dvips

arXiv:2309.05335

Einstein Structure of Squashed Four-Spheres

Jeongwon Ho<EMAIL_ADDRESS>Kyung Kiu Kim
<EMAIL_ADDRESS>and Hyun Seok Yang<EMAIL_ADDRESS>
a Center for Quantum Spacetime, Sogang University, Seoul 121-741, Korea

b College of General Education, Kookmin University, Seoul 02707, Korea

c Department of Physics and Photon Science, Gwangju Institute of Science and
Technology, Gwangju 61005, Korea

ABSTRACT

It is known that the moduli space of Einstein structures is an isolated point
so that an Einstein structure has no infinitesimal Einstein deformations. We
examine the rigidity of the Einstein structure by considering deformations of
the round four-sphere $\mathbb{S}^{4}$. We show that any deformation of the
round four-sphere $\mathbb{S}^{4}$ causes it to deviate from the Einstein
structure, except trivial deformations at most only changing the size of the
sphere.

Keywords: Einstein Structure, Squashed Four-Sphere, Moduli Space

##  1 Introduction

An Einstein manifold has the property that its Ricci curvature tensor is
proportional to its metric tensor. Finding a metric that satisfies this
condition while being nonhomogeneous and compact is quite challenging. It
requires careful balancing of the geometric properties to achieve the desired
curvature. Hence, it is difficult to find explicit solutions to Einstein’s
equations for compact nonhomogeneous manifolds since the field equations for
an appropriate metric ansatz are highly nonlinear partial differential
equations. For a compact Einstein manifold with positive Ricci scalar, the
manifold must be simply connected. For example,
$\mathbb{S}^{1}\times\mathbb{S}^{3}$ does not admit Einstein metrics [1]. It
turns out that there are not many simply connected compact manifolds. Compact
manifolds have a finite volume, which introduces constraints on their topology
and geometry. The moduli space of Einstein structures is, up to a discrete
group action, a finite-dimensional manifold (see Chap. 12 in [2]). The
rigidity of Einstein structures implies that under specific conditions, the
only solutions to Einstein’s equations are homogeneous ones, leaving little
room for nonhomogeneous constructions. For example, G. Jensen proved that a
homogeneous four-dimensional Einstein manifold is isometric to a Riemannian
symmetric space [3]. While some nonhomogeneous examples exist [4, 5], they are
generally not easy to find or generalize. A nonhomogeneous space possesses
less global symmetries, making it more difficult to find suitable geometric
structures that satisfy Einstein equations.

There are few known examples of compact four-dimensional symmetric Einstein
manifolds with positive scalar curvature:
$\mathbb{S}^{4},\,\mathbb{C}\mathbb{P}^{2},\,\mathbb{S}^{2}\times\mathbb{S}^{2}$.
These examples are all homogeneous manifolds. It was shown in [1] that a
connected sum such as $M=n\mathbb{C}\mathbb{P}^{2}$ allows an Einstein metric
only for $1\leq n\leq 3$.111The explicit metrics on
$n\mathbb{C}\mathbb{P}^{2}$ in [6] are four-dimensional Kähler metrics with
vanishing Ricci scalar, which is not a vacuum solution but a solution in
Einstein-Maxwell theory. There is also a compact nonhomogeneous Einstein
manifold $(M,g)$ known as the Page space [4]. It is a solution of Einstein
equations, $R_{ab}=\lambda\delta_{ab}$ with $\lambda>0$, which is a nontrivial
$\mathbb{S}^{2}$ bundle over $\mathbb{S}^{2}$. It can be constructed as
$M=\mathbb{C}\mathbb{P}^{2}\sharp\,\overline{\mathbb{C}\mathbb{P}}^{2}$ by
attaching the complex projective space and its complex conjugate, removing a
4-ball from each manifold, and gluing them together along the resulting
$\mathbb{S}^{3}$ boundaries [2]. This construction was generalized to
$M=\mathbb{C}\mathbb{P}^{2}\sharp\,n\overline{\mathbb{C}\mathbb{P}}^{2}$ for
$1\leq n\leq 8$ [7, 8]. Their explicit metrics are not known in most cases
except for $n=1$ [4]. Only a finite number of compact four-manifolds may carry
an Einstein metric, which is conformal to an extremal Kähler metric [2, 9].

In this paper, we will examine the rigidity of the Einstein structure by
considering deformations of the round sphere $\mathbb{S}^{4}$. The rigidity
means that an Einstein structure has no infinitesimal Einstein deformations
and so it is an isolated point of the moduli space. See $\mathbf{12.H}$ in [2]
and the review [10] for the moduli space of Einstein structures. Although
there are currently no rigidity or uniqueness results for Einstein manifolds
with positive Ricci scalar in contrast to the Ricci-flat and Ricci-negative
Einstein manifolds, it is known [2, 11] (see also [12, 13]) that the standard
metrics on $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2}$ are locally rigid,
i.e. the metrics are isolated points in the moduli space of Einstein metrics.
In order to study the moduli space of Einstein structures, we will use the
irreducible decomposition of curvature tensors [14] (see also Chaps.
$\mathbf{1.G}$ and $\mathbf{1.H}$ in Ref. [2] and Secs. 1.1 and 2.1 in Ref.
[15]). In section 2, we recapitulate the canonical decomposition of the
curvature tensor in Refs. [16, 17, 18, 19, 20] and provide a simple proof of
the Hitchin-Thorpe inequality [1, 10, 21]. We apply the canonical
decomposition to the Page metric [4] which is the first example of a
nonhomogeneous compact Einstein manifold and is a Hermitian-Einstein,
conformal to a Kähler metric, but is not Kähler itself [2, 10]. In section 3,
we examine the rigidity of the Einstein structure by considering deformations
of the round four-sphere $\mathbb{S}^{4}$. We show that any deformation of the
round four-sphere $\mathbb{S}^{4}$ causes it to deviate from the Einstein
structure, except trivial deformations at most only changing the size of the
sphere. The purpose of this analysis is to get some insight into why the
moduli space of Einstein structures is an isolated point so that an Einstein
structure has no infinitesimal Einstein deformations. In section 4, we discuss
some generalization of our approach and the physical origin of the rigidity of
Einstein structures. In appendix A, we will prove the strong version of the
Hitchin-Thorpe inequality (Theorem $\mathbf{2}$ in [1]) in order to
demonstrate the effectiveness of our approach.

##  2 Einstein Structure of Four-Manifolds

On an orientable Riemannian four-manifold, the vector space of 2-forms
$\Omega^{2}(M)=\Lambda^{2}T^{*}M$ is decomposed into the space of self-dual
and anti-self-dual 2-forms [15]

$\Omega^{2}(M)=\Omega_{+}^{2}\oplus\Omega_{-}^{2}$ (2.1)

using the projection operators

$P_{\pm}=\frac{1}{2}(1\pm*),\qquad P_{\pm}^{2}=P_{\pm},\qquad P_{+}+P_{-}=1,$
(2.2)

where $\Omega_{\pm}^{2}\equiv P_{\pm}\Omega^{2}$ is the $\pm 1$ eigenspaces of
the Hodge star operator $*:\Omega^{2}\to\Omega^{2}$. The Lorentz group
$G=Spin(4)$ is a double cover of the four-dimensional Euclidean Lorentz group
$SO(4)$, i.e., $SO(4)\cong Spin(4)/\mathbb{Z}_{2}$. The Lorentz group
$Spin(4)$ splits into a product of two groups [15]

$G=Spin(4)\cong SU(2)_{+}\times SU(2)_{-}.$ (2.3)

The product structure (2.3) also leads to the splitting of the Lie algebra

$\mathfrak{g}=spin(4)\cong so(4)\cong su(2)_{+}\oplus su(2)_{-}.$ (2.4)

The splitting of the vector space $\mathfrak{g}=so(4)$ (or $spin(4)$) is
defined by the chiral operator
$\gamma_{5}=-\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}$ in the Clifford algebra
which obeys $\gamma_{5}^{2}=1$. Therefore one can construct the projection
operator

$P_{\pm}=\frac{1}{2}(1\pm\gamma_{5}),\qquad P_{\pm}^{2}=P_{\pm},\qquad
P_{+}+P_{-}=1,$ (2.5)

acting on the vector space $\mathfrak{g}=so(4)$. Using the projection operator
(2.5), the explicit realization of the splitting (2.4) reads as

$J_{ab}=J_{ab}^{+}\oplus J_{ab}^{-}$ (2.6)

where $J_{ab}=\frac{1}{4}[\gamma_{a},\gamma_{b}]$ are Lie algebra generators
of $so(4)$ and $J_{ab}^{\pm}\equiv P_{\pm}J_{ab}$. Each part in Eq. (2.6)
consists of three $(2\times 2)$ anti-Hermitian traceless matrices. So they can
be expanded in the basis of the Pauli matrices $\tau^{i}\;(i=1,2,3)$ as

$J_{ab}^{+}=\frac{i}{2}\eta^{i}_{ab}\tau^{i}\in su(2)_{+},\qquad
J_{ab}^{-}=\frac{i}{2}\bar{\eta}^{i}_{ab}\tau^{i}\in su(2)_{-}.$ (2.7)

The expansion coefficients in (2.7), the so-called ’t Hooft symbols [23], are
given by

$\eta^{i}_{ab}=-i\mathrm{Tr}\big{(}J_{ab}^{+}\tau^{i}\big{)},\qquad\bar{\eta}^{i}_{ab}=-i\mathrm{Tr}\big{(}J_{ab}^{-}\tau^{i}\big{)}$
(2.8)

and they satisfy the self-duality relation

$\eta^{i}_{ab}=\frac{1}{2}\varepsilon_{abcd}\eta^{i}_{cd},\qquad\bar{\eta}^{i}_{ab}=-\frac{1}{2}\varepsilon_{abcd}\bar{\eta}^{i}_{cd}.$
(2.9)

