# Rigidity of $SU_{n}$-type symmetric spaces

Wafaa Batat  Ecole Nationale Polytechnique d’Oran, B.P 1523 El M’naouar, 31000
Oran, Algeria<EMAIL_ADDRESS>,  Stuart James Hall  School of Mathematics
and Statistics, Herschel Building, Newcastle University, Newcastle-upon-Tyne,
NE1 7RU<EMAIL_ADDRESS>https://www.ncl.ac.uk/maths-
physics/staff/profile/stuarthall ,  Thomas Murphy  Department of Mathematics,
California State University Fullerton, 800 N. State College Bld., Fullerton,
CA 92831, USA<EMAIL_ADDRESS>http://www.fullerton.edu/math/faculty/tmurphy/ and  James Waldron  School of
Mathematics and Statistics, Herschel Building, Newcastle University,
Newcastle-upon-Tyne, NE1 7RU<EMAIL_ADDRESS>https://www.ncl.ac.uk/maths-physics/staff/profile/jameswaldron

###### Abstract.

We prove that the bi-invariant Einstein metric on $SU_{2n+1}$ is isolated in
the moduli space of Einstein metrics, even though it admits infinitesimal
deformations. This gives a non-Kähler, non-product example of this phenomenon
adding to the famous example of $\mathbb{CP}^{2n}\times\mathbb{CP}^{1}$ found
by Koiso. We apply our methods to derive similar solitonic rigidity results
for the Kähler–Einstein metrics on ‘odd’ Grassmannians. We also make explicit
a connection between non-integrable deformations and the dynamical instability
of metrics under Ricci flow.

##  1\. Introduction

###  1.1. Rigidity of Einstein metrics

For a fixed manifold $M$, a central object of study is the set of Einstein
metrics viewed as a subset of all Riemannian metrics on $M$. In general,
giving a full description of the set of Einstein metrics on $M$ is an
intractable project; a more reasonable goal is to produce a local description
of this set around some fixed Einstein metric $g$.

The foundational work on the local structure theory of the set of Einstein
metrics was carried out by Koiso [16] who showed that the premoduli space of
Einstein metrics about $g$ has the structure of an analytic subset of a smooth
manifold $\mathcal{Z}$. Furthermore, the tangent space of $\mathcal{Z}$ at $g$
is naturally isomorphic to a certain eigenspace of the Lichnerowicz Laplacian.
Hence, if the eigenspace is zero-dimensional, $g$ must be isolated in the
premoduli space; isolated Einstein metrics are often referred to as being
rigid. For spaces where the dimension of the eigenspace is positive, an
interesting problem arises: do any of the tangent vectors (so-called
infinitesimal deformations of the Einstein metric) come from genuine
deformations of $g$ through Einstein metrics?

Koiso investigated this question for compact symmetric spaces in [14] and
[15]. We will explain Koiso’s findings in detail in Section 2. A key result is
that all but one of the symmetric spaces where the dimension of the relevant
eigenspace of the Lichnerowicz Laplacian is non-zero are associated to
$SU_{n}$; they include the bi-invariant metric on $SU_{n}$ for $n>2$, and the
Kähler–Einstein metric on the Grassmannians ${SU_{n}/S(U_{n-k}\times U_{k})}$
for $n>3$ and $k>1$. To investigate the rigidity of these metrics we need to
investigate whether any infinitesimal deformations integrate up to genuine
curves of Einstein metrics. Koiso developed an obstruction to integrability
and used this to show that the canonical metric on
$\mathbb{CP}^{2n}\times\mathbb{CP}^{1}$ is isolated in the moduli space but
admits infinitesimal deformations, none of which are integrable. This was the
first such example of this phenomenon in the literature.

One difficulty in computing Koiso’s obstruction is finding a concrete
description of the space of infinitesimal deformations beyond its formal
definition as an eigenspace of the Lichnerowicz Laplacian. For the
$SU_{n}$-type spaces, Gasqui and Goldschmidt found such descriptions in [8]
and [9]. Exploiting these methods leads to the main result.

