# Wavelets Beyond Admissibility

Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds LS2 9JT,
UK
<EMAIL_ADDRESS>
###### Abstract

The purpose of this paper is to articulate an observation that many
interesting type of _wavelets_ (or _coherent states_) arise from group
representations which _are not_ square integrable or vacuum vectors which _are
not_ admissible.

###### keywords:

Wavelets, coherent states, group representations, Hardy space, functional
calculus, Berezin calculus, Radon transform, Möbius map, maximal function,
affine group, special linear group, numerical range.

##  1 Covariant Transform

A general group-theoretical construction [1, 2, 3, 4, 5, 6] of _wavelets_ (or
_coherent states_) starts from an square integrable (_s.i._) representation.
However, such a setup is restrictive and is not necessary, in fact.

###### Definition 1

Let ${\rho}$ be a representation of a group $G$ in a space $V$ and $F$ be an
operator from $V$ to a space $U$. We define a _covariant transform_
$\mathcal{W}$ from $V$ to the space $L{}(G,U)$ of $U$-valued functions on $G$
by the formula:

$\mathcal{W}:v\mapsto\hat{v}(g)=F({\rho}(g^{-1})v),\qquad v\in V,\ g\in G.$
(1)

###### Remark 1

We do not require that operator $F$ shall be linear.

###### Remark 2

Usefulness of the covariant transform is in the reverse proportion to the
dimensionality of the space $U$. The covariant transform encodes properties of
$v$ in a function $\mathcal{W}v$ on $G$. For a low dimensional $U$ this
function can be ultimately investigated by means of harmonic analysis. Thus
$\dim U=1$ is the ideal case, however, it is unattainable sometimes, see Ex.
2.4 below.

###### Theorem 1

The covariant transform $\mathcal{W}$ (1) intertwines ${\rho}$ and the left
regular representation $\Lambda$ on $L{}(G,U)$:

$\Lambda(g):f(h)\mapsto f(g^{-1}h).$ (2)

######  Proof 1.1.

We have a calculation similar to wavelet transform [3, Prop. 2.6]:

$[\mathcal{W}({\rho}(g)v)](h)=F({\rho}(h^{-1}){\rho}(g)v)=[\mathcal{W}v](g^{-1}h)=\Lambda(g)[\mathcal{W}v](h).$

######  Corollary 2.

The image space $\mathcal{W}(V)$ is invariant under the left shifts on $G$.

##  2 Examples of Covariant Transform

######  Example 2.1.

Let $V$ be a Hilbert space with an inner product
$\left\langle\cdot,\cdot\right\rangle$ and ${\rho}$ be a unitary
representation. Let $F:V\rightarrow\mathbb{C}{}$ be a functional
$v\mapsto\left\langle v,v_{0}\right\rangle$ defined by a vector $v_{0}\in V$.
Then the transformation (1) is the well-known expression for a _wavelet
transform_ [4, (7.48)] (or _representation coefficients_):

$\mathcal{W}:v\mapsto\hat{v}(g)=\left\langle{\rho}(g^{-1})v,v_{0}\right\rangle=\left\langle
v,{\rho}(g)v_{0}\right\rangle,\qquad v\in V,\ g\in G.$ (3)

The family of vectors $v_{g}={\rho}(g)v_{0}$ is called _wavelets_ or _coherent
states_. In this case we obtain scalar valued functions on $G$, thus the
fundamental rôle of this example is explained in Rem. 2.

This scheme is typically carried out for a s.i. representation ${\rho}$ and
$v_{0}$ being an admissible vector[1, 2, 4, 5, 6]. In this case the wavelet
(covariant) transform is a map into the s.i. functions [7] with respect to the
left Haar measure.

However s.i. representations and admissible vectors does not cover all
interesting cases.

######  Example 2.2.

Let $G$ be the “$ax+b$” (or _affine_) group [4, § 8.2]: the set of points
$(a,b)$, $a\in\mathbb{R}_{+}{}$, $b\in\mathbb{R}{}$ in the upper half-plane
with the group law:

$(a,b)*(a^{\prime},b^{\prime})=(aa^{\prime},ab^{\prime}+b)$ (4)

and left invariant measure $a^{-2}\,da\,db$. Its isometric representation on
$V=L_{p}{}(\mathbb{R}{})$ is given by the formula:

$[{\rho_{p}}(a,b)\,f](x)=a^{\frac{1}{p}}f\left(ax+b\right).$ (5)

We consider the operators
$F_{\pm}:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ defined by:

$F_{\pm}(f)=\frac{1}{2\pi
i}\int_{\mathbb{R}{}}\frac{f(t)\,dt}{t\mp\mathrm{i}}.$ (6)

Then the covariant transform (1) is the Cauchy integral from
$L_{2}{}(\mathbb{R}{})$ to the Hardy space in the upper/lower half-plane
$H_{2}{}(\mathbb{R}^{2}_{\pm}{})$. Although the representation (5) is s.i. for
$p=2$, the function $\frac{1}{t\pm\mathrm{i}}$ is not an admissible vacuum
vector. Thus the complex analysis become decoupled from the traditional
wavelets theory. As a result the application of wavelet theory shall relay on
an extraneous mother wavelets [8].

However many important objects in complex analysis are generated by
inadmissible mother wavelets like (6). For example, if
$F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}{}$ is defined by $F:f\mapsto
F_{+}f+F_{-}f$ then the covariant transform (1) is simply the _Poisson
integral_. If $F:L_{2}{}(\mathbb{R}{})\rightarrow\mathbb{C}^{2}{}$ is defined
by $F:f\mapsto(F_{+}f,F_{-}f)$ then the covariant transform (1) represents a
function on the real line as a jump between functions analytic in the upper
and the lower half-planes. This makes a decomposition of
$L_{2}{}(\mathbb{R}{})$ into irreducible components of the representation (5).
Another interesting but non-admissible vector is the Gaussian $e^{-x^{2}}$.

######  Example 2.3.

For the group $G=SL_{2}{}(\mathbb{R}{})$ [14] let us consider the unitary
representation ${\rho}$ on the space of s.i. function
$L_{2}{}(\mathbb{R}^{2}_{+}{})$ on the upper half-plane through the Möbius
transformations:

${\rho}(g):f(z)\mapsto\frac{1}{(cz+d)^{2}}\,f\left(\frac{az+b}{cz+d}\right),\qquad
g^{-1}=\ \begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

Let $F_{i}$ be the functional
$L_{2}{}(\mathbb{R}^{2}_{+}{})\rightarrow\mathbb{C}{}$ of pairing with the
lowest/highest $i$-weight vector in the corresponding irreducible component of
the discrete series [14, Ch. VI]. Then we can build an operator $F$ from
various $F_{i}$ similarly to the previous example, e.g. this generalises the
representation of an s.i. function as a sum of analytic ones from different
irreducible subspaces.

Covariant transform is also meaningful for principal and complementary series
of representations of the group $SL_{2}{}(\mathbb{R}{})$[9], which are not
s.i.

######  Example 2.4.

A straightforward generalisation of Ex.2.1 is obtained if $V$ is a Banach
space and $F:V\rightarrow\mathbb{C}{}$ is an element of $V^{*}$. Then the
covariant transform coincides with the construction of wavelets in Banach
spaces [3].

The next stage of generalisation is achieved if $V$ is a Banach space and
$F:V\rightarrow\mathbb{C}^{n}{}$ be a linear operator. Then the corresponding
covariant transform is a map $\mathcal{W}:V\rightarrow
L{}(G,\mathbb{C}^{n}{})$. This is closely related to M.G. Krein’s works on
_directing functionals_ [10], see also _multiresolution wavelet analysis_
[11], Clifford-valued Bargmann spaces [12] and [4, Thm. 7.3.1].

######  Example 2.5.

A step in a different direction is a consideration of non-linear operators.
Take again the “$ax+b$” group and its representation (5). We define $F$ to be
a homogeneous but non-linear functional $V\rightarrow\mathbb{R}_{+}{}$:

$F(f)=\frac{1}{2}\int\limits_{-1}^{1}\left|f(x)\right|\,dx.$

The covariant transform (1) becomes:

$\displaystyle[\mathcal{W}_{p}f](a,b)=\frac{1}{2}\int\limits_{-1}^{1}\left|a^{\frac{1}{p}}f\left(ax+b\right)\right|\,dx=a^{\frac{1}{p}}\frac{1}{2a}\int\limits^{b+a}_{b-a}\left|f\left(x\right)\right|\,dx.$

Obviously $M_{f}(b)=\max_{a}[\mathcal{W}_{\infty}f](a,b)$ coincides with the
Hardy _maximal function_ , which contains important information on the
original function $f$. However, the full covariant transform is even more
detailed. For example,
$\left\|f\right\|=\max_{b}[\mathcal{W}_{\infty}f](\frac{1}{2},b)$ is the shift
invariant norm [13].

From the Cor. 2 we deduce that the operator $M:f\mapsto M_{f}$ intertwines
${\rho_{p}}$ with itself ${\rho_{p}}M=M{\rho_{p}}$.

######  Example 2.6.

Let $V=L_{c}{}(\mathbb{R}^{2}{})$ be the space of compactly supported bounded
functions on the plane. We take $F$ be the linear operator
$V\rightarrow\mathbb{C}{}$ of integration over the real line:

$F:f(x,y)\mapsto F(f)=\int_{\mathbb{R}{}}f(x,0)\,dx.$

Let $G$ be the group of Euclidean motions of the plane represented by ${\rho}$
on $V$ by a change of variables. Then the wavelet transform $F({\rho}(g)f)$ is
the _Radon transform_.

######  Example 2.7.

Let a representation ${\rho}$ of a group $G$ act on a space $X$. Then there is
an associated representation ${\rho_{B}}$ of $G$ on a space $V=B{}(X,Y)$ of
linear operators $X\rightarrow Y$ defined by the identity:

$({\rho_{B}}(g)A)x=A({\rho}(g)x),\qquad x\in X,\ g\in G,\ A\in B{}(X,Y).$

Following the Remark 2 we take $F$ to be a functional
$V\rightarrow\mathbb{C}{}$, for example $F$ can be defined from a pair $x\in
X$, $l\in Y^{*}$ by the expression $F:A\mapsto\left\langle Ax,l\right\rangle$.
Then the covariant transform:

$\mathcal{W}:A\mapsto\hat{A}(g)=F({\rho_{B}}(g)A),\qquad$

this is an example of _covariant calculus_ [3, 15].