The decomposition (2.1) is that the six-dimensional vector space of two-forms
canonically splits into the sum of three-dimensional vector spaces of self-
dual and anti-self-dual two-forms. Canonical basis elements of self-dual and
anti-self-dual two forms are given by

$\zeta^{i}_{+}=\frac{1}{2}\eta^{i}_{ab}e^{a}\wedge
e^{b}\in\Omega^{2}_{+},\qquad\zeta^{i}_{-}=\frac{1}{2}\bar{\eta}^{i}_{ab}e^{a}\wedge
e^{b}\in\Omega^{2}_{-}.$ (2.10)

Explicitly, they are given by

$\zeta^{1}_{\pm}=e^{2}\wedge e^{3}\pm e^{1}\wedge
e^{4},\quad\zeta^{2}_{\pm}=e^{3}\wedge e^{1}\pm e^{2}\wedge
e^{4},\quad\zeta^{3}_{\pm}=e^{1}\wedge e^{2}\pm e^{3}\wedge e^{4}.$ (2.11)

They satisfy the Hodge-duality equation

$*\zeta^{i}_{\pm}=\pm\zeta^{i}_{\pm}.$ (2.12)

and also obey the intersection relation [15]

$\zeta^{i}_{\pm}\wedge\zeta^{j}_{\pm}=\pm
2\delta^{ij}d\mu,\qquad\zeta^{i}_{\pm}\wedge\zeta^{j}_{\mp}=0,$ (2.13)

where $d\mu\equiv e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4}=\sqrt{g}d^{4}x$ is
a volume form on a four-manifold $M$.

Consider a Riemannian manifold $(M,g)$. The metric on $M$ takes the form

$ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}=e^{a}\otimes e^{a}.$ (2.14)

Using the metric, one can determine the spin connections
${\omega^{a}}_{b}={\omega^{a}}_{b\mu}dx^{\mu}$ and curvature tensors
${R^{a}}_{b}=\frac{1}{2}{R^{a}}_{b\mu\nu}dx^{\mu}\wedge dx^{\nu}$ by solving
the structure equations [23, 24]

$\displaystyle T^{a}=de^{a}+{\omega^{a}}_{b}\wedge e^{b}=0,$ (2.15)
$\displaystyle{R^{a}}_{b}=d{\omega^{a}}_{b}+{\omega^{a}}_{c}\wedge{\omega^{c}}_{b}.$
(2.16)

An important point is that the spin connections ${\omega^{a}}_{b}$ and Riemann
curvature tensors ${R^{a}}_{b}$ are $so(4)$-valued one-forms and two-forms in
$\Omega^{p}(M)=\Lambda^{p}T^{*}M$, respectively. Thus one can apply the
decompositions in (2.1) and (2.6) to spin connections and curvature tensors
[17, 18, 19]. The decomposition (2.4) implies that the spin connections can be
split into a pair of $SU(2)_{+}$ and $SU(2)_{-}$ connections according to the
Lie algebra splitting (2.6):

$\omega_{ab}=A^{(+)i}\eta^{i}_{ab}+A^{(-)i}\overline{\eta}^{i}_{ab}.$ (2.17)

The Riemann curvature tensor $R_{ab}=\frac{1}{2}R_{ab\mu\nu}dx^{\mu}\wedge
dx^{\nu}=\frac{1}{2}R_{abcd}e^{c}\wedge e^{d}\in
C^{\infty}(\mathfrak{g}\otimes\Omega^{2})$ is $so(4)$-valued 2-forms. Thus it
is involved with two vector spaces $\mathfrak{g}=so(4)$ in (2.4) and
$\Omega^{2}(M)$ in (2.1). Let us apply the decompositions of the vector spaces
$\mathfrak{g}$ and $\Omega^{2}(M)$ to Riemann curvature tensors. The first
decomposition is that the Riemann tensor can be split into a pair of
$SU(2)_{+}$ and $SU(2)_{-}$ curvatures according to the Lie algebra splitting
(2.6):

$R_{ab}=F^{(+)i}\eta^{i}_{ab}+F^{(-)i}\bar{\eta}^{i}_{ab},$ (2.18)

where $SU(2)_{\pm}$ curvatures are 2-forms on $M$ defined by

$\displaystyle F^{(\pm)i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}F^{(\pm)i}_{cd}e^{c}\wedge e^{d}$ (2.19)
$\displaystyle=$ $\displaystyle dA^{(\pm)i}-\varepsilon^{ijk}A^{(\pm)j}\wedge
A^{(\pm)k}.$

The second decomposition is that the $SU(2)_{\pm}$ curvatures in Eq. (2.19)
can be decomposed as

$F^{(+)i}=f^{ij}_{(++)}\zeta^{j}_{+}+f^{ij}_{(+-)}\zeta^{j}_{-},\qquad
F^{(-)i}=f^{ij}_{(-+)}\zeta^{j}_{+}+f^{ij}_{(--)}\zeta^{j}_{-}.$ (2.20)

Using the intersection pairing (2.13), the coefficients are determined as

$\displaystyle
f^{ij}_{(\pm\pm)}d\mu=\pm\frac{1}{4}\left(F^{(\pm)i}\wedge\zeta^{j}_{\pm}+F^{(\pm)j}\wedge\zeta^{i}_{\pm}\right),$
(2.21) $\displaystyle
f^{ij}_{(\pm\mp)}d\mu=\mp\frac{1}{4}\left(F^{(\pm)i}\wedge\zeta^{j}_{\mp}-F^{(\mp)j}\wedge\zeta^{i}_{\pm}\right).$
(2.22)

Combining the two decompositions (2.18) and (2.20) leads to an irreducible
decomposition of the general Riemann curvature tensor [17, 18]:

$R_{abcd}=f^{ij}_{(++)}\eta^{i}_{ab}\eta^{j}_{cd}+f^{ij}_{(+-)}\eta^{i}_{ab}\bar{\eta}^{j}_{cd}+f^{ij}_{(-+)}\bar{\eta}^{i}_{ab}\eta^{j}_{cd}+f^{ij}_{(--)}\bar{\eta}^{i}_{ab}\bar{\eta}^{j}_{cd}.$
(2.23)

Recently, the decomposition (2.23) has been used significantly in [25] to
study the scalar invariants of the curvature tensor.

Now we impose the first Bianchi identity

$R_{abcd}+R_{acdb}+R_{adbc}=0$ (2.24)

to the decomposition (2.23). The Bianchi identity, being totally 16
conditions, can equivalently be stated as

$\displaystyle R_{abcd}=R_{cdab},$ (2.25)
$\displaystyle\varepsilon^{abcd}R_{abcd}=0.$ (2.26)

Eq. (2.25) requires the symmetry property

$f^{ij}_{(++)}=f^{ji}_{(++)},\quad f^{ij}_{(--)}=f^{ji}_{(--)},\quad
f^{ij}_{(+-)}=f^{ji}_{(-+)}.$ (2.27)

whereas Eq. (2.26) imposes an additional constraint

$f^{ij}_{(++)}\delta^{ij}=f^{ij}_{(--)}\delta^{ij}.$ (2.28)

The “trace-free part” of the Riemann curvature tensor is called the Weyl
tensor defined by [23, 24]

$W_{abcd}=R_{abcd}-\frac{1}{2}(\delta_{ac}R_{bd}-\delta_{ad}R_{bc}-\delta_{bc}R_{ad}+\delta_{bd}R_{ac})+\frac{1}{6}(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc})R.$
(2.29)

The Weyl tensor shares all the symmetry structures of the curvature tensor and
all its traces with the metric are zero. Then the Weyl tensor can be
decomposed as [17, 18]

$\displaystyle
W_{abcd}={\widetilde{f}}^{ij}_{(++)}\eta^{i}_{ab}\eta^{j}_{cd}+{\widetilde{f}}^{ij}_{(--)}\bar{\eta}^{i}_{ab}\bar{\eta}^{j}_{cd},$
(2.30)

where ${\widetilde{f}}^{ij}_{(\pm\pm)}\equiv
f^{ij}_{(\pm\pm)}-\frac{1}{3}\delta^{ij}\big{(}f^{kl}_{(\pm\pm)}\delta^{kl}\big{)}$
are symmetric, traceless $3\times 3$ matrices.

The decomposition (2.23) can be applied to the Ricci tensor $R_{ab}\equiv
R_{acbc}$ and the Ricci scalar $R\equiv R_{aa}$ to yield

$\displaystyle R_{ab}$ $\displaystyle=$
$\displaystyle\big{(}f^{ij}_{(++)}\delta^{ij}+f^{ij}_{(--)}\delta^{ij}\big{)}\delta_{ab}+2f^{ij}_{(+-)}\eta^{i}_{ac}\bar{\eta}^{j}_{bc},$
$\displaystyle R$ $\displaystyle=$ $\displaystyle
4\big{(}f^{ij}_{(++)}\delta^{ij}+f^{ij}_{(--)}\delta^{ij}\big{)}.$ (2.31)

For Einstein manifolds satisfying the equations, $R_{ab}=\lambda\delta_{ab}$,
with $\lambda$ a cosmological constant, one can show [17, 18] that

$f^{ij}_{(+-)}=f^{ji}_{(-+)}=0.$ (2.32)

As a result, the curvature tensor for Einstein manifolds takes the simple form

$R_{abcd}=f^{ij}_{(++)}\eta^{i}_{ab}\eta^{j}_{cd}+f^{ij}_{(--)}\bar{\eta}^{i}_{ab}\bar{\eta}^{j}_{cd}.$
(2.33)

Therefore, we arrive at the following Lemma, proved in [3]:

$\mathbf{Lemma\;2.1}$: The Einstein equations,
$R_{ab}=\lambda\delta_{ab},\;\lambda\in\mathbb{R}$, for an orientable
Riemannian manifold $(M,g)$ are equivalent to the self-duality equations of
Yang-Mills instantons,

$F^{(\pm)i}=\pm*F^{(\pm)i},$ (2.34)

where $F^{(\pm)i}\wedge\zeta^{i}_{\pm}=\pm\lambda d\mu$.