######  Theorem A.

The bi-invariant Einstein metric on $SU_{2n+1}$ is isolated in the moduli
space of Einstein metrics.

This theorem provides an example of a family of non-product and non-Kähler
Einstein metrics admitting infinitesimal deformations, none of which are
integrable. In the course of the proof of Theorem A we will show that almost
all infinitesimal variations of $SU_{2n}$ are obstructed and give a precise
characterisation of those which are not. However, we suspect the unobstructed
variations are not integrable and that a higher order obstruction will
demonstrate this.

To prove Theorem A we follow the representation theoretic methods used by
Koiso [14], [15], and Gasqui–Goldschmidt [9]. For a symmetric space $G/K$, one
can view Koiso’s obstruction as an element of the space
$\mathrm{Hom}_{G}(s^{2}(\mathfrak{g}),\mathfrak{g})$; an infinitesimal
deformation is unobstructed if it is a zero of the associated map. As the
relevant Hom-spaces are one or two dimensional, it is possible to argue that
the obstruction is a multiple of a particular map. The theorem follows by
showing that the map has no zeros and the obstruction is a non-zero multiple
of the map.

The strategy employed in the proof is quite general and should be applicable
to similar rigidity questions for the symmetric spaces $SU_{n}/SO_{n}$,
$SU_{2n}/Sp_{n}$, and $SU_{p+q}/S(U_{p}\times U_{q})$ which have been open for
forty years since Koiso’s work. We give a brief discussion of this question
and related issues in Section 6.

###  1.2. Rigidity of Ricci solitons

Einstein metrics can be thought of as fixed points of the Ricci flow

$\frac{\partial g}{\partial t}=-2\mathrm{Ric}(g).$

Viewed thus, the Einstein metrics are a special case of a type of metric
called a Ricci soliton. One can then ask whether a given Einstein metric $g$
can be deformed through Ricci solitons. The structure theory of the moduli
space of Ricci solitons was developed by Podesta and Spiro in [25] and we
refer the reader to Section 5 for more details. All Hermitian symmetric spaces
admit infinitesimal solitonic deformations. We are able to prove that for the
‘odd’ Grassmannians, none of these deformations can be integrated to give a
non-Einstein Ricci soliton.

######  Theorem B.

The Kähler–Einstein metric on the Grassmannian $Gr_{k}(\mathbb{C}^{2n+1})$ is
weakly solitonically rigid.

Our method also yields a new proof of a theorem of Kröncke. This strengthens
the previous result in the case when $k=1$. It concerns the solitonic rigidity
of the Fubini–Study metric on $\mathbb{CP}^{2n}$.

######  Theorem C (Kröncke, [18]).

The Fubini–Study metric on $\mathbb{CP}^{2n}$ is isolated in the moduli space
of Ricci solitons.

###  1.3. Stability of the Ricci flow

The relationship between infinitesimal deformations of Einstein metrics, their
integrability, and the dynamical stability of the Ricci flow has been known
for quite some time through the works of Sesum [30], Haslhofer and Müller
[12], and Kröncke [17]. In Section 6 we compute the third variation of
Perelman’s $\nu$-functional for infinitesimal deformations of an Einstein
metric (this is related to the calculation made for conformal variations by
Kröncke [20] \- see also the calculation of Knopf and Sesum [13]). The next
theorem shows that the variation is a multiple of Koiso’s obstruction to
integrability $\mathcal{I}$ (see Section 2 for the precise definition of this
obstruction).

######  Theorem D.

Let $(M^{m},g)$ be an Einstein metric with positive Einstein constant
$\frac{1}{2\tau}>0$ and let $h\in s^{2}(T^{\ast}M)$ be an EID. Then if
$g(s):=g+sh$,

$\frac{d^{3}\nu}{ds^{3}}\bigg{|}_{s=0}=\frac{-\tau}{2(4\pi\tau)^{\frac{m}{2}}}\mathcal{I}(h).$

This yields a new proof of the following result which was first demonstrated
by Cao and He in [4].

######  Theorem E (Cao–He [4]).

For $n>2$, the bi–invariant metric on $SU_{n}$ is dynamically unstable as a
fixed point of the Ricci flow.