######  Example 2.8.

A modification of the previous construction is obtained if we have two groups
$G_{1}$ and $G_{2}$ represented by ${\rho_{1}}$ and ${\rho_{2}}$ on $X$ and
$Y^{*}$ respectively. Then we have a covariant transform $B{}(X,Y)\rightarrow
L{}(G_{1}\times G_{2},\mathbb{C}{})$ defined by the formula:

$\mathcal{W}:A\mapsto\hat{A}(g_{1},g_{2})=\left\langle
A{\rho_{1}}(g_{1})x,{\rho_{2}}(g_{2})l\right\rangle.$

This generalises _Berezin functional calculi_ [3].

######  Example 2.9.

Let us restrict the previous example to the case when $X=Y$ is a Hilbert
space, ${\rho_{1}}{}={\rho_{2}}{}={\rho}$ and $x=l$ with $\left\|x\right\|=1$.
Than the range of the covariant transform:

$\mathcal{W}:A\mapsto\hat{A}(g)=\left\langle
A{\rho}(g)x,{\rho}(g)x\right\rangle$

is a subset of the _numerical range_ of the operator $A$.

######  Example 2.10.

The group $SL_{2}{}(\mathbb{R}{})$ consists of $2\times 2$ matrices of the
form $\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}$
with the unit determinant [14, § IX.1]. Let $A$ be an operator with the
spectral radius less than $1$. Then the associated Möbius transformation

$g:A\mapsto g\cdot A=\frac{\alpha A+\beta
I}{\bar{\beta}A+\bar{\alpha}I},\qquad\text{where}\quad
g^{-1}=\begin{pmatrix}\alpha&\beta\\\ \bar{\beta}&\bar{\alpha}\end{pmatrix}\in
SL_{2}{}(\mathbb{R}{}),\ $

produces a well-defined operator with the spectral radius less than $1$ as
well. Thus we have a representation of $SL_{2}{}(\mathbb{R}{})$. A choise of
an operator $F$ will define the corresponding covariant transform. In this way
we obtain generalisations of _Riesz–Dunford functional calculus_ [15].

##  3 Inverse Covariant Transform

An object invariant under the left action $\Lambda$ (2) is called _left
invariant_. For example, let $L$ and $L^{\prime}$ be two left invariant spaces
of functions on $G$. We say that a pairing
$\left\langle\cdot,\cdot\right\rangle:L\times
L^{\prime}\rightarrow\mathbb{C}{}$ is _left invariant_ if

$\left\langle\Lambda(g)f,\Lambda(g)f^{\prime}\right\rangle=\left\langle
f,f^{\prime}\right\rangle,\quad\textrm{ for all }\quad f\in L,\ f^{\prime}\in
L^{\prime}.$ (7)

######  Remark 3.

  1. 1.

We do not require the pairing to be linear in general.

  2. 2.

If the pairing is invariant on space $L\times L^{\prime}$ it is not
necessarily invariant (or even defined) on the whole $C{}(G)\times C{}(G)$.

  3. 3.

In a more general setting we shall study an invariant pairing on a homogeneous
spaces instead of the group. However due to length constraints we cannot
consider it here beyond the Example 3.2.

  4. 4.

An invariant pairing on $G$ can be obtained from an invariant functional $l$
by the formula $\left\langle f_{1},f_{2}\right\rangle=l(f_{1}\bar{f}_{2})$.

For a representation ${\rho}$ of $G$ in $V$ and $v_{0}\in V$ we fix a function
$w(g)={\rho}(g)v_{0}$. We assume that the pairing can be extended in its
second component to this $V$-valued functions, say, in the weak sense.

######  Definition 4.

Let $\left\langle\cdot,\cdot\right\rangle$ be a left invariant pairing on
$L\times L^{\prime}$ as above, let ${\rho}$ be a representation of $G$ in a
space $V$, we define the function $w(g)={\rho}(g)v_{0}$ for $v_{0}\in V$. The
_inverse covariant transform_ $\mathcal{M}$ is a map $L\rightarrow V$ defined
by the pairing:

$\mathcal{M}:f\mapsto\left\langle f,w\right\rangle,\qquad\text{ where }f\in
L.$ (8)

######  Example 3.1.

Let $G$ be a group with a unitary s.i. representation $\rho$. An invariant
pairing of two s.i. functions is obviously done by the integration over the
Haar measure:

$\left\langle f_{1},f_{2}\right\rangle=\int_{G}f_{1}(g)\bar{f}_{2}(g)\,dg.$

For an admissible vector $v_{0}$ [7], [4, Chap. 8] the inverse covariant
transform is known in this setup as _reconstruction formula_.

######  Example 3.2.

Let $\rho$ be a s.i. representation of $G$ modulo a subgroup $H\subset G$ and
let $X=G/H$ be the corresponding homogeneous space with a quasi-invariant
measure $dx$. Then integration over $dx$ with an appropriate weight produces
an invariant pairing. The inverse covariant transform is a more general
version [4, (7.52)] of the _reconstruction formula_ mentioned in the previous
example.

Let $\rho$ be not a s.i. representation (even modulo a subgroup) or let
$v_{0}$ be inadmissible vector of a s.i. representation $\rho$. An invariant
pairing in this case is not associated with an integration over any non
singular invariant measure on $G$. In this case we have a _Hardy pairing_. The
following example explains the name.

######  Example 3.3.

Let $G$ be the “$ax+b$” group and its representation ${\rho}$ (5) from Ex.
2.2. An invariant pairing on $G$, which is not generated by the Haar measure
$a^{-2}da\,db$, is:

$\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b)\,\bar{f}_{2}(a,b)\,db.$ (9)

For this pairing we can consider functions $\frac{1}{2\pi i(x+i)}$ or
$e^{-x^{2}}$, which are not admissible vectors in the sense of s.i.
representations. Then the inverse covariant transform provides an _integral
resolutions_ of the identity.

Similar pairings can be defined for other semi-direct products of two groups.
We can also extend a Hardy pairing to a group, which has a subgroup with such
a pairing.

######  Example 3.4.

Let $G$ be the group $SL_{2}{}(\mathbb{R}{})$ from the Ex. 2.3. Then the
“$ax+b$” group is a subgroup of $SL_{2}{}(\mathbb{R}{})$, moreover we can
parametrise $SL_{2}{}(\mathbb{R}{})$ by triples $(a,b,\theta)$,
$\theta\in(-\pi,\pi]$ with the respective Haar measure [14, III.1(3)]. Then
the Hardy pairing

$\left\langle f_{1},f_{2}\right\rangle=\lim_{a\rightarrow
0}\int\limits_{-\infty}^{\infty}f_{1}(a,b,\theta)\,\bar{f}_{2}(a,b,\theta)\,db\,d\theta.$
(10)

is invariant on $SL_{2}{}(\mathbb{R}{})$ as well. The corresponding inverse
covariant transform provides even a finer resolution of the identity which is
invariant under conformal mappings of the Lobachevsky half-plane.

A further study of covariant transform and its inverse shall be continued
elsewhere.

## References

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# Rigidity of generalized Thurston’s sphere packings on 3-dimensional
manifolds with boundary

Xu Xu, Chao Zheng  School of Mathematics and Statistics, Wuhan University,
Wuhan, 430072, P.R. China<EMAIL_ADDRESS>School of Mathematics and
Statistics, Wuhan University, Wuhan 430072, P.R. China<EMAIL_ADDRESS>
###### Abstract.

Motivated by Guo-Luo’s generalized circle packings on surfaces with boundary
[7], we introduce the generalized Thurston’s sphere packings on 3-dimensional
manifolds with boundary. Then we investigate the rigidity of the generalized
Thurston’s sphere packings. We prove that the generalized Thurston’s sphere
packings are locally determined by the combinatorial scalar curvatures. We
further prove the infinitesimal rigidity that the generalized Thurston’s
sphere packings can not be deformed while keeping the combinatorial Ricci
curvatures fixed.

###### Key words and phrases:

Generalized Thurston’s sphere packings; Rigidity; 3-manifolds with boundary

##  1\. Introduction

This is a continuation of [15], in which we investigated the rigidity and
deformation of the generalized tangential sphere packings on 3-dimensional
manifolds with boundary. In this paper, motivated by Guo-Luo’s generalized
circle packings on surfaces with boundary [7], we introduce the generalized
Thurston’s sphere packings on 3-dimensional manifolds with boundary. These are
analogues of Thurston’s Euclidean sphere packings on 3-dimensional closed
manifolds studied in [3, 6, 8] and natural generalizations of the generalized
tangential sphere packings on 3-dimensional manifolds with boundary studied in
[15]. The main focus of this paper is to study the rigidity of the generalized
Thurston’s sphere packings on 3-dimensional manifolds with boundary.

Suppose $\widetilde{\Sigma}$ is a compact 3-dimensional manifolds with the
boundary $\partial\widetilde{\Sigma}$ consisting of $N$ connected components.
By coning off each boundary component of $\widetilde{\Sigma}$ to be a point,
one can obtain a compact 3-dimensional space, denoted by $\Sigma$. There are
exactly $N$ cone points $\\{v_{1},...,v_{N}\\}$ in $\Sigma$ and
$\Sigma-\\{v_{1},...,v_{N}\\}$ is homeomorphic to
$\widetilde{\Sigma}-\partial\widetilde{\Sigma}$. An ideal triangulation
$\mathcal{T}$ of $\widetilde{\Sigma}$ is a triangulation $\mathcal{T}$ of
$\Sigma$ such that the vertices of the triangulation are exactly the cone
points $\\{v_{1},...,v_{N}\\}$. Then $(\Sigma,\mathcal{T})$ is called a closed
3-dimensional manifold with ideal triangulation. Replacing each tetrahedron in
$\mathcal{T}$ by a hyper-ideal tetrahedron and replacing the affine gluing
homeomorphisms by isometries preserving the corresponding hyperbolic hexagonal
faces, we obtain a hyper-ideal polyhedral metric on $(\Sigma,\mathcal{T})$. We
call $(\Sigma,\mathcal{T})$ an ideally triangulated compact 3-dimensional
manifold with boundary.