It should be pointed out that the equivalence of the equations does not mean
the equivalence of solution spaces because the Einstein equations are the
second-order differential equations of an Einstein metric while the self-
duality equations are the first-order differential equations of an $SU(2)$
connection. For example, if an Einstein metric is given, then $SU(2)$
connections are determined up to gauge transformations by solving Eq. (2.17)
[16]. But the converse is not true. In order to determine the metric from
given $SU(2)$ connections which determine the spin connection via Eq. (2.17),
it is necessary to solve the torsion-free equation (2.15). Moreover, it is
necessary to specify the pair of instantons and anti-instantons to determine
the spin connections. Therefore the moduli space of Yang-Mills instantons is
not directly related to the moduli space of Einstein metrics. Nevertheless,
since non-Einstein metrics imply non-instanton connections, the Lemma
$\mathbf{2.1}$ will be very useful for studying the moduli space of Einstein
structures.

The above understanding of orientable four-manifolds comes via the geometry of
connections or gauge fields where Riemannian manifolds are described as a
gauge theory of $SU(2)_{\pm}$ connections in (2.17) and their curvatures in
(2.18). Therefore Riemannian manifolds encode a topological information in the
form of Yang-Mills instantons of $SU(2)_{\pm}$ gauge fields. Hence it is
natural to expect that the topological invariants of a Riemannian manifold
$(M,g)$ will be determined by the configuration of $SU(2)_{\pm}$ Yang-Mills
instantons. For a general closed Riemannian manifold $M$ without boundaries,
the Euler characteristic $\chi(M)$ and the Hirzebruch signature $\tau(M)$ are
defined by [2, 22, 23]

$\displaystyle\chi(M)$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi^{2}}\int_{M}\varepsilon^{abcd}R_{ab}\wedge
R_{cd},$ (2.35) $\displaystyle\tau(M)$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{2}}\int_{M}R_{ab}\wedge R_{ab}.$ (2.36)

The topological invariants can be expressed in terms of $SU(2)_{\pm}$
connections using the decompositions (2.18) and (2.20) [17, 19, 20]

$\displaystyle\chi(M)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\int_{M}\big{(}F^{(+)i}\wedge
F^{(+)i}-F^{(-)i}\wedge F^{(-)i}\big{)},$ (2.37) $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\int_{M}\left((f^{ij}_{(++)})^{2}+(f^{ij}_{(--)})^{2}-2(f^{ij}_{(+-)})^{2}\right)d\mu,$
$\displaystyle\tau(M)$ $\displaystyle=$
$\displaystyle\frac{1}{6\pi^{2}}\int_{M}\big{(}F^{(+)i}\wedge
F^{(+)i}+F^{(+)i}\wedge F^{(+)i}\big{)},$ (2.38) $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}}\int_{M}\left((f^{ij}_{(++)})^{2}-(f^{ij}_{(--)})^{2}\right)d\mu,$

where we used the intersection relation (2.13).

An Einstein manifold has curvature tensors given by (2.33). In this case, the
Euler characteristic $\chi(M)$ is determined by the sum of instantons and
anti-instantons whereas the Hirzebruch signature $\tau(M)$ is their
difference. Then it becomes easy to verify the important inequalities for the
topological invariants. The first inequality is $\chi(M)\geq 0$ with equality
only if $f^{ij}_{(++)}=f^{ij}_{(--)}=0$, i.e., $M$ is flat ($\mathbf{6.32}$ in
[2] and Sect. 10.4 in [23]). The second inequality is the Hitchin-Thorpe
inequality [1, 21] stating that

$\chi(M)\pm\frac{3}{2}\tau(M)=\frac{1}{\pi^{2}}\int_{M}\big{(}f^{ij}_{(\pm\pm)}\big{)}^{2}d\mu\geq
0$ (2.39)

where the equality holds only if $f^{ij}_{(++)}=0$ or $f^{ij}_{(--)}=0$, i.e.,
$M$ is half-flat (a gravitational instanton). A direct consequence of the
first inequality is that $\mathbb{S}^{1}\times\mathbb{S}^{3}$ does not admit
Einstein metrics [2]. On the other hand, the second inequality leads to the
result that $n\mathbb{C}\mathbb{P}^{2}\;(n\geq 4)$ is a simply connected
compact manifold which does not carry an Einstein metric [1], since $\chi=n+2$
and $\tau=n$. For the rational surfaces $M_{k}=\mathbb{C}\mathbb{P}^{2}\sharp
k\overline{\mathbb{C}\mathbb{P}}^{2}$, one has $\chi(M_{k})=3+k$ and
$\tau(M_{k})=1-k$, so the Hitchin-Thorpe inequality implies that $M_{k}$ does
not admit an Einstein metric if $k\geq 9$ [10]. The four-manifold $M_{k}$ does
admit an Einstein metric only for $k\leq 8$. Some examples of four-dimensional
compact Einstein manifolds are shown up in Fig. 1.

If an Einstein manifold with positive scalar curvature is further assumed to
have a nonnegative sectional curvature, more refined inequalities can be
stated [1]. A compact Einstein manifold $M$ with nonnegative sectional
curvature must have its Euler characteristic $\chi(M)$ bounded by
$1\leq\chi(M)\leq 9$ and its Hirzebruch signature $\tau(M)$ should satisfy the
inequality

$|\tau(M)|\leq\left(\frac{2}{3}\right)^{\frac{3}{2}}\chi(M).$ (2.40)

We will prove this inequality in appendix A. Since
$\left(\frac{2}{3}\right)^{\frac{3}{2}}\approx 0.544$ is an irrational number,
the equality can only occur if $M$ is flat. Then we see that
$|\tau(M)|\leq\left(\frac{2}{3}\right)^{\frac{3}{2}}\times 9\approx 4.899$,
i.e., $|\tau(M)|\leq 4$. These results show that most four-manifolds cannot
carry any Einstein structure with positive or nonnegative sectional curvature.
This is the reason why the red dots in Fig. 1 are mostly located in small
topological numbers except the $K3$ surface which is a half-flat manifold.

Figure 1:  Topological numbers of closed Einstein manifolds [20]

There is an important theorem on compact complex surfaces $(M,J)$ [7, 8] where
$J$ is the complex structure on $M$:

$\mathbf{Theorem\;2.2}$: A compact complex surface $(M,J)$ admits a Kähler-
Einstein metric with $\lambda>0$ if and only if the first Chern class
$c_{1}(M)$ is positive and the Lie algebra of holomorphic vector fields is
reductive. This occurs exactly on
$\mathbb{C}\mathbb{P}^{2},\;\mathbb{S}^{2}\times\mathbb{S}^{2}$, or
$\mathbb{C}\mathbb{P}^{2}\sharp k\overline{\mathbb{C}\mathbb{P}}^{2}\;(3\leq
k\leq 8)$ which is the blow-up of $\mathbb{C}\mathbb{P}^{2}$ at general $k$
points.

Thus the Page space
$\mathbb{C}\mathbb{P}^{2}\sharp\,\overline{\mathbb{C}\mathbb{P}}^{2}$, which
will be considered below, is not a Kähler manifold. Another non-Kähler metric
on $\mathbb{C}\mathbb{P}^{2}\sharp 2\overline{\mathbb{C}\mathbb{P}}^{2}$ was
constructed in [5]. Note that there are relatively few Kähler-Einstein metrics
when $\lambda\geq 0$; most all have $\lambda<0$, in analogy to the case of
Riemann surfaces.

It will be instructive to analyze the Einstein structure of the Page space [4]
in detail since it is the first example of nonhomogeneous Einstein manifolds
and its explicit metric is known. The Page space
$\mathbb{C}\mathbb{P}^{2}\sharp\,\overline{\mathbb{C}\mathbb{P}}^{2}$ can be
constructed by the blow-up of $\mathbb{C}\mathbb{P}^{2}$ at a point
$p\in\mathbb{C}\mathbb{P}^{2}$ [2]. The metric on the Page space takes on a
complicated form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle e^{a}\otimes e^{a}$
$\displaystyle=$
$\displaystyle\frac{3}{\lambda}\frac{1+\nu^{2}}{3+\nu^{2}}\left(U^{-1}d\chi^{2}+\frac{U}{4}\sin^{2}\chi(d\psi+\cos\theta
d\phi)^{2}+\frac{1-\nu^{2}\cos^{2}\chi}{4\nu}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right),$

where

$U(\chi)=1-\frac{2\nu^{2}\sin^{2}\chi}{(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)}$

and $\nu$ is the positive root of $\nu^{4}+4\nu^{3}-6\nu^{2}+12\nu-3=0$ which
works out to be $0.2817$. The spin connections for the Page metric (2) can be
determined by solving the torsion-free condition (2.15) as

$\displaystyle\omega_{12}=-\sqrt{\frac{4\nu}{1-\nu^{2}\cos^{2}\chi}}\cot\theta
e^{2}+fe^{3},\quad\omega_{13}=fe^{2},\quad\omega_{23}=-fe^{1},$ (2.42)
$\displaystyle\omega_{14}=ge^{1},\quad\omega_{24}=ge^{2},\quad\omega_{34}=\left(\sqrt{U}\cot\chi+\frac{2(\nu^{2}-1)}{U(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)}g\right)e^{3},$

where

$f=\frac{\sqrt{U}\nu}{1-\nu^{2}\cos^{2}\chi}\sin\chi,\qquad g=f\nu\cos\chi.$

Here we chose $\lambda=3\frac{1+\nu^{2}}{3+\nu^{2}}$ for simplicity. The
corresponding $SU(2)$ connections are given by [26]