Note that this result does not depend upon the parity of $n$. This is because,
to show instability, we do not require all infinitesimal Einstein deformations
to be obstructed, but just a single one. We can find such a deformation for
all $n>2$.

###  1.4. Existing work on rigidity and stability

As mentioned already, the foundational work on the rigidity of Einstein
metrics on symmetric spaces was carried out by Koiso in the papers [14], [15],
and [16]. In particular, Koiso classified the compact irreducible spaces
admitting infinitesimal Einstein deformations. Geometric constuctions of the
infinitesimal deformations were given by Gasqui and Goldschmidt in [8] and
[9]. Recently, Derdzinski and Gal demonstrated the bi-invariant Einstein
metric on $SU_{n}$ is isolated within the space of left-invariant Einstein
metrics on $SU_{n}$ [6]; Theorem A removes the restriction that the
deformations are through left-invariant metrics in the case when $n$ is odd.

The calculation of the second variation of Perelman’s $\nu$-entropy and the
link between the spectrum of the Lichnerowicz Laplacian and the stability of
the Ricci flow appeared in [3] (with a detailed proof given in [5]). The
dynamical stability of compact symmetric spaces was considered by Cao and He
in [4] where the authors also addressed the stability of compact irreducible
symmetric spaces with respect to the Einstein–Hilbert functional (where
deformations are through TT tensors). The work of Semmelmann and Weingart [29]
and Schwahn [26] now gives a complete understanding of the Einstein–Hilbert
picture.

There have also been other interesting developments concerning deformability
of Einstein metrics and stability questions. For example, there is recent work
by Kröncke on sine-cones [19]. The interaction of rigidity and stability
questions with other geometric stuctures (e.g. special holonomy) has been
taken up by Wang and Wang in [32] and [33] as well as the joint works with
Semmelmann [27] [28] . The relationship between deformability of $G_{2}$
structures and Einstein metrics is investigated by Nagy and Semmelmann in
[22].

###  1.5. Conventions

All the manifolds we consider in this paper will be smooth and closed. We
follow the convention that the rough Laplacian is $\Delta
h=\mathrm{tr}_{12}(\nabla^{2}h)$ and it therefore has a non-positive spectrum.
The curvature tensor is

$\mathrm{R}(X,Y)Z=\nabla^{2}_{X,Y}Z=\nabla_{[X,Y]}Z+\nabla_{Y}\nabla_{X}Z-\nabla_{X}\nabla_{Y}Z.$

The curvature operator on symmetric tensors
${\mathrm{Rm}:s^{2}(T^{\ast}M)\rightarrow s^{2}(T^{\ast}M)}$ is defined by

$\mathrm{Rm}(h)(X,Y)=\sum_{k}h(R(X,E_{k})Y,E_{k}),$

where $\\{E_{i}\\}$ is a local orthonormal frame. When referring to symmetric
spaces we adopt the convention of only referring to the manifold with the
Riemannian symmetric space metric implicitly understood. Occasionally we will
perform tensor calculations using indices and we will use the Einstein
summation convention that repeated indices are to be summed over.

Acknowledgements:

We would like to thank Paul-Andi Nagy for his interest in our work,
discussions about the paper [25], and for bringing the reference [6] to our
attention.

##  2\. Deformations of Einstein metrics

###  2.1. General Theory

We give the essential definitions and theory in this section. Almost all of
what is written here is expanded upon and well explained in the book [1].

######  Definition 2.1 (Premoduli space).

Let $(M,g)$ be an Einstein manifold scaled to have unit volume, let
$\mathcal{M}_{1}$ be the space of all metrics on $M$ with unit volume, and let
$\mathfrak{S}_{g}\subset\mathcal{M}_{1}$ be the slice to the action of the
diffeomorphism group $D$ on $\mathcal{M}_{1}$. The subset of Einstein metrics
in $\mathfrak{S}_{g}$ is the premoduli space of Einstein structures about $g$.
We denote this set by $\mathcal{E}(g)$.

