertex points.

The next step is to extend the conformal factor $\phi$ across the faces of the cubic region $\mathcal{B}_{A}$ in a way that a) satisfies the boundary conditions given in Eqs. (19) and (20) along each edge, and b) ensures that it is continuous across the interface with the neighboring cubic region. The conformal factor $\phi_{A\{\alpha\}}$ on the $A\{\alpha\}$ face satisfies the boundary conditions $\phi_{A\{\alpha\beta\}}=0$ and Eq. (20) along the $A\{\alpha\beta\}$ edge. Analogous conditions must also be imposed on each edge of this face. Together, these conditions constitute Dirichlet and Neumann boundary conditions for $\phi_{A\{\alpha\}}$ on the $A\{\alpha\}$ face. One convenient way to find $\phi_{A\{\alpha\}}$ that satisfy these boundary conditions is to solve the bi-harmonic equation for $\phi_{A\{\alpha\}}$ on this face:

Solutions to the bi-harmonic equation are uniquely determined by specifying both Dirichlet and Neumann conditions on the boundary of a compact domain [16]. This approach can then be used to determine the surface values of $\phi$ on each face of each region in the multicube structure. The pseudo-spectral numerical methods used to solve this equation for this study are described briefly in B.

The boundary conditions that determine the solution to Eq. (21) only depend on the intrinsic components of the metric, $\hat{g}_{\beta\beta}$, $\hat{g}_{\beta\gamma}$ and $\hat{g}_{\gamma\gamma}$, on the $A\{\alpha\}$ face. These intrinsic metric components were constructed to be continuous across this interface boundary in Sec. 2. It follows that boundary conditions used to determine $\phi_{A\{\alpha\}}$ will be the same on both sides of the interface boundary. Since the solution to Eq. (21) with Dirichlet and Neumann boundary conditions is unique [16], it follows that the $\phi_{A\{\alpha\}}$ determined in this way will be the same on both sides of the interface.

The method describe above can be used to determine the surface values $\phi_{A\{\alpha\}}$ on each face of each cubic region. These solutions provide Dirichlet boundary conditions for the full conformal factor $\phi_{A}$ within each region. The normal derivatives of $\phi_{A\{\alpha\}}$ are unconstrained, however, beyond the requirement that those derivatives agree along the edges with the tangential derivatives from the neighboring faces. The conformal factor $\phi$ within the cubic region can therefore be determined in any number of ways. For example, it could be determined by solving the three-dimensional Laplace equation with the Dirichlet boundary conditions $\phi_{A\{\alpha\}}$.

A computationally more efficient approach has been adopted for this study. Begin by defining a set of coordinates $s^{\alpha}_{A}$ that measure the relative distance between a point inside region $\mathcal{B}_{A}$ and the $\partial_{-\alpha}\mathcal{B}_{A}$ face of that region. The $s^{\alpha}_{A}$ are normalized so that $s^{\alpha}_{A}=0$ on the $\partial_{\alpha}\mathcal{B}_{A}$ face, while $s^{\alpha}_{A}=1$ on the opposite $\partial_{-\alpha}\mathcal{B}_{A}$ face. In particular

where $\vec{x}=(x,y,z)$ are the global Cartesian coordinates of the multicube structure, $\vec{c}_{A}=(c^{x}_{A},c^{y}_{A},c^{z}_{A})$ are the coordinates of the center, and $L$ is the coordinate size of region $\mathcal{B}_{A}$.

The conformal factor $\phi_{A\{\alpha\}}$ on the $A\{\alpha\}$ face, constructed by solving Eq. (21), can now be extrapolated into the interior using the $h(s)$ functions defined in Eq. (5). Consider the extrapolation $\phi^{\alpha}_{A}=h(s^{\alpha}_{A})\,\phi_{A\{\alpha\}}$. The $\phi_{A\{\alpha\}}$ vanish identically along the edges because of the boundary conditions used to solve Eq. (21). Therefore the $\phi_{A\{\alpha\}}$ extrapolated in this way do not modify the $\phi_{A\{\beta\}}$ on the adjacent faces. It does not modify $\phi_{A\{-\alpha\}}$ on the opposite face, either, because $h(s^{\alpha}_{A})$ vanishes there. The complete conformal factor $\phi_{A}$ in the interior of region $\mathcal{B}_{A}$ can therefore be determined by combining the extrapolations from all the cube faces:

The resulting $\phi_{A}$ automatically satisfies the Dirichlet conditions $\phi_{A\{\alpha\}}$ on each of the faces.