$\displaystyle A^{(\pm)1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}(-f\pm
g)e^{1},\qquad A^{(\pm)2}=\frac{1}{2}(-f\pm g)e^{2},$ $\displaystyle
A^{(\pm)3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\sqrt{\frac{4\nu}{1-\nu^{2}\cos^{2}\chi}}\cot\theta
e^{2}+fe^{3}\pm\Big{(}\sqrt{U}\cot\chi+\frac{2(\nu^{2}-1)}{U(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)}g\Big{)}e^{3}\right).$

The corresponding $SU(2)$ curvatures have the form,
$f^{ij}_{(\pm\pm)}=\pm\psi^{i}_{(\pm)}\delta^{ij}$, i.e.,

$\displaystyle F^{(\pm)1}=\pm\psi^{1}_{(\pm)}\zeta^{1}_{\pm},\qquad
F^{(\pm)2}=\pm\psi^{2}_{(\pm)}\zeta^{2}_{\pm},\qquad
F^{(\pm)1}=\pm\psi^{3}_{(\pm)}\zeta^{3}_{\pm},$ (2.43)

where $\psi^{1}_{(\pm)}=\psi^{2}_{(\pm)}$ and

$\displaystyle\psi^{1}_{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{U}{2}\frac{\nu(1-\nu^{2})\cos\chi\mp\nu^{2}(\cos^{2}\chi-\sin^{2}\chi)\pm\nu^{4}\cos^{4}\chi}{(1-\nu^{2}\cos^{2}\chi)^{2}}$
$\displaystyle-\frac{\nu^{3}(1-\nu^{2})\cos\chi\sin^{2}\chi}{(3+\nu^{2})(1-\nu^{2}\cos^{2}\chi)^{3}}(1\mp\nu\cos\chi),$
$\displaystyle\psi^{3}_{(\pm)}$ $\displaystyle=$
$\displaystyle-\frac{\nu}{2(1-\nu^{2}\cos^{2}\chi)}\left(U^{\prime}\sin\chi+\frac{2(1-\nu^{2})U\cos\chi}{1-\nu^{2}\cos^{2}\chi}\right)$
(2.44) $\displaystyle\pm\frac{\nu}{2(1-\nu^{2}\cos^{2}\chi)}\left(4-\frac{\nu
U\sin^{2}\chi(3+\nu^{2}\cos^{2}\chi)}{1-\nu^{2}\cos^{2}\chi}\right).$

The result (2.43) shows that the $SU(2)_{+}\;(SU(2)_{-})$ curvatures of the
Page metric (2) are self-dual (anti-self-dual), so the Page space must be an
Einstein manifold according to the Lemma $\mathbf{2.1}$. One can explicitly
check that the Ricci tensor is given by (after recovering the factor
$\frac{3}{\lambda}\frac{1+\nu^{2}}{3+\nu^{2}}$)

$\displaystyle R_{ab}=\lambda\,\text{diag}\left(c,c,1,1\right),$ (2.45)

where
$c=\frac{\nu^{4}-8\nu^{3}+9\nu^{2}-24\nu+3\left(\nu^{2}+1\right)\nu^{2}\cos
2\chi}{3\nu^{4}-3\nu^{2}-6+3\left(\nu^{2}+1\right)\nu^{2}\cos 2\chi}$. Using
the quartic equation $\nu^{4}+4\nu^{3}-6\nu^{2}+12\nu-3=0$, one can show that
$c=1$. We show in appendix A that (anti-)self-dual Weyl tensors have only two
distinct eigenvalues at each point. So, according to Proposition 5 in [9], the
Page metric (2) is locally conformal to a Kähler metric although it is not a
Kähler metric in itself.

The explicit calculation using the result (2.43) gives us $\chi(M)=4$ and
$\tau(M)=0$ for the Page space [22] as shown up in Fig. 1. It should be
$\tau(M)=0$ if $M$ is the manifold obtained by blowing up one point in
$\mathbb{C}\mathbb{P}^{2}$ (see $\mathbf{11.82\;Example}$ in [2]).

##  3 Squashed Four-Spheres

Let us define the instanton density as
$\rho^{(\pm)}_{n}=-\frac{1}{4}\mathrm{Tr}F^{(\pm)}_{\alpha\beta}F^{(\pm)\alpha\beta}$,
which describes a curvature distribution over $M$. The following formula is
useful;

$F^{(\pm)i}\wedge F^{(\pm)i}=\pm 2\rho^{(\pm)}_{n}d\mu.$ (3.1)

If $M$ is a homogeneous Einstein manifold, the left-hand side in Eq. (3.1)
must be a constant multiple of the volume form. In other words, the instanton
density (3.1) characterizes the homogeneity of an underlying Einstein
manifold. One can check that the Page space (2) has a nontrivial distribution
of the instanton density [26], so confirm the nonhomogeneity. Using the
definition (3.1), the topological invariants in (2.37) and (2.38) can be
written as

$\displaystyle\chi(M)$ $\displaystyle=$
$\displaystyle\int_{M}\rho_{\chi}(M)d\mu,$ (3.2) $\displaystyle\tau(M)$
$\displaystyle=$ $\displaystyle\int_{M}\rho_{\tau}(M)d\mu,$ (3.3)

where

$\displaystyle\rho_{\chi}(M)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}(\rho^{(+)}_{n}+\rho^{(-)}_{n}),$ (3.4)
$\displaystyle\rho_{\tau}(M)$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}}(\rho^{(+)}_{n}-\rho^{(-)}_{n}).$ (3.5)

Thus
$\rho_{\chi}(M)\pm\frac{3}{2}\rho_{\tau}(M)=\frac{1}{\pi^{2}}\rho^{(\pm)}_{n}\geq
0$.

We now examine another nonhomogeneous space obtained by deforming a round
sphere $\mathbb{S}^{4}$ whose deformation is parameterized by three axis-
length parameters $(r,l,\tilde{l})$. The four-dimensional ellipsoids are
defined by

$\frac{x_{0}^{2}}{r^{2}}+\frac{x_{1}^{2}+x_{2}^{2}}{l^{2}}+\frac{x_{3}^{2}+x_{4}^{2}}{\tilde{l}^{2}}=1.$
(3.6)

The metric of squashed 4-sphere is given by $ds^{2}=e^{a}\otimes e^{a}$ with
the vierbeins [27]

$\left\\{e^{1}=\sin\rho\tilde{\sigma}^{1},\;e^{2}=\sin\rho\tilde{\sigma}^{2},\;e^{3}=\sin\rho\tilde{\sigma}^{3}+hd\rho,\;e^{4}=gd\rho\right\\},$
(3.7)

where

$\displaystyle f$ $\displaystyle\equiv$
$\displaystyle\sqrt{l^{2}\sin^{2}\theta+\tilde{l}^{2}\cos^{2}\theta},$
$\displaystyle g$ $\displaystyle\equiv$
$\displaystyle\sqrt{r^{2}\sin^{2}\rho+l^{2}\tilde{l}^{2}f^{-2}\cos^{2}\rho},$
$\displaystyle h$ $\displaystyle\equiv$
$\displaystyle\frac{\tilde{l}^{2}-l^{2}}{f}\cos\rho\sin\theta\cos\theta,$

and $\tilde{\sigma}^{i}\,(i=1,2,3)$ are dreibeins of the three-dimensional
ellipsoid given in polar coordinates $(\varphi,\chi,\theta)$ by

$\tilde{\sigma}^{1}=l\cos\theta
d\varphi,\quad\tilde{\sigma}^{2}=\tilde{l}\sin\theta
d\chi,\quad\tilde{\sigma}^{3}=fd\theta.$

The ranges of coordinates are $\rho\in[0,\pi],\;\theta\in[0,\frac{\pi}{2}]$
and $\varphi,\chi\in[0,2\pi]$.

It is straightforward to calculate the $SU(2)$ connections

$\displaystyle A^{(\pm)1}$ $\displaystyle=$
$\displaystyle\pm\frac{\tilde{l}^{2}}{2gf^{2}}\cot\rho
e^{1}+\frac{\cot\theta}{2f\sin\rho}e^{2},$ $\displaystyle A^{(\pm)2}$
$\displaystyle=$ $\displaystyle\pm\frac{l^{2}}{2gf^{2}}\cot\rho
e^{2}+\frac{\tan\theta}{2f\sin\rho}e^{1},$ (3.8) $\displaystyle A^{(\pm)3}$
$\displaystyle=$
$\displaystyle\pm\frac{l^{2}\tilde{l}^{2}}{2gf^{4}}\cot\rho\left(e^{3}-\frac{h}{g}e^{4}\right),$