A hyper-ideal tetrahedron is a compact convex polyhedron in $\mathbb{H}^{3}$
that is diffeomorphic to a truncated tetrahedron in $\mathbb{E}^{3}$, which
has four right-angled hyperbolic hexagonal faces and four hyperbolic
triangular faces. Any triangular face is required to be orthogonal to its
three adjacent hexagonal faces. The four triangular faces isometric to
hyperbolic triangles are called vertex triangles. An edge in a hyper-ideal
tetrahedron is the intersection of two hexagonal faces and a vertex edge is
the intersection of a hexagonal face and a vertex triangle. We use $E$, $F$
and $T$ to represent the set of edges, faces and hyper-ideal tetrahedra in the
triangulation $\mathcal{T}$ respectively. A hyper-ideal tetrahedron with four
vertex triangles $\triangle_{\nu},\nu=i,j,k,h$ is denoted by
$\sigma=\\{ijkh\\}$. Using the Klein model of $\mathbb{H}^{3}$, the hyper-
ideal tetrahedron $\sigma=\\{ijkh\\}$ corresponds to a Euclidean tetrahedron
with four vertices $i,j,k,h$. We call these vertices as hyper-ideal vertices
of the hyper-ideal tetrahedron $\sigma=\\{ijkh\\}$, which corresponds to
vertex triangles $\triangle_{\nu},\nu=i,j,k,h$. The set of all hyper-ideal
vertices in the hyper-ideal tetrahedra is denoted by $V$. The edge joining
$\triangle_{i}$ to $\triangle_{j}$ is denoted by $\\{ij\\}\in E$. The
hexagonal face adjacent to $\\{ij\\}$, $\\{jk\\}$ and $\\{ik\\}$ is denoted by
$\\{ijk\\}\in F$. The length of the edge $\\{ij\\}\in E$ is denoted by
$l_{ij}$. The length of the vertex edge $\triangle_{i}\bigcap\\{ijk\\}$ is
denoted by $x^{i}_{jk}$. Please refer to Figure 1.

\begin{overpic}[scale={1}]{Figure_2.pdf} \put(100.0,42.0){$\triangle_{h}$}
\put(48.0,-5.0){$\triangle_{k}$} \put(-10.0,45.0){$\triangle_{j}$}
\put(60.0,96.0){$\triangle_{i}$} \put(48.0,97.0){$x^{i}_{jh}$}
\put(56.0,80.0){$x^{i}_{kh}$} \put(37.0,80.0){$x^{i}_{jk}$}
\put(18.0,72.0){$l_{ij}$} \put(55.0,60.0){$l_{ik}$} \put(16.0,20.0){$l_{jk}$}
\put(80.0,72.0){$l_{ih}$} \put(37.0,40.0){$l_{jh}$} \put(82.0,19.0){$l_{kh}$}
\end{overpic} Figure 1. Hyper-ideal tetrahedron $\sigma=\\{ijkh\\}$

######  Definition 1.1.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary. Let $\Phi:E\rightarrow[0,\frac{\pi}{2}]$
be a weight defined on the edges of $(\Sigma,\mathcal{T})$. A generalized
Thurston’s sphere packing metric on $(\Sigma,\mathcal{T},\Phi)$ is defined to
be a map $r:V\rightarrow(0,+\infty)$ such that

(1):

The length $l_{ij}$ of the edge $\\{ij\\}\in E$ between two vertex triangles
$\triangle_{i}$ and $\triangle_{j}$ is given by

(1) $l_{ij}=\cosh^{-1}(\sinh r_{i}\sinh r_{j}+\cos\Phi_{ij}\cosh r_{i}\cosh
r_{j}).$

(2):

The lengths $l_{ij},l_{ik},l_{ih},l_{jk},l_{jh},l_{kh}$ determine a hyper-
ideal tetrahedron $\\{ijkh\\}$.

The function $l:E\rightarrow(0,+\infty)$ determined by (1) is a discrete
hyperbolic metric on $(\Sigma,\mathcal{T})$. The condition (2) in Definition
1.1 is called the non-degenerate condition. Intuitively, a degenerate hyper-
ideal tetrahedron is a collapsed octagon, which was called a flat hyper-ideal
tetrahedron by Luo-Yang [10]. Furthermore, Luo-Yang [10] showed that a hyper-
ideal tetrahedron is non-degenerate if and only if the four vertex triangles
are non-degenerate, i.e., triangle inequalities hold for any vertex triangle.
Please see Section 2 or [10] for more details.

If $\Phi\equiv 0$, then the generalized Thurston’s sphere packing metric is
exactly the generalized tangential sphere packing metric introduced in [15].
We give a geometric explanation of the generalized Thurston’s sphere packing
metric. Given a generalized Thurston’s sphere packing metric $r$ on
$(\Sigma,\mathcal{T},\Phi)$, a sphere $B_{i}$ of radius $r_{i}$ is attached to
each hyper-ideal vertex $i\in V$, and the intersection angle of $B_{i}$ and
$B_{j}$ attached to the hyper-ideal vertices $i,j\in V$ is $\Phi_{ij}$. Then
the length $l_{ij}$ of the edge $\\{ij\\}$ is a function of $r_{i},\ r_{j}$
and the fixed $\Phi_{ij}$ defined by (1) by the cosine law for hyperbolic
pentagons. Please refer to Figure 2.

\begin{overpic}[scale={1}]{Figure_1.pdf} \put(60.0,32.0){$r_{j}$}
\put(30.0,32.0){$r_{i}$} \put(0.0,0.0){$i$} \put(98.0,0.0){$j$}
\put(45.0,-2.0){$l_{ij}$} \put(50.0,46.0){$\Phi_{ij}$}
\put(17.0,6.0){$\theta_{ij}$} \put(67.0,7.0){$\theta_{ji}$} \end{overpic}
Figure 2. Hyperbolic pentagon

For each $\sigma\in T$, the generalized Thurston’s sphere packing metric
determines a non-degenerate hyper-ideal tetrahedron by Definition 1.1. Gluing
these hyper-ideal tetrahedra along the faces by isometries may produce
singularities. We use the following combinatorial Ricci curvature to describe
the singularities along the edges.

######  Definition 1.2 ([6]).

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary and $l:E\rightarrow(0,+\infty)$ is a
discrete hyperbolic metric on $(\Sigma,\mathcal{T})$. The combinatorial Ricci
curvature of $l$ along the edge $\\{ij\\}\in E$ is defined to be

(2) $K_{ij}=2\pi-\sum_{\\{ijkh\\}\in T}\beta_{ij,kh},$

where $\beta_{ij,kh}$ is the dihedral angle of the hyper-ideal tetrahedron
$\\{ijkh\\}$ at the edge $\\{ij\\}\in E$ and the summation is taken over the
hyper-ideal tetrahedra with $\\{ij\\}$ as a common edge.

We have the following infinitesimal rigidity for the generalized Thurston’s
sphere packing metrics.

######  Theorem 1.3.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary. Let
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$ be a weight defined on the
edges of $(\Sigma,\mathcal{T})$. Then the generalized Thurston’s sphere
packing metric on $(\Sigma,\mathcal{T},\Phi)$ can not be deformed while
keeping the combinatorial Ricci curvatures along the edges fixed.

Similar to the combinatorial scalar curvature for Thurston’s Euclidean sphere
packing metrics on 3-dimensional closed manifolds [6, 8], we introduce the
following combinatorial scalar curvature for the generalized Thurston’s sphere
packing metrics to describe the singularities at the hyper-ideal vertices.

######  Definition 1.4.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary. Let $\Phi:E\rightarrow[0,\frac{\pi}{2}]$
be a weight defined on the edges of $(\Sigma,\mathcal{T})$. The combinatorial
scalar curvature at a hyper-ideal vertex $i\in V$ for a generalized Thurston’s
sphere packing metric $r$ is defined to be

$K_{i}=\sum_{j;j\sim i}K_{ij}\cosh\theta_{ij},$

where $\theta_{ij}$ is the edge length opposite to the edge with length
$r_{j}$ in the hyperbolic pentagon formed by three edges with lengths $r_{i},\
r_{j},\ l_{ij}$ as shown in Figure 2.

######  Remark 1.5.

If $\Phi_{ij}=0$ for any edge $\\{ij\\}\in E$, then $\cosh\theta_{ij}=1$ and
hence

$K_{i}=\sum_{j;j\sim i}\bigg{(}2\pi-\sum_{\\{ijkh\\}\in
T}\beta_{ij,kh}\bigg{)}=2\pi\chi(\Sigma_{i})+\mathrm{Area}(\Sigma_{i}),$

where $\Sigma_{i}$ is a connected component of the boundary $\partial\Sigma$
and $\chi(\Sigma_{i})$ is the Euler number of $\Sigma_{i}$. This is the
combinatorial scalar curvature for the generalized tangential sphere packing
metrics introduced in [15].

We have the following local rigidity for the generalized Thurston’s sphere
packing metrics.

######  Theorem 1.6.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary. Let
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$ be a weight defined on the
edges of $(\Sigma,\mathcal{T})$. If $\overline{r}$ is a non-degenerate
generalized Thurston’s sphere packing metric on $(\Sigma,\mathcal{T},\Phi)$
with $K_{ij}(\overline{r})\sin^{2}\Phi_{ij}\geq 0$ for any edge $\\{ij\\}\in
E$, then there exists a neighborhood $U$ of $\overline{r}$ such that the
generalized Thurston’s sphere packing metric in $U$ is determined by its
combinatorial scalar curvature.

######  Remark 1.7.

As mentioned in [8], there exists a subtle difference between the notion of
the infinitesimal rigidity in Theorem 1.3 and the notion of the local rigidity
in Theorem 1.6. In fact, the local rigidity implies the infinitesimal
rigidity, but the reverse is not true due to dimensional differences. Note
that the condition $K_{ij}\sin^{2}\Phi_{ij}\geq 0$ in Theorem 1.6 includes the
case that $K_{ij}(r)<0$ with $\Phi_{ij}=0$ for $\\{ij\\}\in E$. Specially, if
$\Phi\equiv 0$, then Theorem 1.6 is reduced to the local rigidity of the
generalized tangential sphere packing. And the global rigidity of the
generalized tangential sphere packings was proved in [15].

As a direct consequence of Theorem 1.6, we have the following corollary.

######  Corollary 1.8.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary. Let
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$ be a weight defined on the
edges of $(\Sigma,\mathcal{T})$. If $\overline{r}$ is a non-degenerate
generalized Thurston’s sphere packing metric on $(\Sigma,\mathcal{T},\Phi)$
with $K_{ij}(\overline{r})\geq 0$ for any edge $\\{ij\\}\in E$, then there
exists a neighborhood $U$ of $\overline{r}$ such that the generalized
Thurston’s sphere packing metric in $U$ is determined by its combinatorial
scalar curvature.

The paper is organized as follows. In Section 2, we prove the simply
connectedness of the admissible space of the generalized Thurston’s sphere
packing metrics for a truncated tetrahedron, i.e., Theorem 2.10. In Section 3,
we prove Theorem 1.3 and Theorem 1.6.

Acknowledgements
The first author thanks Professor Tian Yang at Texas A&M University for
helpful communications.