######  Definition 2.2 (Einstein Infinitesimal Deformation (EID)).

Let $(M,g)$ be an Einstein manifold and let ${h\in s^{2}(T^{\ast}M)}$ satisfy

${}\mathrm{tr}(h)=0,$ (2.1) ${}\mathrm{div}(h)=0,$ (2.2)

and

${}\Delta h+2\mathrm{Rm}(h)=0.$ (2.3)

The tensor $h$ is referred to as an essential Einstein infinitesimal
deformation (EID). We denote the space of such $h$ by $\varepsilon(g)$.

As Equation (2.3) is elliptic, we see that the space $\varepsilon(g)$ is
always finite dimensional. Tensors that satisfy Equations (2.1) and (2.2) are
often referred to as transverse trace-free or TT tensors. The foundational
structure theorem for $\mathcal{E}(g)$ is due to Koiso.

######  Theorem 2.3 (Koiso [16]).

Let $(M,g)$ be an Einstein manifold. Then within the slice $\mathfrak{S}_{g}$
there exists a finite dimensional real analytic submanifold $\mathcal{Z}$ such
that:

  1. (i)

The tangent space of $\mathcal{Z}$ at $g$ is the space of essential Einstein
infinitesimal deformations $\varepsilon(g)$.

  2. (ii)

The manifold $\mathcal{Z}$ contains the premoduli space $\mathcal{E}(g)$ as a
real analytic subset.

To try to understand what spaces arise as $\mathcal{E}(g)$, we consider
whether the elements of $\varepsilon(g)$ are the genuine tangents to a curve
of metrics in $\mathcal{E}(g)$ passing through the Einstein metric $g$. To
make this precise, we introduce the Einstein operator
${E:\mathcal{M}_{1}\rightarrow s^{2}(T^{\ast}M)}$ given by

$E(g)=\mathrm{Ric}(g)-\left(\frac{\int_{M}\mathrm{S}(g)d\mathrm{Vol}_{g}}{\dim(M)}\right)g,$
(2.4)

where $\mathrm{Ric}(g)$ and $\mathrm{S}(g)$ are respectively the Ricci
curvature and scalar curvature of $g$. When $\dim(M)>2$, solutions of the
equation $E(g)=0$ are Einstein metrics.

If $g$ is an Einstein metric and $g(t)$ is a curve of metrics in
$\mathcal{M}_{1}$ such that $g(0)=g$ and $\dot{g}(0)=h\in\varepsilon(g)$, it
is straightforward to check that

$\frac{d}{dt}E(g(t))\bigg{|}_{t=0}=0.$

######  Definition 2.4 (Integrability of EIDs).

Let $(M,g)$ be an Einstein manifold and let $k\in\mathbb{N}$. We say that an
element $h\in\varepsilon(g)$ is integrable to order $k$ if there exist
${h_{2},h_{3},\ldots,h_{k}\in s^{2}(T^{\ast}M)}$ such that the curve

$g_{k}(t):=g+th+\sum_{j=2}^{j=k}\frac{t^{j}}{j!}h_{j},$

satisfies

$\frac{d^{j}}{dt^{j}}E(g_{k}(t))\bigg{|}_{t=0}=0,$

for all $j=1,\ldots,k$. An element $h\in\varepsilon(g)$ is formally integrable
if there is a formal power series

$g(t):=g+th+\sum_{j=2}^{\infty}\frac{t^{j}}{j!}h_{j},$

such that $E(g(t))=0$.

An immediate consequence of Koiso’s structure Theorem 2.3 is that if
${h\in\varepsilon(g)}$ is formally integrable then there is a smooth curve of
Einstein metrics in $\mathcal{E}(g)$ such that $\dot{g}(0)=h$ (in other words
a convergent power series solution can be constructed). It is also clear that
if $h\in\varepsilon(g)$ is formally integrable, then it must be integrable to
order $k$ for all $k\in\mathbb{N}$. Hence it is natural to investigate the
obstruction to EIDs being integrable to order two. Formally we see that

$\frac{d^{2}}{dt^{2}}E(g(t))\bigg{|}_{t=0}=E^{\prime\prime}(h,h)+E^{\prime}(\ddot{g}(0)),$

where $E^{\prime}$ and $E^{\prime\prime}$ are the first and second derivatives
(suitably interpreted) of the Einstein operator $E$ defined by (2.4). Koiso
proved a useful way to check integrability to order two.