and their curvatures

$\displaystyle F^{(\pm)1}$ $\displaystyle=$
$\displaystyle\pm\frac{\tilde{l}^{2}}{2f^{3}\sin^{2}\rho}\left(\frac{h}{g^{3}f^{3}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)-\frac{(\tilde{l}^{2}-l^{2})}{2}\psi\sin
2\theta\right)e^{3}\wedge e^{1}$
$\displaystyle\pm\frac{\tilde{l}^{2}}{2gf^{2}\sin^{2}\rho}\left(\frac{r^{2}}{g^{3}}\sin^{2}\rho+\frac{h^{2}}{g^{3}f^{4}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)-\frac{(\tilde{l}^{2}-l^{2})h}{2f}\psi\sin
2\theta\right)e^{1}\wedge e^{4}$
$\displaystyle+\frac{l^{2}r^{2}}{2f^{4}g^{2}}e^{2}\wedge
e^{3}-\frac{l^{2}r^{2}h}{2f^{4}g^{3}}e^{2}\wedge e^{4}$ $\displaystyle
F^{(\pm)2}$ $\displaystyle=$
$\displaystyle\pm\frac{l^{2}}{2f^{3}\sin^{2}\rho}\left(-\frac{h}{g^{3}f^{3}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)+\frac{(\tilde{l}^{2}-l^{2})}{2}\psi\sin
2\theta\right)e^{2}\wedge e^{3}$
$\displaystyle\pm\frac{l^{2}}{2gf^{2}\sin^{2}\rho}\left(\frac{r^{2}}{g^{3}}\sin^{2}\rho+\frac{h^{2}}{g^{3}f^{4}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)-\frac{(\tilde{l}^{2}-l^{2})h}{2f}\psi\sin
2\theta\right)e^{2}\wedge e^{4}$
$\displaystyle+\frac{\tilde{l}^{2}r^{2}}{2f^{4}g^{2}}e^{3}\wedge
e^{1}+\frac{\tilde{l}^{2}r^{2}h}{2f^{4}g^{3}}e^{1}\wedge e^{4}$ $\displaystyle
F^{(\pm)3}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{2f^{2}g^{2}}\left(e^{1}\wedge
e^{2}\pm\frac{l^{2}\tilde{l}^{2}}{f^{2}g^{2}}e^{3}\wedge e^{4}\right),$

where $\psi\equiv\frac{\cos\rho}{gf^{2}}$.

The squashed four-sphere is not an Einstein manifold. According to the Lemma
$\mathbf{2.1}$, $SU(2)$ connections constructed from a four-manifold $M$
become self-dual or anti-self-dual only if $M$ is Einstein, i.e., obeying the
vacuum Einstein equations, $R_{\mu\nu}=\lambda g_{\mu\nu}$. One can check that
the $SU(2)$ curvatures in Eq. (3) do not satisfy the self-duality (2.34). The
instanton density for the metric (3.7) exhibits a nontrivial distribution

$\displaystyle\rho^{(\pm)}_{n}=\frac{3}{4}\frac{l^{2}\tilde{l}^{2}r^{4}}{f^{6}g^{6}}.$
(3.10)

Therefore the squashed $\mathbb{S}^{4}$ is nonhomogeneous as expected. When
$l=\tilde{l}=r$, it becomes the round four-sphere $\mathbb{S}^{4}$ which is an
Einstein manifold. In this case, the $SU(2)$ connections in (3) become
instanton connections222The connections on $\mathbb{S}^{4}$ can be mapped to
the gauge fields of a self-dual SU(2) instanton on $\mathbb{R}^{4}$ by using
the $\mathbb{R}^{4}$-coordinates for the left-invariant one-forms $\sigma^{i}$
on $\mathbb{S}^{3}$ [26]. and the instanton density (3.10) becomes constant,
i.e., $F^{(\pm)i}\wedge F^{(\pm)i}=\pm\frac{3}{2r^{4}}d\mu$.

The deformation of Einstein structures is captureed by the coefficients
$f^{ij}_{(\pm\mp)}$ which can be computed by Eq. (2.22):

$\displaystyle f^{11}_{(\pm\mp)}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{l}^{2}}{4gf^{2}\sin^{2}\rho}\left(\frac{r^{2}}{g^{3}}\sin^{2}\rho+\frac{h^{2}}{g^{3}f^{4}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)-\frac{(\tilde{l}^{2}-l^{2})h}{2f}\psi\sin
2\theta\right)+\frac{l^{2}r^{2}}{4f^{4}g^{2}},$ $\displaystyle
f^{12}_{(\pm\mp)}$ $\displaystyle=$
$\displaystyle\pm\frac{(l^{2}+\tilde{l}^{2})}{8f^{3}\sin^{2}\rho}\left(\frac{h}{g^{3}f^{3}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)-\frac{(\tilde{l}^{2}-l^{2})}{2}\psi\sin
2\theta+\frac{r^{2}h}{fg^{3}}\sin^{2}\rho\right),$ $\displaystyle
f^{22}_{(\pm\mp)}$ $\displaystyle=$
$\displaystyle-\frac{l^{2}}{4gf^{2}\sin^{2}\rho}\left(\frac{r^{2}}{g^{3}}\sin^{2}\rho+\frac{h^{2}}{g^{3}f^{4}}(2g^{2}f^{2}-l^{2}\tilde{l}^{2}\cos^{2}\rho)-\frac{(\tilde{l}^{2}-l^{2})h}{2f}\psi\sin
2\theta\right)+\frac{\tilde{l}^{2}r^{2}}{4f^{4}g^{2}},$ $\displaystyle
f^{33}_{(\pm\mp)}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{4f^{2}g^{2}}\left(1-\frac{l^{2}\tilde{l}^{2}}{f^{2}g^{2}}\right),$
(3.11)

and
$f^{13}_{(\pm\mp)}=f^{23}_{(\pm\mp)}=f^{31}_{(\pm\mp)}=f^{32}_{(\pm\mp)}=0$.
If the coefficients in Eq. (3) vanish, the metric (3.7) now becomes an
Einstein metric. Therefore a question is whether there is another Einstein
manifold obtained by continuous deformations of the round four-sphere. Note
that the deformations in (3.6) correspond to the deformation of Einstein
structures within a fixed conformal class. Hence, the rigidity of the Einstein
structure [2, 11] implies that there is no solution of the vanishing condition
except as $l=\tilde{l}=r$. It is easy to see that the coefficients in Eq. (3)
vanish only if $r=l=\tilde{l}$.

The above result implies that there is no continuous deformation of Einstein
structures. Therefore we confirm that the moduli space of Einstein structures
is an isolated point. Now we will generalize the result by considering the
family of squashed 4-spheres. The metric is defined by [28]

$ds^{2}=dr^{2}+\frac{f(r)^{2}}{4}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})+\frac{g(r)^{2}}{4}(d\psi+\cos\theta d\phi)^{2}$ (3.12)

and vierbein one-forms are

$e^{a}=\left\\{\frac{f(r)}{2}d\theta,\frac{f(r)}{2}\sin\theta
d\phi,\frac{g(r)}{2}(d\psi+\cos\theta d\phi),dr\right\\}$

where $f(r)$ and $g(r)$ are smooth functions of $r$.333For the metric (3.12),
$0\leq r\leq\pi,\;0\leq\theta\leq\pi,\;0\leq\phi\leq 2\pi,\;0\leq\psi\leq
4\pi$. See Eq. (55) in [28]. The above metric has $SU(2)\times U(1)$ isometry
[29]. The round four-sphere $\mathbb{S}^{4}$ corresponds to $f(r)=g(r)=\sin
r$. Note that deformations described by the functions $f(r)$ and $g(r)$ can
also include a deformation of conformal structures of an Einstein metric [30,
31, 32]. However, we will show that there exists only a trivial deformation,
at most only changing the size of the sphere, if the Einstein structure is
maintained.

It is straightforward to calculate the $SU(2)$ connections from the metric
(3.12)

$\displaystyle A^{(\pm)1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{g}{f^{2}}\pm\frac{f^{\prime}}{f}\right)e^{1},\qquad
A^{(\pm)2}=\frac{1}{2}\left(-\frac{g}{f^{2}}\pm\frac{f^{\prime}}{f}\right)e^{2},$
$\displaystyle A^{(\pm)3}$ $\displaystyle=$
$\displaystyle-\frac{\cot\theta}{f}e^{2}+\frac{1}{2}\left(\frac{g}{f^{2}}\pm\frac{g^{\prime}}{g}\right)e^{3}$
(3.13)

and their curvatures

$\displaystyle F^{(\pm)1}$ $\displaystyle=$
$\displaystyle\frac{1}{2f^{3}}(fg^{\prime}-f^{\prime}g)(e^{1}\wedge e^{4}\pm
e^{2}\wedge e^{3})\mp\frac{f^{\prime\prime}}{2f}e^{1}\wedge
e^{4}-\frac{1}{2f}\left(\frac{f^{\prime}g^{\prime}}{g}-\frac{g^{2}}{f^{3}}\right)e^{2}\wedge
e^{3},$ $\displaystyle F^{(\pm)2}$ $\displaystyle=$
$\displaystyle\frac{1}{2f^{3}}(fg^{\prime}-f^{\prime}g)(e^{2}\wedge e^{4}\pm
e^{3}\wedge e^{1})\mp\frac{f^{\prime\prime}}{2f}e^{2}\wedge
e^{4}-\frac{1}{2f}\left(\frac{f^{\prime}g^{\prime}}{g}-\frac{g^{2}}{f^{3}}\right)e^{3}\wedge
e^{1},$ (3.14) $\displaystyle F^{(\pm)3}$ $\displaystyle=$
$\displaystyle-\frac{1}{f^{3}}(fg^{\prime}-f^{\prime}g)(e^{3}\wedge e^{4}\pm
e^{1}\wedge e^{2})\mp\frac{g^{\prime\prime}}{2g}e^{3}\wedge
e^{4}-\frac{1}{2f^{4}}(f^{2}f^{\prime 2}+3g^{2}-4f^{2})e^{1}\wedge e^{2}.$

The instanton density on the four-manifold $(M,g)$ is given by

$\displaystyle F^{(\pm)i}\wedge F^{(\pm)i}$ $\displaystyle=$
$\displaystyle\left[-\frac{2}{f^{4}}(fg^{\prime}-f^{\prime}g)f^{\prime\prime}+\frac{2g^{\prime\prime}}{f^{3}g}(fg^{\prime}-f^{\prime}g)\right.$
$\displaystyle\;\;\left.\pm\left(\frac{3}{f^{6}}(fg^{\prime}-f^{\prime}g)^{2}+\Big{(}\frac{f^{\prime\prime}}{f}\Big{)}^{2}+\frac{1}{2}\Big{(}\frac{g^{\prime\prime}}{g}\Big{)}^{2}\right)\right]d\mu$
$\displaystyle=$
$\displaystyle\left[\frac{1}{f^{4}g}\frac{d}{dr}(fg^{\prime}-f^{\prime}g)^{2}\pm\left(\frac{3}{f^{6}}(fg^{\prime}-f^{\prime}g)^{2}+\Big{(}\frac{f^{\prime\prime}}{f}\Big{)}^{2}+\frac{1}{2}\Big{(}\frac{g^{\prime\prime}}{g}\Big{)}^{2}\right)\right]d\mu.$

For the round four-sphere $\mathbb{S}^{4}$, $f(r)=g(r)=\sin r$ and
$F^{(\pm)i}\wedge F^{(\pm)i}=\pm\frac{3}{2}d\mu$. But the squashed
$\mathbb{S}^{4}$ has $f(r)\neq g(r)$ and the instanton density is not uniform
in general because the underlying manifold is not homogeneous.