##  2\. The admissible space of generalized Thurston’s sphere packing metrics
on a truncated tetrahedron

In this section, we give an explicit characterization of the admissible space
of generalized Thurston’s sphere packing metrics on a truncated tetrahedron by
solving the global version of triangle inequalities for the vertex triangles
of a truncated tetrahedron. The admissible space is proved to be homotopy
equivalent to $\mathbb{R}^{4}_{>0}$ and therefore simply connected. The
approach is taken from He-Xu’s work on Thurston’s sphere packing metrics for a
Euclidean tetrahedron [8]. Similar approaches are also used in [12, 13, 14]
and others.

Suppose $\sigma=\\{ijkh\\}$ is a truncated tetrahedron with four vertex
triangles $\triangle_{\nu},\nu=i,j,k,h$ as shown in Figure 1. The admissible
space $\Omega_{ijkh}$ of the generalized Thurston’s sphere packing metrics on
$\sigma=\\{ijkh\\}$ is the set of
$(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}^{4}_{>0}$ such that the hyper-ideal
tetrahedron with edge lengths $l_{ij}$, $l_{ik}$, $l_{ih}$, $l_{jk}$,
$l_{jh}$, $l_{kh}$ is non-degenerate. For a non-degenerate hyper-ideal
tetrahedron, Luo-Yang [10] gave the following characterization.

######  Proposition 2.1.

([10], Proposition 4.4) The truncated tetrahedron $\sigma=\\{ijkh\\}$ with
edge lengths
$(l_{ij},l_{ik},l_{ih},l_{jk},l_{jh},l_{kh})\in\mathbb{R}^{6}_{>0}$ determines
a non-degenerate hyper-ideal tetrahedron if and only if the four vertex
triangles $\triangle_{\nu},\nu=i,j,k,h,$ are non-degenerate, i.e., the
triangle inequalities hold for any vertex triangle $\triangle_{\nu}$.

We first consider the vertex triangle $\triangle_{i}$ with edge lengths
$x^{i}_{jk},x^{i}_{jh},x^{i}_{kh}$. To simplify the notations, we set

$\displaystyle\sinh l_{ij}=s_{ij},\ \cosh l_{ij}=c_{ij},\ \tanh r_{i}=t_{i},\
$ $\displaystyle\cos\Phi_{ij}=a,\ \cos\Phi_{ik}=b,\ \cos\Phi_{ih}=c,\ $
$\displaystyle\cos\Phi_{jk}=d,\ \cos\Phi_{jh}=e,\ \cos\Phi_{kh}=f.$

Some of the following formulas have been calculated in [15]. For completeness,
we include them here. By the cosine law for a right-angled hyperbolic hexagon,
we have

(3) $\displaystyle\cosh x^{i}_{jk}=$ $\displaystyle\frac{\cosh l_{ij}\cosh
l_{ik}+\cosh l_{jk}}{\sinh l_{ij}\sinh
l_{ik}}=\frac{c_{ij}c_{ik}+c_{jk}}{s_{ij}s_{ik}},$ $\displaystyle\cosh
x^{i}_{jh}=$ $\displaystyle\frac{\cosh l_{ij}\cosh l_{ih}+\cosh l_{jh}}{\sinh
l_{ij}\sinh l_{ih}}=\frac{c_{ij}c_{ih}+c_{jh}}{s_{ij}s_{ih}},$
$\displaystyle\cosh x^{i}_{kh}=$ $\displaystyle\frac{\cosh l_{ik}\cosh
l_{ih}+\cosh l_{kh}}{\sinh l_{ik}\sinh
l_{ih}}=\frac{c_{ik}c_{ih}+c_{kh}}{s_{ik}s_{ih}}.$

Note that $x^{i}_{jk},x^{i}_{jh},x^{i}_{kh}$ satisfy the triangle inequality
if and only if

$\sinh\frac{x^{i}_{jk}+x^{i}_{jh}+x^{i}_{kh}}{2}\sinh\frac{x^{i}_{jk}+x^{i}_{jh}-x^{i}_{kh}}{2}\sinh\frac{x^{i}_{jk}-x^{i}_{jh}+x^{i}_{kh}}{2}\sinh\frac{-x^{i}_{jk}+x^{i}_{jh}+x^{i}_{kh}}{2}>0.$

By direct calculations, we have

$\displaystyle
4\sinh\frac{x^{i}_{jk}+x^{i}_{jh}+x^{i}_{kh}}{2}\sinh\frac{x^{i}_{jk}+x^{i}_{jh}-x^{i}_{kh}}{2}\sinh\frac{x^{i}_{jk}-x^{i}_{jh}+x^{i}_{kh}}{2}\sinh\frac{-x^{i}_{jk}+x^{i}_{jh}+x^{i}_{kh}}{2}$
$\displaystyle=$ $\displaystyle 1+2\cosh x^{i}_{jk}\cosh x^{i}_{jh}\cosh
x^{i}_{kh}-\cosh^{2}x^{i}_{jk}-\cosh^{2}x^{i}_{jh}-\cosh^{2}x^{i}_{kh}$
$\displaystyle=$ $\displaystyle
1+2\frac{c_{ij}c_{ik}+c_{jk}}{s_{ij}s_{ik}}\cdot\frac{c_{ij}c_{ih}+c_{jh}}{s_{ij}s_{ih}}\cdot\frac{c_{ik}c_{ih}+c_{kh}}{s_{ik}s_{ih}}-(\frac{c_{ij}c_{ik}+c_{jk}}{s_{ij}s_{ik}})^{2}$
$\displaystyle-(\frac{c_{ij}c_{ih}+c_{jh}}{s_{ij}s_{ih}})^{2}-(\frac{c_{ik}c_{ih}+c_{kh}}{s_{ik}s_{ih}})^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{s^{2}_{ij}s^{2}_{ik}s^{2}_{ih}}[s^{2}_{ij}s^{2}_{ik}s^{2}_{ih}+2(c_{ij}c_{ik}+c_{jk})(c_{ij}c_{ih}+c_{jh})(c_{ik}c_{ih}+c_{kh})$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(c_{ij}c_{ik}+c_{jk})^{2}s_{ih}^{2}-(c_{ij}c_{ih}+c_{jh})^{2}s_{ik}^{2}-(c_{ik}c_{ih}+c_{kh})^{2}s_{ij}^{2}],$

where (3) is used in the last line. Furthermore, using the formula
$s^{2}_{ij}=c_{ij}^{2}-1$, we have

(4) $\displaystyle Q_{1}:=$ $\displaystyle
s^{2}_{ij}s^{2}_{ik}s^{2}_{ih}+2(c_{ij}c_{ik}+c_{jk})(c_{ij}c_{ih}+c_{jh})(c_{ik}c_{ih}+c_{kh})$
$\displaystyle-(c_{ij}c_{ik}+c_{jk})^{2}s_{ih}^{2}-(c_{ij}c_{ih}+c_{jh})^{2}s_{ik}^{2}-(c_{ik}c_{ih}+c_{kh})^{2}s_{ij}^{2}$
$\displaystyle=$
$\displaystyle(c_{ij}^{2}-1)(c_{ik}^{2}-1)(c_{ih}^{2}-1)+2(c_{ij}c_{ik}+c_{jk})(c_{ij}c_{ih}+c_{jh})(c_{ik}c_{ih}+c_{kh})$
$\displaystyle-(c_{ij}c_{ik}+c_{jk})^{2}(c_{ih}^{2}-1)-(c_{ij}c_{ih}+c_{jh})^{2}(c_{ik}^{2}-1)-(c_{ik}c_{ih}+c_{kh})^{2}(c_{ij}^{2}-1)$
$\displaystyle=$ $\displaystyle
c_{ij}^{2}+c_{ik}^{2}+c_{ih}^{2}+c_{jk}^{2}+c_{jh}^{2}+c_{kh}^{2}-c_{ij}^{2}c_{kh}^{2}-c_{ik}^{2}c_{jh}^{2}-c_{ih}^{2}c_{jk}^{2}$
$\displaystyle+2c_{ij}c_{ik}c_{jk}+2c_{ik}c_{ih}c_{kh}+2c_{jk}c_{jh}c_{kh}+2c_{ij}c_{ih}c_{jh}$
$\displaystyle+2c_{ik}c_{ih}c_{jk}c_{jh}+2c_{ij}c_{ik}c_{jh}c_{kh}+2c_{ij}c_{ih}c_{jk}c_{kh}-1.$

Substituting (1) into (4) gives

(5) $\displaystyle Q_{2}=$
$\displaystyle\frac{1}{\cosh^{2}r_{i}\cosh^{2}r_{j}\cosh^{2}r_{k}\cosh^{2}r_{h}}Q_{1}$
$\displaystyle=$ $\displaystyle
t_{i}^{2}(1-d^{2}-e^{2}-f^{2}-2def)+t_{j}^{2}(1-b^{2}-c^{2}-f^{2}-2bcf)$
$\displaystyle+t_{k}^{2}(1-a^{2}-c^{2}-e^{2}-2ace)+t_{h}^{2}(1-a^{2}-b^{2}-d^{2}-2abd)$
$\displaystyle+2t_{i}t_{j}[(1-f^{2})a+bd+ce+bef+cdf]$
$\displaystyle+2t_{i}t_{k}[(1-e^{2})b+ad+cf+aef+cde]$
$\displaystyle+2t_{i}t_{h}[(1-d^{2})c+ae+bf+adf+bde]$
$\displaystyle+2t_{j}t_{k}[(1-c^{2})d+ab+ef+acf+bce]$
$\displaystyle+2t_{j}t_{h}[(1-b^{2})e+ac+df+abf+bcd]$
$\displaystyle+2t_{k}t_{h}[(1-a^{2})f+bc+de+abe+acd]$
$\displaystyle+2abef+2acdf+2bcde+2abd+2ace+2bcf+2def$
$\displaystyle+a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}-a^{2}f^{2}-b^{2}e^{2}-c^{2}d^{2}-1.$

Note that $Q_{2}$ is symmetric in $i,j,k,h$, hence the triangle inequalities
hold for $\triangle_{i},\triangle_{j},\triangle_{k},\triangle_{h}$ if and only
if $Q_{2}>0$. In summary, we have the following result.

######  Lemma 2.2.

A hyper-ideal tetrahedron $\sigma=\\{ijkh\\}$ generated by a generalized
Thurston’s sphere packing metric is non-degenerate in $\mathbb{H}^{3}$ if and
only if $Q_{2}>0$.