The conditions in Eqs. (19) and (20) ensure that the edges of each cubic region are geodesics of the metric $\bar{g}_{ab}=e^{\phi}\hat{g}_{ab}$. The continuity across the interface boundaries of $\phi_{A}$ ensures that the global solution for $\phi$ is continuous across those boundaries. And this in turn ensures that the intrinsic components of the $\bar{g}_{ab}$ metric are continuous across all the interface boundaries as well.

Let $\bar{g}_{ab}$ denote the global $C^{\,0}$ metric constructed in Sec. 3.1. The goal of this second step is to convert $\bar{g}_{ab}$ into a metric $\bar{\bar{g}}_{ab}$ having two important properties: first, the extrinsic curvatures $\bar{\bar{K}}^{\{\alpha\}}_{ab}$ associated with $\bar{\bar{g}}_{ab}$ must vanish identically on each edge of each cubic region; and second, the intrinsic parts of $\bar{\bar{g}}_{ab}$ must be identical to those of $\bar{g}_{ab}$. We note that while $\bar{g}_{ab}$ was constructed to have intrinsic parts that are only $C^{\,0}$ across the interface boundaries, within each region $\bar{g}_{ab}$ is actually smooth.

Consider the interface boundary $A\{\alpha\}$ of cubic region $\mathcal{B}_{A}$. In the global Cartesian coordinates of our multicube structure, the boundary $A\{\alpha\}$ is a level surface of the coordinate $x^{\alpha}$. The unit normal co-vector field to the foliation of constant $x^{\alpha}$ surfaces, $\bar{\bar{n}}^{\{\alpha\}}_{a}$, is given by

where $\bar{\bar{N}}^{\{\alpha\}}=\epsilon^{\{\alpha\}}\left(\bar{\bar{g}}^{\alpha\alpha}\right)^{-1/2}$. The constant $\epsilon^{\{\alpha\}}=\pm 1$ determines the sign of $\bar{\bar{N}}^{\{\alpha\}}$, and is chosen to ensure that $\bar{\bar{n}}^{\{\alpha\}}_{a}$ is the outward directed unit normal on the $A\{\alpha\}$ face. The extrinsic curvature $\bar{\bar{K}}^{\{\alpha\}}_{ab}$ of this surface is given by

where $\bar{\bar{\nabla}}_{c}$ is the $\bar{\bar{g}}_{ab}$ metric-compatible covariant derivative, and $\bar{\bar{P}}^{\{\alpha\}\,c}_{a}$ is the projection tensor $\bar{\bar{P}}^{\{\alpha\}\,c}_{a}=\delta_{a}^{c}-\bar{\bar{n}}^{\{\alpha\}}_{a}\bar{\bar{n}}^{\{\alpha\}\,c}$. Note that the term proportional to $\partial_{c}\bar{\bar{n}}^{\{\alpha\}}_{d}=\partial_{c}\bar{\bar{N}}^{\{\alpha\}}\partial_{d}x^{\alpha}$ in Eq. (26) vanishes identically because $\bar{\bar{P}}^{\{\alpha\}\,d}_{b}\partial_{d}x^{\alpha}=0$.