One can see that the self-duality (2.34) is satisfied if the functions $f(r)$
and $g(r)$ obey the differential equations

$\frac{f^{\prime\prime}}{f}=\frac{f^{\prime}g^{\prime}}{fg}-\frac{g^{2}}{f^{4}},\qquad\frac{g^{\prime\prime}}{g}=\frac{f^{\prime
2}-4}{f^{2}}+\frac{3g^{2}}{f^{4}}.$ (3.16)

Then the squashed $\mathbb{S}^{4}$ becomes an Einstein manifold according to
the Lemma $\mathbf{2.1}$. Consider infinitesimal deformations from the round
sphere:

$f(r)=\sin r+\epsilon p(r),\qquad g(r)=\sin r+\epsilon q(r),$ (3.17)

where $\epsilon$ is an infinitesimal deformation parameter and $p(r)$ and
$q(r)$ are smooth functions. Then the differential equations (3.16) at the
first order of $\epsilon$ reduce to

$\displaystyle p^{\prime\prime}-q=(p^{\prime}+q^{\prime})\cot
r+\frac{3}{\sin^{2}r}(p-q),$ $\displaystyle
q^{\prime\prime}+q-2p=2p^{\prime}\cot r-\frac{6}{\sin^{2}r}(p-q).$ (3.18)

Define $h\equiv p-q$. Then, $h^{\prime\prime}+h^{\prime}\cot
r+\left(2-\frac{9}{\sin^{2}r}\right)h=0$. A general solution is

$h(r)=\frac{1}{\sin^{3}r}\big{(}a_{1}\cos
r+a_{2}(4\sin^{2}r+\sin^{4}r-8)\big{)}.$ (3.19)

Since $h(r)$ must be a smooth and finite function on $r\in[0,\pi]$, we require
$a_{1}=a_{2}=0$, i.e., $p=q$. With this condition, a general solution of Eq.
(3) is

$p(r)=q(r)=c_{1}\cos r+c_{2}(\sin r-r\cos r),$ (3.20)

where $c_{1}$ and $c_{2}$ are arbitrary constants.

The existence of the solution (3.20) appears as if a new Einstein manifold has
been obtained. At first sight, it seems to contradict the rigidity of the
Einstein structure [2, 11]. However, it turns out to be a trivial solution, at
most equivalent to changing only the size of the four-sphere. First let us
examine the Euler characteristic $\chi(M)$ and the Hirzebruch signature
$\tau(M)$ by expanding them for the infinitesimal deformation (3.17) up to the
first order such that $\chi(M)=2+\epsilon\chi_{1}(M)+\cdots$ and
$\tau(M)=0+\epsilon\tau_{1}(M)+\cdots$. It is easy to see that $\tau_{1}(M)=0$
since $f(r)=g(r)$ in (3.17), so $F^{(\pm)i}\wedge
F^{(\pm)i}=\pm\frac{3}{2}\Big{(}\frac{f^{\prime\prime}}{f}\Big{)}^{2}d\mu$. It
is also easy to verify that $\chi_{1}(M)=0$. Thus the topological numbers do
not change under the deformation (3.17), as they should be.

The $SU(2)$ connections (3) under the infinitesimal deformation (3.17) are
changed as

$\displaystyle A^{\prime(\pm)1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(-1\pm\cos
r)d\theta\pm\frac{\epsilon}{4}p^{\prime}(r)d\theta,$ $\displaystyle
A^{\prime(\pm)2}$ $\displaystyle=$ $\displaystyle\frac{1}{4}(-1\pm\cos
r)\sin\theta d\phi\pm\frac{\epsilon}{4}p^{\prime}(r)\sin\theta d\phi,$ (3.21)
$\displaystyle A^{\prime(\pm)3}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\cos\theta d\phi+\frac{1}{4}(1\pm\cos
r)(d\psi+\cos\theta d\phi)\pm\frac{\epsilon}{4}p^{\prime}(r)(d\psi+\cos\theta
d\phi),$

where $p^{\prime}(r)=(c_{2}r-c_{1})\sin r$. Since the added terms linear in
$\epsilon$ simply correspond to a replacement $f(r)=g(r)=\sin
r\to\tilde{f}(r)=\tilde{g}(r)=\sin r+\epsilon p(r)$ in the metric (3.12), we
can use Eq. (3) to calculate the curvatures for the deformed connections (3).
The result is444The reason why $\chi_{1}(M)=0$ is that the volume form also
has a linear correction,
$d\mu=d\mu(\mathbb{S}^{4})\left(1+3\epsilon\frac{p(r)}{\sin r}\right)$, which
exactly cancels the linear term in (3.22).

$F^{\prime(\pm)i}=\left(\frac{1}{2}-\epsilon c_{2}\right)\zeta^{i}_{\pm}.$
(3.22)

The result (3.22) shows that the smooth deformations (3.17) preserve the
Einstein structure since $f^{ij}_{(\pm\mp)}$ identically vanish. It turns out
that even the Einstein-Weyl structure [30, 31, 32] has not been deformed by
the solutions (3.20). The Einstein-Weyl structure deformations are captured by
${\widetilde{f}}^{ij}_{(\pm\pm)}\equiv
f^{ij}_{(\pm\pm)}-\frac{1}{3}\delta^{ij}\big{(}f^{kl}_{(\pm\pm)}\delta^{kl}\big{)}$
in Eq. (2.30). For the curvature (3.22),
$f^{ij}_{(\pm\pm)}=\left(\frac{1}{2}-\epsilon c_{2}\right)\delta^{ij}$ and so
${\widetilde{f}}^{ij}_{(\pm\pm)}=0$. This means that the Einstein metric
determined by the functions in (3.17) is still conformally flat. This in turn
implies that the Einstein-Weyl structure of a round four-sphere is preserved,
while maintaining the Einstein structure. It is allowed to change at most only
the size of the sphere. Such a constant scaling is regarded as trivial (see
Theorem $\mathbf{5.41}$ in [2]). This property was crucial in [26, 33] to
realize the instanton-induced inflation of four-dimensional spacetime and the
dynamical compactification of internal space in higher-dimensional Einstein-
Yang-Mills theory.

##  4 Discussion

We expect that $\mathbb{C}\mathbb{P}^{2}$ will exhibit a similar feature as
$\mathbb{S}^{4}$ for the Einstein deformations [2, 11]. The Theorem
$\mathbf{2.2}$ also implies that any continuous deformations of
$\mathbb{C}\mathbb{P}^{2}$ will break the Einstein structure. However, some
details are worth studying for the case of $\mathbb{C}\mathbb{P}^{2}$.
Interestingly, $\mathbb{S}^{2}\times\mathbb{S}^{2}$ would reveal a very
different aspect of Einstein structures since all two-dimensional Riemannian
manifolds are Einstein spaces. So any smooth deformations of $\mathbb{S}^{2}$
(e.g., ellipsoids) will not change the underlying Einstein structure. This
reasoning is consistent with the Proposition 12.75 (the case $SU(m)/SO(m)$ for
$m=2$) in [2].

Let us consider from a physical perspective why the deformation of Einstein
structures is so rigid. The Einstein equations with matters take a more
general form

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\lambda g_{\mu\nu}=\frac{8\pi
G}{c^{4}}T_{\mu\nu},$ (4.1)

where the right-hand side refers to the energy-momentum tensor of matters.
Without matters, the Einstein equations reduce to $R_{\mu\nu}=\lambda
g_{\mu\nu}$, whose solutions are called Einstein manifolds. Therefore, the
moduli space of Einstein structures refers to the space of all possible
solutions of Einstein equations without matters (i.e., in vacuum). This system
is not elliptic due to the diffeomorphism invariance of the equations.
However, for a suitable gauge choice (e.g., the de Donder gauge), the
restricted equations become elliptic [10]. To uniquely determine a solution to
Einstein equations, it is necessary to specify suitable initial conditions or
boundary conditions. The uniqueness of solutions can depend on several factors
such as the choice of boundary conditions and the presence of symmetries,
which are usually asymptotic and global conditions. Given a solution under the
specified condition, local deformations of the given solution cannot change
such an asymptotic and global condition underling the solution. Therefore, the
existence of a nontrivial moduli space of Einstein structures depends on
whether a new Einstein metric can be generated by local deformations while
maintaining such asymptotic and global conditions. Considering that spacetime
behaves like a metrical elasticity with tension, delicate balances are
required to maintain such deformations. For example, the Page metric is
obtained as a limiting case of the Kerr-de Sitter solution [4, 22]. This may
be a reason why the Page space could be an Einstein metric even though it is
nonhomogeneous. Furthermore, it is a nontrivial $\mathbb{S}^{2}$ bundle over
$\mathbb{S}^{2}$ which is more flexible against (local) deformations than
other Einstein spaces. For this reason, it seems very unlikely that one
parameter family of Einstein structure deformations exists.