If we assume the weight $\Phi:E\rightarrow[0,\frac{\pi}{2}]$ on
$(\Sigma,\mathcal{T})$, then there are some trivial cases that, for any
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}^{4}_{>0}$, the hyper-ideal
tetrahedron generated by the generalized Thurston’s sphere packing metric
defined by (1) degenerates. For example, if the intersection angles on the
hyper-ideal tetrahedron satisfies $a=f=1$ and $b=c=d=e=0$, then it is direct
to check that $Q_{2}=0$ for any
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}^{4}_{>0}$, which implies that the
hyper-ideal tetrahedron degenerates by Lemma 2.2. Please refer to Figure 3 for
a configuration of the four spheres.

\begin{overpic}[scale={0.5}]{Figure_3.pdf} \put(16.0,24.0){$i$}
\put(37.0,56.0){$h$} \put(37.0,8.0){$k$} \put(72.0,24.0){$j$} \end{overpic}
Figure 3. Degenerate sphere packing metric

One can also choose another pair of opposite edges with weight $1$ and the
other edges with weight $0$, in which case the hyper-ideal tetrahedron also
degenerates for any $r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}^{4}_{>0}$.

Let $S$ be the set of weights on the edges of the hyper-ideal tetrahedron with
value $1$ on a pair of opposite edges and value $0$ on the other edges, i.e.

(6) $\displaystyle\begin{aligned}
S=&\\{(\Phi_{ij},\Phi_{ik},\Phi_{ih},\Phi_{jk},\Phi_{jh},\Phi_{kh})\in[0,\frac{\pi}{2}]^{6}|\\\
&(0,\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},0),(\frac{\pi}{2},0,\frac{\pi}{2},\frac{\pi}{2},0,\frac{\pi}{2}),(\frac{\pi}{2},\frac{\pi}{2},0,0,\frac{\pi}{2},\frac{\pi}{2})\\}.\end{aligned}$

Set

(7) $\displaystyle h_{i}=$ $\displaystyle
t_{i}(1-d^{2}-e^{2}-f^{2}-2def)+t_{j}[(1-f^{2})a+bd+ce+bef+cdf]$
$\displaystyle+t_{k}[(1-e^{2})b+ad+cf+aef+cde]+t_{h}[(1-d^{2})c+ae+bf+adf+bde],$
$\displaystyle h_{j}=$ $\displaystyle
t_{j}(1-b^{2}-c^{2}-f^{2}-2bcf)+t_{i}[(1-f^{2})a+bd+ce+bef+cdf]$
$\displaystyle+t_{k}[(1-c^{2})d+ab+ef+acf+bce]+t_{h}[(1-b^{2})e+ac+df+abf+bcd],$
$\displaystyle h_{k}=$ $\displaystyle
t_{k}(1-a^{2}-c^{2}-e^{2}-2ace)+t_{i}[(1-e^{2})b+ad+cf+aef+cde]$
$\displaystyle+t_{j}[(1-c^{2})d+ab+ef+acf+bce]+t_{h}[(1-a^{2})f+bc+de+abe+acd],$
$\displaystyle h_{h}=$ $\displaystyle
t_{h}(1-a^{2}-b^{2}-d^{2}-2abd)+t_{i}[(1-d^{2})c+ae+bf+adf+bde]$
$\displaystyle+t_{j}[(1-b^{2})e+ac+df+abf+bcd]+t_{k}[(1-a^{2})f+bc+de+abe+acd].$

Then

(8) $Q_{2}=t_{i}h_{i}+t_{j}h_{j}+t_{k}h_{k}+t_{h}h_{h}+Q_{3},$

where

$\displaystyle Q_{3}=$ $\displaystyle 2abef+2acdf+2bcde+2abd+2ace+2bcf+2def$
$\displaystyle+a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}-a^{2}f^{2}-b^{2}e^{2}-c^{2}d^{2}-1.$

Set

$\displaystyle A_{i}=d^{2}+e^{2}+f^{2}+2def-1,\
A_{j}=b^{2}+c^{2}+f^{2}+2bcf-1,$ $\displaystyle
A_{k}=a^{2}+c^{2}+e^{2}+2ace-1,\ A_{h}=a^{2}+b^{2}+d^{2}+2abd-1.$

######  Lemma 2.3.

If there exists at least one $\nu\in\\{i,j,k,h\\}$ such that $A_{\nu}\geq 0$,
then $Q_{3}\geq 0$. In particular, if $A_{\nu}>0$, then $Q_{3}>0$.

###### Proof.

Note that $Q_{3}$ is increasing in $a,b,c,d,e,f\in[0,1]$. If
$A_{i}=d^{2}+e^{2}+f^{2}+2def-1\geq 0$, then

$\displaystyle Q_{3}=$
$\displaystyle(d^{2}+e^{2}+f^{2}+2def-1)+a^{2}(1-f^{2})+b^{2}(1-e^{2})+c^{2}(1-d^{2})$
$\displaystyle+2abef+2acdf+2bcde+2abd+2ace+2bcf$ $\displaystyle\geq$
$\displaystyle 0.$

Specially, if $A_{i}>0$, then $Q_{3}>0$. Similar results apply to
$A_{j},A_{k},A_{h}$. Q.E.D.

By Lemma 2.2, $r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a
degenerate generalized Thurston’s sphere packing metric if and only if
$t=(t_{i},t_{j},t_{k},t_{h})=(\tanh r_{i},\tanh r_{j},\tanh r_{k},\tanh
r_{h})\in(0,1)^{4}$ satisfies

(9) $Q_{2}=t_{i}h_{i}+t_{j}h_{j}+t_{k}h_{k}+t_{h}h_{h}+Q_{3}\leq 0,$

which is equivalent to

$A_{i}t^{2}_{i}+B_{i}t_{i}+C_{i}\geq 0$

with

(10) $\displaystyle A_{i}=$ $\displaystyle d^{2}+e^{2}+f^{2}+2def-1,$
$\displaystyle B_{i}=$
$\displaystyle-\big{[}2t_{j}\big{(}(1-f^{2})a+bd+ce+bef+cdf\big{)}$
$\displaystyle\ \ \ \ +2t_{k}\big{(}(1-e^{2})b+ad+cf+aef+cde\big{)}$
$\displaystyle\ \ \ \ +2t_{h}\big{(}(1-d^{2})c+ae+bf+adf+bde\big{)}\big{]},$
$\displaystyle C_{i}=$
$\displaystyle-\big{[}t_{j}^{2}\big{(}1-b^{2}-c^{2}-f^{2}-2bcf\big{)}+t_{k}^{2}\big{(}1-a^{2}-c^{2}-e^{2}-2ace\big{)}$
$\displaystyle\ \ \ \
+t_{h}^{2}\big{(}1-a^{2}-b^{2}-d^{2}-2abd\big{)}+2t_{j}t_{k}\big{(}(1-c^{2})d+ab+ef+acf+bce\big{)}$
$\displaystyle\ \ \ \ +2t_{j}t_{h}\big{(}(1-b^{2})e+ac+df+abf+bcd\big{)}$
$\displaystyle\ \ \ \
+2t_{k}t_{h}\big{(}(1-a^{2})f+bc+de+abe+acd\big{)}+Q_{3}\big{]}.$

######  Lemma 2.4.

If $A_{i}=d^{2}+e^{2}+f^{2}+2def-1>0$, then
$\Delta_{i}=B^{2}_{i}-4A_{i}C_{i}>0$.

###### Proof.

Lengthy and direct calculations give

(11) $\displaystyle\Delta_{i}=$
$\displaystyle-4[t_{j}^{2}(1-f^{2})+t_{k}^{2}(1-e^{2})+t_{h}^{2}(1-d^{2})$
$\displaystyle\ \ \ \ \
+2t_{j}t_{k}(d+ef)+2t_{j}t_{h}(e+df)+2t_{k}t_{h}(f+de)]$
$\displaystyle\times(1-a^{2}-b^{2}-c^{2}-d^{2}-e^{2}-f^{2}+a^{2}f^{2}+b^{2}e^{2}+c^{2}d^{2}$
$\displaystyle\ \ \ \ \ -2ace-2abd-2bcf-2def-2abef-2acdf-2bcde)$
$\displaystyle+4(d^{2}+e^{2}+f^{2}+2def-1)Q_{3}$ $\displaystyle=$
$\displaystyle
4Q_{3}[(d^{2}+e^{2}+f^{2}+2def-1)+t_{j}^{2}(1-f^{2})+t_{k}^{2}(1-e^{2})+t_{h}^{2}(1-d^{2})$
$\displaystyle+2t_{j}t_{k}(d+ef)+2t_{j}t_{h}(e+df)+2t_{k}t_{h}(f+de)].$

If $A_{i}>0$, then $Q_{3}>0$ by Lemma 2.3. This implies $\Delta_{i}>0$ by
(11). Q.E.D.

######  Remark 2.5.

If we take $Q_{2}$ as a quadratic function of $t_{\nu}$ for
$\nu\in\\{j,k,h\\}$, results similar to Lemma 2.4 hold for
$\nu\in\\{j,k,h\\}$.

######  Lemma 2.6.

Let $\sigma=\\{ijkh\\}$ be a truncated tetrahedron with the weight
$\Phi\in[0,\frac{\pi}{2}]^{6}\setminus S$. If
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a degenerate
generalized Thurston’s sphere packing metric, then there exists no subset
$\\{\mu,\nu\\}\subseteq\\{i,j,k,h\\}$ such that $h_{\mu}\leq 0$ and
$h_{\nu}\leq 0$.

###### Proof.

Without loss of generality, we assume $h_{i}\leq 0$ and $h_{j}\leq 0$. By (7),
we have

(12) $\displaystyle t_{i}(d^{2}+e^{2}+f^{2}+2def-1)\geq$ $\displaystyle
t_{j}[(1-f^{2})a+bd+ce+bef+cdf]$
$\displaystyle+t_{k}[(1-e^{2})b+ad+cf+aef+cde]$
$\displaystyle+t_{h}[(1-d^{2})c+ae+bf+adf+bde]$

and

(13) $\displaystyle t_{j}(b^{2}+c^{2}+f^{2}+2bcf-1)\geq$ $\displaystyle
t_{i}[(1-f^{2})a+bd+ce+bef+cdf]$
$\displaystyle+t_{k}[(1-c^{2})d+ab+ef+acf+bce]$
$\displaystyle+t_{h}[(1-b^{2})e+ac+df+abf+bcd].$

This implies

$A_{i}=d^{2}+e^{2}+f^{2}+2def-1\geq 0,\quad A_{j}=b^{2}+c^{2}+f^{2}+2bcf-1\geq
0,$

and

$\displaystyle M:=$
$\displaystyle(d^{2}+e^{2}+f^{2}+2def-1)(b^{2}+c^{2}+f^{2}+2bcf-1)$
$\displaystyle-[(1-f^{2})a+bd+ce+bef+cdf]^{2}$ $\displaystyle\geq$
$\displaystyle 0.$

However, direct calculations give

$M=-(1-f^{2})Q_{3}\leq 0$

by Lemma 2.3. Therefore, $M\equiv 0$, which implies $f=1$ or $Q_{3}=0$.