We define the difference between the $\bar{\bar{g}}_{ab}$ and the $\bar{g}_{ab}$ metrics, $\delta\bar{g}_{ab}$, and the associated differences between extrinsic curvatures, $\delta\bar{K}^{\{\alpha\}}_{ab}$:

Note that these differences are not necessarily infinitesimal. To ensure that the intrinsic parts of $\bar{\bar{g}}_{ab}$ are identical to those of $\bar{g}_{ab}$, we choose $\delta\bar{g}_{ab}$ to be a smooth tensor in the interior of each cubic region that satisfies,

on each cube face $A\{\alpha\}$. Note that the projection tensor $\bar{\bar{P}}_{a}^{\{\alpha\}\,c}$ is identical to $\bar{P}_{a}^{\{\alpha\}\,c}$ on $A\{\alpha\}$ because $\bar{\bar{P}}_{a}^{\{\alpha\}\,c}\partial_{c}x^{\alpha}=\bar{P}_{a}^{\{\alpha\}\,c}\partial_{c}x^{\alpha}=0$. Therefore the metric continuity condition on $\delta\bar{g}_{ab}$ is equivalent to

which is easier to enforce since the metric $\bar{g}_{ab}$, and consequently the normal vector $\bar{n}^{\{\alpha\}}_{a}$, is already known. The condition that the extrinsic curvature $\bar{\bar{K}}^{\{\alpha\}}_{ab}$ vanish along each edge of the $A\{\alpha\}$ face can be expressed as the following condition on $\delta\bar{K}^{\{\alpha\}}_{ab}$,

on each edge of each cube face $A\{\alpha\}$.

To determine exactly what restrictions are placed on $\delta\bar{g}_{ab}$ by the intrinsic metric and extrinsic curvature continuity conditions, Eqs. (31)–(32), we examine those conditions expressed in the Cartesian coordinates $\{x^{\alpha},x^{\beta},x^{\gamma}\}$ of region $\mathcal{B}_{A}$. The $A\{\alpha\}$ face of this region is defined by an $x^{\alpha}$=constant surface, while the $x^{\beta}$ and $x^{\gamma}$ coordinates label points on that face. In these coordinates the intrinsic metric continuity condition, Eq. (31), implies that all the $x^{\beta}$ and $x^{\gamma}$ components of the metric perturbation vanish everywhere on that face: $\delta\bar{g}_{\beta\beta}=\delta\bar{g}_{\beta\gamma}=\delta\bar{g}_{\gamma\gamma}=0$. Similarly on the adjacent $A\{\beta\}$ face, all the $x^{\alpha}$ and $x^{\gamma}$ components of the metric perturbation vanish $\delta\bar{g}_{\alpha\alpha}=\delta\bar{g}_{\alpha\gamma}=\delta\bar{g}_{\gamma\gamma}=0$. It follows that all the components of $\delta\bar{g}_{ab}$ except $\delta\bar{g}_{\alpha\beta}$ must vanish along the $A\{\alpha\beta\}$ edge: $\delta\bar{g}_{\alpha\alpha}=\delta\bar{g}_{\alpha\gamma}=\delta\bar{g}_{\beta\beta}=\delta\bar{g}_{\beta\gamma}=\delta\bar{g}_{\gamma\gamma}=0$.

The $C^{\,0}$ metric $\bar{g}_{ab}$ was constructed so that the dihedral angle between the $A\{\alpha\}$ and $A\{\beta\}$ faces, $\cos\psi_{\{A\alpha\beta\}}=-\bar{g}^{\alpha\beta}/\sqrt{\bar{g}^{\alpha\alpha}\bar{g}^{\beta\beta}}$, is constant along the edge between these faces. This was done to ensure there is no conical singularity along this edge. To ensure that the $\bar{\bar{g}}_{ab}$ metric has no conical singularity there, we keep this dihedral angle fixed in this metric along this edge as well. This can be done by imposing the additional constraint $\delta\bar{g}_{\alpha\beta}=0$ along this edge. This makes all the components $\delta\bar{g}_{ab}=0$, and consequently $\delta\bar{g}^{ab}=0$ which keeps the dihedral angles fixed. Thus the intrinsic metric continuity conditions, along with the conditions to ensure there are no conical singularities along the edges, require that all the components of the metric perturbation vanish along each edge: $\delta\bar{g}_{ab}=0$.