Recent progresses in four-dimensional topology has shown that many four-
dimensional manifolds have exotic differential structures [15]. However, the
Poincaré conjecture in dimension 4 remains to be an open problem, that is,
whether there exists exotic differential structures on the four-sphere.
According to the Freedman’s classification [34], two smooth, compact, simply-
connected four-manifolds are homeomorphic if and only if they are both spin or
both non-spin and their Euler characteristics and signatures are equal. Four-
sphere is consisted of an $SU(2)_{+}$ instanton and an $SU(2)_{-}$ anti-
instanton which determines $\chi=2$ and $\tau=0$ [17, 26] as shown in Fig. 1.
Thus it is impossible to change the instanton composition in order for two
four-spheres to have the same smooth structure. Perhaps only deforming the
arrangement of two instantons is allowed. But it corresponds to the
deformation of Einstein structures on $\mathbb{S}^{4}$. Therefore the exotic
differential structures on the four-sphere may be deeply related to its
Einstein structures [35].

## Acknowledgments

We are deeply grateful to Prof. Youngjoo Chung for his help on the solution of
differential equations. This work was supported by the National Research
Foundation of Korea (NRF) with grant numbers NRF-2019R1A2C1007396 (KK) and
NRF-2018R1D1A1B0705011314 (HSY). JH was supported by the Ministry of Education
through the Center for Quantum Spacetime (CQUeST) of Sogang University
(NRF-2020R1A6A1A03047877) and NRF-2020R1A2C1014371. We acknowledge the
hospitality at APCTP where part of this work was done.

##  Appendix A Strong version of the Hitchin-Thorpe inequality

The Hitchin–Thorpe inequality [1, 2] provides an important obstruction to the
existence of Einstein metrics on four-manifolds. In this appendix, we will
prove the strong version of the Hitchin-Thorpe inequality (2.40). Here we
assume that a compact Einstein manifold has nonnegative sectional curvature.

The curvature tensors for an Einstein manifold are given by (2.33) where the
coefficients $f^{ij}_{(\pm\pm)}$ are real symmetric $3\times 3$ matrices.
Therefore they can be diagonalized by orthogonal transformations as
[25]555Since $SU(2)_{+}$ and $SU(2)_{-}$ parts are independent of each other,
they can be diagonalized separately.

$(A_{\pm})_{ij}\equiv
f^{ij}_{(\pm\pm)}=\mathrm{diag}(a_{\pm}^{1},a_{\pm}^{2},a_{\pm}^{3}).$ (A.1)

The diagonal elements satisfy the relation

$a_{+}^{1}+a_{+}^{2}+a_{+}^{3}=a_{-}^{1}+a_{-}^{2}+a_{-}^{3}=\frac{R}{8}$
(A.2)

according to Eqs. (2.28) and (2).

We regard the curvature tensor $R$ as a symmetric linear transformation of the
bundle $\Omega^{2}(M)$ of 2-forms defined by [1]

$R(e^{a}\wedge e^{b})=R_{ab}=\frac{1}{2}R_{abcd}e^{c}\wedge e^{d}$ (A.3)

relative to a local orthonormal basis $\\{e^{a}\\}$ of the 1-forms. We choose
the basis of $\Omega^{2}(M)$ as $\\{e^{1}\wedge e^{2},e^{3}\wedge
e^{1},e^{2}\wedge e^{3},e^{3}\wedge e^{4},e^{2}\wedge e^{4},e^{1}\wedge
e^{4}\\}$ according to the canonical pairing (2.11). Then $R$ takes the form
[1]

$R=\left(\begin{array}[]{cc}A&B\\\ B&A\\\ \end{array}\right)$ (A.4)

where

$A=\left(\begin{array}[]{ccc}\lambda_{1}&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}\\\ \end{array}\right),\qquad
B=\left(\begin{array}[]{ccc}\mu_{1}&0&0\\\ 0&\mu_{2}&0\\\ 0&0&\mu_{3}\\\
\end{array}\right),$ (A.5)

and
$\lambda_{1}=a_{+}^{3}+a_{-}^{3},\;\lambda_{2}=a_{+}^{2}+a_{-}^{2},\;\lambda_{3}=a_{+}^{1}+a_{-}^{1}$,
and
$\mu_{1}=a_{+}^{3}-a_{-}^{3},\;\mu_{2}=a_{+}^{2}-a_{-}^{2},\;\mu_{3}=a_{+}^{1}-a_{-}^{1}$.
The condition for the nonnegative sectional curvature means that
$\lambda_{i}\geq 0,\,i=1,2,3$. Eq. (A.2) imposes the conditions
$\lambda_{1}+\lambda_{2}+\lambda_{3}=\frac{R}{4}>0$ and
$\mu_{1}+\mu_{2}+\mu_{3}=0$. Thus we set $\mu_{3}=-(\mu_{1}+\mu_{2})$ and
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq 0$. We regard
$\vec{\lambda}=(\lambda_{1},\lambda_{2},\lambda_{3})$ and
$\vec{\mu}=(\mu_{1},\mu_{2},\mu_{3})$ as vectors in $\mathbb{R}^{3}$. Since
the vector $\vec{\mu}=(\mu_{1},\mu_{2},\mu_{3})$ is constrained by
$\mu_{1}+\mu_{2}+\mu_{3}=0$, if $\vec{\lambda},\;\vec{\mu}$ are nonzero, then
the angle $\theta$ between them must satisfy

$\displaystyle\cos\theta$ $\displaystyle=$
$\displaystyle\frac{\vec{\lambda}\cdot\vec{\mu}}{|\vec{\lambda}||\vec{\mu}|}\leq\frac{1}{|\vec{\lambda}|\sqrt{6\mu_{1}\mu_{2}}}\big{(}(\lambda_{1}-\lambda_{3})\mu_{1}+(\lambda_{2}-\lambda_{3})\mu_{2}\big{)}\leq\sqrt{\frac{2}{3}}\frac{\sqrt{(\lambda_{3}-\lambda_{1})(\lambda_{3}-\lambda_{2})}}{|\vec{\lambda}|}$
(A.6) $\displaystyle\leq$ $\displaystyle\sqrt{\frac{2}{3}},$

using the inequality between arithmetic and geometric means. Therefore

$\vec{\lambda}\cdot\vec{\mu}\leq\sqrt{\frac{2}{3}}|\vec{\lambda}||\vec{\mu}|.$

From the inequality $|\vec{\lambda}|^{2}+|\vec{\mu}|^{2}\geq
2|\vec{\lambda}||\vec{\mu}|$, we get

$|\vec{\lambda}|^{2}+|\vec{\mu}|^{2}\geq
2\sqrt{\frac{3}{2}}(\vec{\lambda}\cdot\vec{\mu}).$ (A.7)

Note that

$|\vec{\lambda}|^{2}+|\vec{\mu}|^{2}=4\pi^{2}\rho_{\chi},\qquad(\vec{\lambda}\cdot\vec{\mu})=3\pi^{2}\rho_{\tau},$

where the densities were defined in (3.4) and (3.5). Then the inequality (A.7)
can be written as

$\rho_{\tau}\leq\left(\frac{2}{3}\right)^{\frac{3}{2}}\rho_{\chi}.$ (A.8)

Integrating and using the expressions for the Euler characteristic and
Hirzebruch signature in (3.2) and (3.3), we obtain the inequality (2.40).
Incidentally, the Hitchin-Thorpe inequality (2.39) can also be derived from
the inequality

$4\pi^{2}\left(\rho_{\chi}\pm\frac{3}{2}\rho_{\tau}\right)=(\vec{\lambda}\pm\vec{\mu})^{2}\geq
0.$

The case of nonpositive sectional curvature is similar.

The Hitchin-Thorpe inequality can be further refined by rewriting the
topological invariants in Eqs. (2.37) and (2.38) as follows:

$\displaystyle\chi(M)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\int_{M}\left((\widetilde{f}^{ij}_{(++)})^{2}+(\widetilde{f}^{ij}_{(--)})^{2}+\frac{R^{2}}{96}\right)d\mu,$
(A.9) $\displaystyle\tau(M)$ $\displaystyle=$
$\displaystyle\frac{1}{3\pi^{2}}\int_{M}\left((\widetilde{f}^{ij}_{(++)})^{2}-(\widetilde{f}^{ij}_{(--)})^{2}\right)d\mu,$
(A.10)

where
$\widetilde{f}^{ij}_{(\pm\pm)}=f^{ij}_{(\pm\pm)}-\frac{R}{24}\delta^{ij}$ are
the coefficients in the expansion of Weyl tensors in Eq. (2.30). Then we get

$\chi(M)\pm\frac{3}{2}\tau(M)=\frac{1}{\pi^{2}}\int_{M}\left(\widetilde{f}^{ij}_{(\pm\pm)}\right)^{2}d\mu+\frac{R^{2}}{192\pi^{2}}\mathrm{vol}(M),$
(A.11)

where $\mathrm{vol}(M)$ is the volume of an Einstein manifold $M$. Since the
scalar curvature is given by $R=4\lambda$, Eq. (A.11) can be written as the
form