In the case of $f=1$, since $M\equiv 0$, then the inequality (12) should be an
equality and the second and third terms in the righthand side of (12) should
be $0$. This implies $b=c=0$ and $a(d+e)=0$. Similarly, we have $d=e=0$ and
$a(b+c)=0$ by (13). By the condition that the weight
$\Phi\in[0,\frac{\pi}{2}]^{6}\setminus S$, we have $a\neq 1$. Substituting
$b=c=d=e=0,f=1,a\neq 1$ into (5) gives

$Q_{2}=t_{k}^{2}(1-a^{2})+t_{h}^{2}(1-a^{2})+2t_{k}t_{h}(1-a^{2})>0.$

This contradicts with the condition that $r$ is a degenerate generalized
Thurston’s sphere packing metric.

In the case of $Q_{3}=0$, we have $A_{\nu}\leq 0$ for any
$\nu\in\\{i,j,k,h\\}$ by Lemma 2.3. By $Q_{2}\leq 0$ and (5), we have
$A_{\nu}\equiv 0$ for any $\nu\in\\{i,j,k,h\\}$ , which implies $Q_{3}>0$ due
to the weight $\Phi\in[0,\frac{\pi}{2}]^{6}\setminus S$. This is a
contradiction. Q.E.D.

By Lemma 2.6, if $r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a
degenerate generalized Thurston’s sphere packing metric on a hyper-ideal
tetrahedron with the weight $\Phi\in[0,\frac{\pi}{2}]^{6}\setminus S$, then
there are only two cases. One is that $h_{\nu}>0$ for all
$\nu\in\\{i,j,k,h\\}$, the other is that one of $h_{i},h_{j},h_{k},h_{h}$ is
negative and the others are positive. To remove the former, we need to add
some restrictions on the weight $\Phi$.

######  Lemma 2.7.

Let $\sigma=\\{ijkh\\}$ be a truncated tetrahedron with the weight
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$. If
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a degenerate
generalized Thurston’s sphere packing metric, then it is impossible that
$h_{i}>0,\ h_{j}>0,\ h_{k}>0$ and $h_{h}>0$.

###### Proof.

Assume $h_{i}>0,\ h_{j}>0,\ h_{k}>0$ and $h_{h}>0$. Since $h_{i}>0$, then

$\displaystyle t_{i}$
$\displaystyle(d^{2}+e^{2}+f^{2}+2def-1)<t_{j}[(1-f^{2})a+bd+ce+bef+cdf]$
$\displaystyle+t_{k}[(1-e^{2})b+ad+cf+aef+cde]+t_{h}[(1-d^{2})c+ae+bf+adf+bde].$

If $A_{i}=d^{2}+e^{2}+f^{2}+2def-1\geq 0$, then $Q_{3}\geq 0$ by Lemma 2.3.
This implies $Q_{2}>0$ by (8), which contradicts with the condition that $r$
is a degenerate generalized Thurston’s sphere packing metric by Lemma 2.2.
Therefore, we only need to consider the case that $A_{i}<0$, $A_{j}<0$,
$A_{k}<0$ and $A_{h}<0$. Note that
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$, i.e.,
$a,b,c,d,e,f\in[\frac{1}{3},1]$. Since $Q_{3}$ is increasing in $a,b,c,d,e,f$,
then $Q_{3}\geq 0$ and $Q_{3}=0$ if and only if $a=b=c=d=e=f=\frac{1}{3}$.
This implies $Q_{2}>0$ by (8), which contradicts with the condition that $r$
is a degenerate generalized Thurston’s sphere packing metric by Lemma 2.2.
Q.E.D.

######  Remark 2.8.

By Lemma 2.3, if $A_{\nu}\geq 0$ for some $\nu\in\\{i,j,k,h\\}$, then
$Q_{3}\geq 0$. However, if $A_{\nu}<0$ for all $\nu\in\\{i,j,k,h\\}$, then the
sign of $Q_{3}$ is unclear, neither is $Q_{2}$ by (8). To ensure $Q_{3}\geq
0$, we add a strong condition that
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$ on the weight. In this case,
the set $S$ defined by (6) is empty. And Lemma 2.3, Lemma 2.4 and Lemma 2.6
still hold.

Combining Lemma 2.6 with Lemma 2.7, we have the following corollary.

######  Corollary 2.9.

Let $\sigma=\\{ijkh\\}$ be a truncated tetrahedron with the weight
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$. If
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a degenerate
generalized Thurston’s sphere packing metric, then one of
$h_{i},h_{j},h_{k},h_{h}$ is negative and the others are positive.

Now we can prove the main result of this section.

######  Theorem 2.10.

Let $\sigma=\\{ijkh\\}$ be a truncated tetrahedron with the weight
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$. Then the admissible space
$\Omega_{ijkh}(\Phi)$ of the generalized Thurston’s sphere packing metric
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a simply connected non-
empty open set.

###### Proof.

Suppose $r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a degenerate
generalized Thurston’s sphere packing metric, then
$t=(t_{i},t_{j},t_{k},t_{h})=(\tanh r_{i},\tanh r_{j},\tanh r_{k},\tanh
r_{h})\in(0,1)^{4}$ satisfies (9). By Corollary 2.9, one of
$h_{i},h_{j},h_{k},h_{h}$ is negative and the others are positive. Without
loss of generality, we assume $h_{i}<0$. Then
$A_{i}=d^{2}+e^{2}+f^{2}+2def-1>0$ by the definition of $h_{i}$. Take
$Q_{2}\leq 0$ as a quadratic inequality $A_{i}t^{2}_{i}+B_{i}t_{i}+C_{i}\geq
0$, where $A_{i},B_{i},C_{i}$ are given by (10). As $A_{i}>0$, we have
$\Delta_{i}=B^{2}_{i}-4A_{i}C_{i}>0$ by Lemma 2.4, which implies

$t_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\quad\text{or}\quad
t_{i}\leq\frac{-B_{i}-\sqrt{\Delta_{i}}}{2A_{i}}.$

The definition of $h_{i}$ in (7) implies
$h_{i}=-\frac{1}{2}(2A_{i}t_{i}+B_{i})$. Then $h_{i}<0$ is equivalent to
$t_{i}>-\frac{B_{i}}{2A_{i}}$, which implies
$t_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}$.

In the case that $A_{i}>0$, we set

(14)
$V_{i}=\\{r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}|t_{i}\geq\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}\\},$

which is bounded by an analytical function defined on $\mathbb{R}_{>0}^{3}$.
For any $(r_{i},r_{j},r_{k},r_{h})\in V_{i}$, it is direct to check that
$A_{i}t^{2}_{i}+B_{i}t_{i}+C_{i}\geq 0$, which implies $Q_{2}\leq 0$, thus
$V_{i}\subseteq\mathbb{R}^{4}\setminus\Omega_{ijkh}(\Phi)$. One can define
$V_{j},V_{k},V_{h}$ similarly, if the corresponding $A_{j}>0,A_{k}>0,A_{h}>0$.
Similar arguments imply
$V_{j},V_{k},V_{h}\subseteq\mathbb{R}^{4}\setminus\Omega_{ijkh}(\Phi)$.

If $A_{\mu}>0$ for $\mu\in P\subseteq\\{i,j,k,h\\}$ and $A_{\omega}\leq 0$ for
$\omega\in\\{i,j,k,h\\}\backslash P$, then the space of degenerate generalized
Thurston’s sphere packing metrics is $\bigcup_{\mu\in P}V_{\mu}$. Therefore,

$\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}\setminus\bigcup_{\mu\in P}{V_{\mu}}.$

For any $r\in V_{i}$, we have $t_{i}>-\frac{B_{i}}{2A_{i}}$, which is
equivalent to $h_{i}<0$. Similarly, for $r\in V_{j}$, we have $h_{j}<0$. Then
Corollary 2.9 implies $V_{i}\bigcap V_{j}=\varnothing$. Similarly, we have
$V_{i},V_{j},V_{k},V_{h}$ are mutually disjoint, if they are non-empty.
Therefore, $\Omega_{ijkh}(\Phi)$ is homotopy equivalent to
$\mathbb{R}_{>0}^{4}$ and hence is a simply connected non-empty open set.
Q.E.D.

By the proof of Theorem 2.10, the admissible space
$\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$ if and only if $A_{i}\leq 0\
,A_{j}\leq 0,\ A_{k}\leq 0,\ A_{h}\leq 0$. Furthermore, we have the following
results.

######  Corollary 2.11.

Let $\sigma=\\{ijkh\\}$ be a truncated tetrahedron with the weight
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$. If any three of
$A_{i},A_{j},A_{k},A_{h}$ are non-positive, then the admissible space
$\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$. Specially, if any three of the
following four formulas hold,

$\displaystyle\Phi_{ij}+\Phi_{jk}+\Phi_{ik}\geq\pi,\
\Phi_{ik}+\Phi_{ih}+\Phi_{kh}\geq\pi,\ $
$\displaystyle\Phi_{ij}+\Phi_{ih}+\Phi_{jh}\geq\pi,\
\Phi_{ij}+\Phi_{ik}+\Phi_{jk}\geq\pi,$

then the admissible space $\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$.

###### Proof.

Without loss of generality, we assume $A_{j}\leq 0,A_{k}\leq 0,A_{h}\leq 0$.
By the definition of $h_{j},h_{k},h_{h}$ in (7) and Lemma 2.6, we have
$h_{j}>0,\ h_{k}>0,\ h_{h}>0$. If
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a degenerate
generalized Thurston’s sphere packing metric, then $h_{i}<0$ by Corollary 2.9.
Then $A_{i}=d^{2}+e^{2}+f^{2}+2def-1>0$ and hence $Q_{3}>0$ by Lemma 2.3. Note
that $\Phi\in[0,\mathrm{arccos}\frac{1}{3}]^{6}$, i.e.,
$a,b,c,d,e,f\in[\frac{1}{3},1]$, then $Q_{3}-A_{i}>0$. This implies

$-B_{i}+\sqrt{\Delta_{i}}-2A_{i}>2\sqrt{A_{i}Q_{3}}-2A_{i}=2\sqrt{A_{i}}\cdot\frac{Q_{3}-A_{i}}{\sqrt{Q_{3}}+\sqrt{A_{i}}}>0.$

Hence, $\frac{-B_{i}+\sqrt{\Delta_{i}}}{2A_{i}}>1$. Combining with
$t_{i}=\tanh r_{i}\in(0,1)$ gives $V_{i}=\emptyset$ by (14). Thus
$r=(r_{i},r_{j},r_{k},r_{h})\in\mathbb{R}_{>0}^{4}$ is a non-degenerate
generalized Thurston’s sphere packing metric and hence
$\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$.

For the second part, note that

$\displaystyle A_{i}=$
$\displaystyle\cos^{2}\Phi_{jk}+\cos^{2}\Phi_{jh}+\cos^{2}\Phi_{kh}+2\cos\Phi_{jk}\cos\Phi_{jh}\cos\Phi_{kh}-1$
$\displaystyle=$ $\displaystyle
4\cos\frac{\Phi_{jk}+\Phi_{jh}+\Phi_{kh}}{2}\cos\frac{\Phi_{jk}+\Phi_{jh}-\Phi_{kh}}{2}\cos\frac{\Phi_{jk}-\Phi_{jh}+\Phi_{kh}}{2}\cos\frac{-\Phi_{jk}+\Phi_{jh}+\Phi_{kh}}{2}.$

Since $\Phi_{ij}\in[0,\mathrm{acrcos}\frac{1}{3}]\subseteq[0,\frac{\pi}{2}]$,
then

$\cos\frac{\Phi_{jk}+\Phi_{jh}-\Phi_{kh}}{2}\geq 0,\
\cos\frac{\Phi_{jk}-\Phi_{jh}+\Phi_{kh}}{2}\geq 0,\
\cos\frac{-\Phi_{jk}+\Phi_{jh}+\Phi_{kh}}{2}\geq 0.$

Therefore, if $\Phi_{jk}+\Phi_{jh}+\Phi_{kh}\geq\pi$, then
$\cos\frac{\Phi_{jk}+\Phi_{jh}+\Phi_{kh}}{2}\leq 0$ and hence $A_{i}\leq 0$.
Similarly, if

$\Phi_{ik}+\Phi_{ih}+\Phi_{kh}\geq\pi,\ \Phi_{ij}+\Phi_{ih}+\Phi_{jh}\geq\pi,\
\Phi_{ij}+\Phi_{ik}+\Phi_{jk}\geq\pi,$

then $A_{j}\leq 0,\ A_{k}\leq 0,\ A_{h}\leq 0$ respectively. Q.E.D.

######  Corollary 2.12.

Let $\sigma=\\{ijkh\\}$ be a truncated tetrahedron with the weight
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$. If $\cos\Phi\equiv
C\in[\frac{1}{3},1]$, then the admissible space
$\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$.

###### Proof.

If $\cos\Phi\equiv C\in[\frac{1}{3},\frac{1}{2}]$, then
$A_{i}=d^{2}+e^{2}+f^{2}+2def-1=3C^{2}+2C^{3}-1\leq 0$. Similarly, $A_{j}\leq
0,\ A_{k}\leq 0,\ A_{h}\leq 0$. Then $\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$
by Corollary 2.11. If $\cos\Phi\equiv C\in[\frac{1}{2},1]$, then
$1-3C^{2}-2C^{3}\leq 0$. Note that $t_{i}=\tanh r_{i}\in(0,1)$, then by (5),
we have

$\displaystyle Q_{2}=$
$\displaystyle(t_{i}^{2}+t_{j}^{2}+t_{k}^{2}+t_{h}^{2})(1-3C^{2}-2C^{3})$
$\displaystyle+2(t_{i}t_{j}+t_{i}t_{k}+t_{i}t_{h}+t_{j}t_{k}+t_{j}t_{h}+t_{k}t_{h})(C^{3}+2C^{2}+C)$
$\displaystyle+3C^{4}+8C^{3}+6C^{2}-1$ $\displaystyle\geq$ $\displaystyle
2(t_{i}t_{j}+t_{i}t_{k}+t_{i}t_{h}+t_{j}t_{k}+t_{j}t_{h}+t_{k}t_{h})(C^{3}+2C^{2}+C)$
$\displaystyle+3C^{4}-6C^{2}+3$ $\displaystyle>$ $\displaystyle 0,$

which implies $\Omega_{ijkh}(\Phi)=\mathbb{R}_{>0}^{4}$ by Lemma 2.2. Q.E.D.

######  Remark 2.13.

As a special case that $\Phi\equiv 0$, Corollary 2.12 shows that the
admissible space $\Omega_{ijkh}(\Phi)$ for the generalized tangential sphere
packing metrics is $\mathbb{R}_{>0}^{4}$, which was proved in [15].

##  3\. Rigidity of generalized Thurston’s sphere packings

Theorem 2.10 shows that the admissible space $\Omega_{ijkh}(\Phi)$ is simply
connected for a weighted truncated tetrahedron $\sigma=\\{ijkh\\}$. Hence, the
admissible space on $(\Sigma,\mathcal{T},\Phi)$ can be defined to be
$\Omega(\Phi)=\bigcap_{\sigma=\\{ijkh\\}}\Omega_{ijkh}(\Phi)$. Note that
$\Omega(\Phi)$ may be not connected.

Denote the volume of a hyper-ideal tetrahedra $\sigma=\\{ijkh\\}$ by $vol$,
which is considered as a function in the dihedral angles. By the Schläfli
formula, we have

(15) $2dvol+\sum_{j\sim i}l_{ij}d\beta_{ij,kh}=0.$

One can refer to [2, 10] for more details on the Schläfli formula. Using the
volume function $vol$ on a singer hyper-ideal tetrahedra $\sigma=\\{ijkh\\}$ ,
we can define the following function

$G=\sum_{j\sim i}K_{ij}l_{ij}-2\sum_{\sigma\in T}vol$

on $(\Sigma,\mathcal{T},\Phi)$, where $K_{ij}$ is the combinatorial Ricci
curvature along the edge $\\{ij\\}\in E$ defined by (2). Then by (15), we have
$dG=\sum_{j\sim i}K_{ij}dl_{ij}$, which implies

(16) $\frac{\partial G}{\partial r_{i}}=\sum_{j;j\sim i}K_{ij}\frac{\partial
l_{ij}}{\partial r_{i}}=\sum_{j;j\sim i}K_{ij}\cosh\theta_{ij}=K_{i}.$

Furthermore,

$\displaystyle\frac{\partial^{2}G}{\partial r_{i}\partial r_{j}}=$
$\displaystyle\sum_{j;j\sim i}\frac{\partial K_{ij}}{\partial
r_{j}}\frac{\partial l_{ij}}{\partial r_{i}}+\sum_{j;j\sim
i}K_{ij}\frac{\partial^{2}l_{ij}}{\partial r_{i}\partial r_{j}}$
$\displaystyle=$ $\displaystyle\sum_{s\sim t}\sum_{u\sim v}\frac{\partial
l_{uv}}{\partial r_{j}}\frac{\partial K_{st}}{\partial l_{uv}}\frac{\partial
l_{st}}{\partial r_{i}}+\sum_{s\sim t}K_{st}\frac{\partial^{2}l_{st}}{\partial
r_{i}\partial r_{j}}$ $\displaystyle=$ $\displaystyle-\sum_{\\{ijkh\\}\in
T}\sum_{s\sim t}\sum_{u\sim v}\frac{\partial l_{uv}}{\partial
r_{j}}\frac{\partial\beta_{st}}{\partial l_{uv}}\frac{\partial
l_{st}}{\partial r_{i}}+\sum_{s\sim t}K_{st}\frac{\partial^{2}l_{st}}{\partial
r_{i}\partial r_{j}},$

where $\beta_{ij}:=\beta_{ij,kh}$ for simplification. This implies

(17) $\mathrm{Hess}_{r}G=-\sum_{\\{ijkh\\}\in T}\bigg{(}\frac{\partial
l}{\partial r}\bigg{)}_{\sigma}\bigg{(}\frac{\partial\beta}{\partial
l}\bigg{)}_{\sigma}\bigg{(}\frac{\partial l}{\partial
r}\bigg{)}_{\sigma}^{T}+\sum_{j\sim i}K_{ij}\mathrm{Hess}_{r}(l_{ij}),$

where

$\bigg{(}\frac{\partial l}{\partial
r}\bigg{)}_{\sigma}=\left(\begin{array}[]{cccccc}\cosh\theta_{ij}&\cosh\theta_{ik}&\cosh\theta_{ih}&0&0&0\\\
\cosh\theta_{ji}&0&0&\cosh\theta_{jk}&\cosh\theta_{jh}&0\\\
0&\cosh\theta_{ki}&0&\cosh\theta_{kj}&0&\cosh\theta_{kh}\\\
0&0&\cosh\theta_{hi}&0&\cosh\theta_{hj}&\cosh\theta_{hk}\end{array}\right)$

and

$\bigg{(}\frac{\partial\beta}{\partial
l}\bigg{)}_{\sigma}=\left(\begin{array}[]{cccccc}\frac{\partial\beta_{ij}}{\partial
l_{ij}}&\frac{\partial\beta_{ij}}{\partial
l_{ik}}&\frac{\partial\beta_{ij}}{\partial
l_{ih}}&\frac{\partial\beta_{ij}}{\partial
l_{jk}}&\frac{\partial\beta_{ij}}{\partial
l_{jh}}&\frac{\partial\beta_{ij}}{\partial l_{kh}}\\\
\frac{\partial\beta_{ik}}{\partial l_{ij}}&\frac{\partial\beta_{ik}}{\partial
l_{ik}}&\frac{\partial\beta_{ik}}{\partial
l_{ih}}&\frac{\partial\beta_{ik}}{\partial
l_{jk}}&\frac{\partial\beta_{ik}}{\partial
l_{jh}}&\frac{\partial\beta_{ik}}{\partial l_{kh}}\\\
\frac{\partial\beta_{ih}}{\partial l_{ij}}&\frac{\partial\beta_{ih}}{\partial
l_{ik}}&\frac{\partial\beta_{ih}}{\partial
l_{ih}}&\frac{\partial\beta_{ih}}{\partial
l_{jk}}&\frac{\partial\beta_{ih}}{\partial
l_{jh}}&\frac{\partial\beta_{ih}}{\partial l_{kh}}\\\
\frac{\partial\beta_{jk}}{\partial l_{ij}}&\frac{\partial\beta_{jk}}{\partial
l_{ik}}&\frac{\partial\beta_{jk}}{\partial
l_{ih}}&\frac{\partial\beta_{jk}}{\partial
l_{jk}}&\frac{\partial\beta_{jk}}{\partial
l_{jh}}&\frac{\partial\beta_{jk}}{\partial l_{kh}}\\\
\frac{\partial\beta_{jh}}{\partial l_{ij}}&\frac{\partial\beta_{jh}}{\partial
l_{ik}}&\frac{\partial\beta_{jh}}{\partial
l_{ih}}&\frac{\partial\beta_{jh}}{\partial
l_{jk}}&\frac{\partial\beta_{jh}}{\partial
l_{jh}}&\frac{\partial\beta_{jh}}{\partial l_{kh}}\\\
\frac{\partial\beta_{kh}}{\partial l_{ij}}&\frac{\partial\beta_{kh}}{\partial
l_{ik}}&\frac{\partial\beta_{kh}}{\partial
l_{ih}}&\frac{\partial\beta_{kh}}{\partial
l_{jk}}&\frac{\partial\beta_{kh}}{\partial
l_{jh}}&\frac{\partial\beta_{kh}}{\partial l_{kh}}\\\ \end{array}\right).$

It is easy to check that the rank of the metric $(\frac{\partial l}{\partial
r})_{\sigma}$ is 4. We have the following result on the matrix
$(\frac{\partial\beta}{\partial l})_{\sigma}$.