$\chi(M)-\frac{\lambda^{2}}{12\pi^{2}}\mathrm{vol}(M)\geq\frac{3}{2}|\tau(M)|,$
(A.12)

where the equality holds only if $\widetilde{f}^{ij}_{(++)}=0$ or
$\widetilde{f}^{ij}_{(--)}=0$, i.e., $M$ is conformally half-flat (such as
$\mathbb{S}^{4}$ and $\mathbb{C}\mathbb{P}^{2}$). Indeed, as
$\frac{\lambda^{2}}{12\pi^{2}}\mathrm{vol}(M)=2$ and $\frac{3}{2}$ for
$\mathbb{S}^{4}$ and $\mathbb{C}\mathbb{P}^{2}$, respectively, the relation
(A.12) holds with the equality. However, for other spaces, we have, for
example,

$\widetilde{f}^{ij}_{(\pm\pm)}=\left\\{\begin{array}[]{ll}\frac{1}{6R^{2}}\,\mathrm{diag}(-1,-1,2)&\hbox{for
$\mathbb{S}^{2}\times\mathbb{S}^{2}$;}\\\
\pm\left(\psi^{i}_{(\pm)}-\frac{1}{3}(2\psi^{1}_{(\pm)}+\psi^{3}_{(\pm)})\right)\delta^{ij}&\hbox{for
Page},\end{array}\right.$ (A.13)

where we used the result (2.43) and $\psi^{1}_{(\pm)}=\psi^{2}_{(\pm)}$ for
the Page space. Since both spaces have $\chi=4$ and $\tau=0$ as shown in Fig.
1, the inequality (A.12), in particular, implies that
$\mathrm{vol}(\mathrm{Page})\leq\frac{48\pi^{2}}{\lambda^{2}}$. One can see
from Eq. (A.13) that the (anti-)self-dual Weyl tensors for the Page space have
only two distinct eigenvalues
$\left(\pm\frac{1}{3}(\psi^{1}_{(\pm)}-\psi^{3}_{(\pm)}),\;\mp\frac{2}{3}(\psi^{1}_{(\pm)}-\psi^{3}_{(\pm)})\right)$
at each point. Therefore, according to Proposition 5 in [9] (see also Theorem
$\mathbf{11.81}$ in [2]), the Page metric (2) must be locally conformal to a
Kähler metric although it is not a Kähler metric in itself.

The inequality (A.12) plays an important role to find infinitely many compact
simply connected smooth four-manifolds which do not admit Einstein metrics,
but nevertheless satisfy the strict Hitchin-Thorpe inequality
$\chi>\frac{3}{2}|\tau|$ [36].

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upstream from the corresponding node of $\widetilde{{\mathcal{A}}}_{2}$ in
${\mathcal{P}}({\mathcal{R}})$. In addition, Lemma B.1 shows that every path
from $\widetilde{{\mathcal{A}}}_{1}$ to $\widetilde{{\mathcal{A}}}_{2}$ in
${\mathcal{P}}({\mathcal{G}})$ containing a super-simple node is equivalent to
every path from the corresponding node of $\widetilde{{\mathcal{A}}}_{1}$ to
the corresponding node of $\widetilde{{\mathcal{A}}}_{2}$ in
${\mathcal{P}}({\mathcal{R}})$ containing a super-simple node in PRN
${\mathcal{R}}$. Using Theorem A.12$(d)$, we conclude this part. Second,
suppose $\widetilde{{\mathcal{A}}}_{1}=\\{\tau\\}$ is a single appendage node
and homeostasis inducing, that is, the homeostasis is induced by either
R-null-degradation $\big{[}f_{\tau^{R},\tau^{R}}\big{]}$ or P-null-degradation
$\big{[}f_{\tau^{P},\tau^{P}}\big{]}$. Since
$\tau^{R}\to\tau^{P}\in{\mathcal{P}}({\mathcal{R}})$ is an appendage path,
from Theorem A.12$(d)$ we get that R-null-degradation doesn’t induce
$\tau^{P}\in{\mathcal{R}}$. On the other side, since $\tau$ is a single
appendage node, $\tau^{P}\to\tau^{R}\notin{\mathcal{P}}({\mathcal{R}})$.
Theorem B.6 and Lemmas C.3 \- C.4 imply that there exists at least one path
from $\tau^{P}$ to $\tau^{R}$ and every such path must contain a super-simple
node in ${\mathcal{P}}({\mathcal{R}})$, therefore P-null-degradation induces
$\tau^{R}\in{\mathcal{R}}$. ∎

The following Lemma shows the connection between the occurrence of homeostasis
on the mRNA node and on the protein node associated to the same gene node in
GRN.

######  Lemma C.7.

Let ${\mathcal{G}}$ be an input-output GRN and ${\mathcal{R}}$ be the
associated input-output PRN. Suppose that infinitesimal homeostasis occurs in
the PRN at ${\mathcal{I}}_{0}$, induced by a homeostasis subnetwork
${\mathcal{K}}$ of ${\mathcal{R}}$. Then

  1. (a)

If $\nu$ is neither a super-simple nor single appendage node in
${\mathcal{G}}$, with $\nu^{R},\nu^{P}\notin{\mathcal{K}}$, then $\nu^{R}$ is
homeostatic if and only if $\nu^{P}$ is homeostatic in ${\mathcal{R}}$.

  2. (b)

If $\rho$ is a super-simple node in ${\mathcal{G}}$, with
$\rho^{R},\rho^{P}\notin{\mathcal{K}}$, then generically $\rho^{R}$ is
homeostatic if and only if $\rho^{P}$ is homeostatic in ${\mathcal{R}}$.

  3. (c)

If $\tau$ is a single appendage node in ${\mathcal{G}}$, with
$\tau^{R},\tau^{P}\notin{\mathcal{K}}$, then generically $\tau^{R}$ is
homeostatic if and only if $\tau^{P}$ is homeostatic in ${\mathcal{R}}$.

###### Proof.

(a) Using Lemma C.1 and the fact that the GRN-node $\nu$ is neither a super-
simple nor single appendage node in ${\mathcal{G}}$, we have that
$\nu^{R},\nu^{P}\in{\mathcal{R}}$ belong to the same homeostasis subnetwork of
${\mathcal{R}}$, distinct from ${\mathcal{K}}$. Thus, we conclude the result
from Theorem C.6.

(b) By Lemma C.1 and Lemma C.2 and the fact that $\rho$ is a super-simple node
in ${\mathcal{G}}$, we have

$\rho^{R}\to{\widetilde{{\mathcal{L}}}}(\rho^{R},\rho^{P})\to\rho^{P}$ (C.22)

where
$\rho^{R},{\widetilde{{\mathcal{L}}}}(\rho^{R},\rho^{P}),\rho^{P}\in{\mathcal{P}}_{\mathcal{S}}({\mathcal{R}})$
and ${\widetilde{{\mathcal{L}}}}(\rho^{R},\rho^{P})=\emptyset$. Assume that
$\rho^{R}$ is homeostatic in the PRN. Then, we have the steady-state equation
$f_{\rho^{P}}(\rho^{R},\rho^{P})=0$ and implicit differentiation gives

$\frac{d}{d{\mathcal{I}}}f_{\rho^{P}}\big{|}_{{\mathcal{I}}={\mathcal{I}}_{0}}=f_{\rho^{P},\rho^{R}}(\rho^{R})^{\prime}+f_{\rho^{P},\rho^{P}}(\rho^{P})^{\prime}=0$
(C.23)

Generically, we can assume $f_{\rho^{P},\rho^{P}}\neq 0$ (i.e. no null
degradation, since $\rho$ is not appendage), and get

$(\rho^{P})^{\prime}=0$

This shows that $\rho^{P}$ is homeostatic in the PRN. On the other hand,
assume that $\rho^{P}$ is homeostatic in the PRN. Again, (C.23) holds.
Generically, we can assume $f_{\rho^{P},\rho^{R}}\neq 0$ (i.e. no Haldane
homeostasis, since $\rho^{R},\rho^{P}\notin{\mathcal{K}}$), and obtain

$(\rho^{R})^{\prime}=0$

This shows that $\rho^{R}$ is homeostatic in the PRN.

We skip the proof of item (c), since it is analogous to the proof of item (b).
∎

######  Remarks C.8.

In Lemma C.7, items (b) and (c) characterize homeostatic nodes that are not
contained in the homeostasis subnetwork ${\mathcal{K}}$ inducing homeostasis.
Now, it is easy to see what happens when they do belong to ${\mathcal{K}}$.

  1. (a)

Suppose that $\rho$ is a super-simple node in ${\mathcal{G}}$. On one hand, if
$\rho^{R}\in{\mathcal{K}}$ then ${\mathcal{R}}$-Haldane homeostasis occurs at
${\mathcal{I}}_{0}$ (i.e. $f_{\rho^{P},\rho^{R}}=0$). In this case, $\rho^{R}$
is not homeostatic but $\rho^{P}$ is homeostatic. On the other hand, if
$\rho^{P}\in{\mathcal{K}}$ then $\rho^{R}$ and $\rho^{P}$ fail to be
simultaneously homeostatic. In both cases, $\rho$ is not GRN-homeostatic.

  2. (b)

Suppose that $\tau$ is a single appendage node in ${\mathcal{G}}$. On one
hand, if $\tau^{P}\in{\mathcal{K}}$ then $P$-null-degradation occurs at
${\mathcal{I}}_{0}$ (i.e. $f_{\tau^{P},\tau^{P}}=0$). In this case, $\tau^{P}$
is not homeostatic but $\tau^{R}$ is homeostatic. On the other hand, if
$\tau^{R}\in{\mathcal{K}}$ ($R$-null-degradation) then $\tau^{R}$ and
$\tau^{P}$ fail to be simultaneously homeostatic. In both cases, $\tau$ is not
GRN-homeostatic. $\Diamond$

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