######  Theorem 3.1 ([9]).

For a hyper-ideal tetrahedron $\sigma=\\{ijkh\\}$ with dihedral angle
$\beta_{ij}$ and length $l_{ij}$ at the edge $\\{ij\\}\in E$, the matrix
$(\frac{\partial\beta}{\partial l})$ is symmetric and strictly positive
definite on $\mathcal{L}$, where $\mathcal{L}$ is the set of vectors
$(l_{ij},l_{ik},l_{ih},l_{jk},l_{jh},l_{kh})\in\mathbb{R}^{6}_{>0}$ such that
there exists a non-degenerate hyper-ideal tetrahedron $\sigma=\\{ijkh\\}$ with
$l_{ij}$ as the length of the edge $\\{ij\\}$.

######  Remark 3.2.

The simply connectedness of $\mathcal{L}$ has been proved in [1, 5, 4, 9].

By Theorem 3.1, the first term in the right of (17) is a symmetric and
strictly negative definite matrix on the admissible space $\Omega(\Phi)$. Now
we consider the second term in the righthand side of (17). Note that in the
special case of generalized sphere packings, i.e., $\Phi\equiv 0$, the second
term equals 0 because of $l_{ij}=r_{i}+r_{j}$. Then $\mathrm{Hess}_{r}G<0$ on
$(\Sigma,\mathcal{T},\Phi)$, which was proved in [15] by direct calculations.
Please refer to [15] for more details. In the general case of
$\Phi:E\rightarrow[0,\frac{\pi}{2}]$, we have

$\displaystyle\frac{\partial^{2}l_{ij}}{\partial r^{2}_{i}}=$
$\displaystyle\frac{\cosh l_{ij}}{\sinh^{3}l_{ij}}[\sinh^{2}l_{ij}-(\cosh
r_{i}\sinh r_{j}+\cos\Phi_{ij}\sinh r_{i}\cosh r_{j})^{2}]$ $\displaystyle=$
$\displaystyle\frac{\cosh l_{ij}}{\sinh^{3}l_{ij}}[(\sinh r_{i}\sinh
r_{j}+\cos\Phi_{ij}\cosh r_{i}\cosh r_{j})^{2}-1$ $\displaystyle-(\cosh
r_{i}\sinh r_{j}+\cos\Phi_{ij}\sinh r_{i}\cosh r_{j})^{2}]$ $\displaystyle=$
$\displaystyle-\frac{\sin^{2}\Phi_{ij}}{\sinh^{3}l_{ij}}\cosh
l_{ij}\cosh^{2}r_{j},$

and

$\displaystyle\frac{\partial^{2}l_{ij}}{\partial r_{i}\partial r_{j}}=$
$\displaystyle\frac{1}{\sinh^{3}l_{ij}}[(\cosh r_{i}\cosh
r_{j}+\cos\Phi_{ij}\sinh r_{i}\sinh r_{j})(\cosh^{2}l_{ij}-1)$
$\displaystyle-\cosh l_{ij}(\sinh r_{i}\cosh r_{j}+\cos\Phi_{ij}\cosh
r_{i}\sinh r_{j})$ $\displaystyle\times(\cosh r_{i}\sinh
r_{j}+\cos\Phi_{ij}\sinh r_{i}\cosh r_{j})]$ $\displaystyle=$
$\displaystyle-\frac{\sin^{2}\Phi_{ij}}{\sinh^{3}l_{ij}}\cosh r_{i}\cosh
r_{j}.$

Then

$\mathrm{Hess}_{r}(l_{ij})=-\frac{\sin^{2}\Phi_{ij}}{\sinh^{3}l_{ij}}\left(\begin{array}[]{cc}\cosh
l_{ij}\cosh^{2}r_{j}&\cosh r_{i}\cosh r_{j}\\\ \cosh r_{i}\cosh r_{j}&\cosh
l_{ij}\cosh^{2}r_{i}\\\ \end{array}\right),$

which is a symmetric and negative definite matrix if $\Phi_{ij}\not\equiv 0$
for any edge $\\{ij\\}\in E$. Therefore,

$\displaystyle\mathrm{Hess}_{r}G=$ $\displaystyle-\sum_{\\{ijkh\\}\in
T}\bigg{(}\frac{\partial l}{\partial
r}\bigg{)}_{\sigma}\bigg{(}\frac{\partial\beta}{\partial
l}\bigg{)}_{\sigma}\bigg{(}\frac{\partial l}{\partial r}\bigg{)}_{\sigma}^{T}$
$\displaystyle-\sum_{j\sim
i}\frac{K_{ij}\sin^{2}\Phi_{ij}}{\sinh^{3}l_{ij}}\left(\begin{array}[]{cc}\cosh
l_{ij}\cosh^{2}r_{j}&\cosh r_{i}\cosh r_{j}\\\ \cosh r_{i}\cosh r_{j}&\cosh
l_{ij}\cosh^{2}r_{i}\\\ \end{array}\right).$

Proof of Theorem 1.6 If $\overline{r}$ is a non-degenerate generalized
Thurston’s sphere packing metric on $(\Sigma,\mathcal{T},\Phi)$ with
$K_{ij}(\overline{r})\sin^{2}\Phi_{ij}\geq 0$ for any edge $\\{ij\\}\in E$,
then $\mathrm{Hess}_{r}G(\overline{r})<0$ and hence the function $G$ is a
strictly concave function on a convex neighborhood $U$ of $\overline{r}$.

Suppose there exist two different generalized Thurston’s sphere packing
metrics $r_{1}$ and $r_{2}$ in $U$ such that $K_{i}(r_{1})=K_{i}(r_{2})$ for
all $i\in V$. Set

$f(t)=G((1-t)r_{1}+tr_{2}),\ t\in[0,1].$

Then $f(t)$ is a $C^{1}$ smooth concave function of $t\in[0,1]$ with

$f^{\prime}(t)=\sum_{i=1}^{N}\nabla_{r_{i}}G|_{(1-t)r_{1}+tr_{2}}\cdot(r_{2,i}-r_{1,i})=\sum_{i=1}^{N}K_{i}|_{(1-t)r_{1}+tr_{2}}\cdot(r_{2,i}-r_{1,i}).$

By the assumption that $K_{i}(r_{1})=K_{i}(r_{2})$, we have
$f^{\prime}(0)=f^{\prime}(1)=0$, which implies $f^{\prime}(t)\equiv 0$ by the
concavity of $f(t)$. Then $f^{\prime\prime}(t)\equiv 0$ for $t\in[0,1]$.
Combining with

$f^{\prime\prime}(t)=(r_{2}-r_{1})^{T}\cdot\mathrm{Hess}_{r}G|_{(1-t)r_{1}+tr_{2}}\cdot(r_{2}-r_{1})$

for $t\in[0,1]$ and $\mathrm{Hess}_{r}G$ is negative definite, we have
$r_{1}=r_{2}$. Q.E.D.

For the combinatorial Ricci curvature, we rewrite Theorem 1.3 as follows.

######  Theorem 3.3.

Suppose $(\Sigma,\mathcal{T})$ is an ideally triangulated compact
3-dimensional manifold with boundary. Let
$\Phi:E\rightarrow[0,\mathrm{arccos}\frac{1}{3}]$ be a weight defined on the
edges of $(\Sigma,\mathcal{T})$. Suppose there exists a neighborhood
$U\subseteq\Omega(\Phi)$ of $\overline{r}$ such that if $r\in U$ has the same
combinatorial Ricci curvature as $\overline{r}$, then $r=\overline{r}$.

###### Proof.

Define the following functional

$\widetilde{G}(r_{1},r_{2},...,r_{N})=G-\sum_{j;j\sim
i}K_{ij}(\overline{r})l_{ij}.$

Then by (16), we have

$\frac{\partial\widetilde{G}}{\partial r_{i}}=K_{i}-\sum_{j;j\sim
i}K_{ij}(\overline{r})\frac{\partial l_{ij}}{\partial r_{i}}=\sum_{j;j\sim
i}(K_{ij}-K_{ij}(\overline{r}))\frac{\partial l_{ij}}{\partial r_{i}},$

which implies $\nabla\widetilde{G}(r)=\nabla\widetilde{G}(\overline{r})$ by
the assumption that $K_{ij}(r)=K_{ij}(\overline{r})$. Furthermore,

$\displaystyle\mathrm{Hess}_{r}\widetilde{G}=$
$\displaystyle-\sum_{\\{ijkh\\}\in T}\bigg{(}\frac{\partial l}{\partial
r}\bigg{)}_{\sigma}\bigg{(}\frac{\partial\beta}{\partial
l}\bigg{)}_{\sigma}\bigg{(}\frac{\partial l}{\partial r}\bigg{)}_{\sigma}^{T}$
$\displaystyle-\sum_{j\sim
i}(K_{ij}-K_{ij}(\overline{r}))\frac{\sin^{2}\Phi_{ij}}{\sinh^{3}l_{ij}}\left(\begin{array}[]{cc}\cosh
l_{ij}\cosh^{2}r_{j}&\cosh r_{i}\cosh r_{j}\\\ \cosh r_{i}\cosh r_{j}&\cosh
l_{ij}\cosh^{2}r_{i}\\\ \end{array}\right),$

which implies that $\mathrm{Hess}_{r}\widetilde{G}|_{r=\overline{r}}<0$ and
hence the function $\widetilde{G}$ is a strictly concave function on a convex
neighborhood $U$ of $\overline{r}$. Set

$f(t)=\widetilde{G}((1-t)r+t\overline{r}),\ t\in[0,1].$

Then $f(t)$ is a $C^{1}$ smooth concave function of $t\in[0,1]$. The rest of
the proof is similar to that of Theorem 1.6, we omit it here. Q.E.D.

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