######  Lemma 2.5 (Koiso, Lemma 4.7 in [15]).

Let $(M,g)$ be an Einstein manifold. Then $h\in\varepsilon(g)$ is integrable
to order two if and only if $E^{\prime\prime}(h,h)\in\varepsilon(g)^{\perp}$.
Here the orthogonal complement is with respect to the $L^{2}$-inner product on
$s^{2}(T^{\ast}M)$ induced by $g$.

Using this result, we see that a necessary condition for $h$ to be integrable
is the vanishing of the quantity $\langle
E^{\prime\prime}(h,h),h\rangle_{L^{2}}$. This quantity was also computed by
Koiso.

######  Proposition 2.6 (Koiso [15]).

Let $(M,g)$ be an Einstein metric with Einstein constant $\lambda>0$ and let
$h\in\varepsilon(g)$. Then an obstruction to the integrability of $h$ to order
two is given by the nonvanishing of the quantity

$\mathcal{I}(h):=2\lambda\langle
h_{i}^{k}h_{kj},h_{ij}\rangle+3\langle\nabla_{i}\nabla_{j}h_{kl},h_{ij}h_{kl}\rangle-6\langle\nabla_{i}\nabla_{j}h_{kl},h_{ik}h_{jl}\rangle,$
(2.5)

where each of the brackets denotes the $L^{2}$-inner product induced by the
metric $g$ on the appropriate bundle.

###  2.2. Deformations of compact symmetric spaces

For compact irreducible symmetric spaces, the space of Einstein infinitesimal
deformations $\varepsilon(g)$ for the canonical Einstein metrics can be
computed using representation theoretic methods.

######  Theorem 2.7 (Koiso, Theorem 1.1 in [14] \- see also [8] Proposition
2.40).

Let $(M,g)$ be a compact irreducible symmetric space. Then the space of EIDs
$\varepsilon(g)$ is $\\{0\\}$, except in the following cases:

  1. (i)

$M=SU_{n}$ with $n\geq 3$, here
$\varepsilon(g)\cong\mathfrak{su}(n)\oplus\mathfrak{su}_{n}$,

  2. (ii)

$M=SU_{n}/SO_{n}$ with $n\geq 3$, here $\varepsilon(g)\cong\mathfrak{su}_{n}$,

  3. (iii)

$M=SU_{2n}/Sp_{n}$ with $n\geq 3$ here
$\varepsilon(g)\cong\mathfrak{su}_{2n}$,

  4. (iv)

$M=SU_{p+q}/S(U_{p}\times U_{q})$ with $p\geq q\geq 2$, here
$\varepsilon(g)\cong\mathfrak{su}_{p+q}$,

  5. (v)

$M=E_{6}/F_{4}$, here $\varepsilon(g)\cong\mathfrak{e}_{6}$.

For the spaces with non-zero $\varepsilon(g)$, if $M=G/K$ with $G$ the
isometry group of $g$, then $\varepsilon(g)\cong\mathfrak{g}$. Of course, this
isomorphism is really just a statement about the dimension of $\varepsilon(g)$
and part of the difficulty in determining whether the infinitesimal
deformations are integrable is describing them geometrically. Fortunately, for
spaces (i)-(iii) in Theorem 2.7, there is a straightforward way describing the
tensors in $\varepsilon(g)$ due to Gasqui and Goldschmidt. This construction
is detailed in the memoir [9]; we give explicit differential geometric proofs
to show the construction really does produce EIDs.

Let $M$ be a symmetric space of compact type; $M$ is thus the coset space of a
Riemannian symmetric pair $(G,K)$ where $G$ is a compact, semi-simple Lie
group and $K$ is a closed subgroup of $G$. We denote the Lie algebras of $G$
and $K$ by $\mathfrak{g}$ and $\mathfrak{k}$ respectively and note that the
inner product on $\mathfrak{g}$ induced by the Killing form yields the
$\mathrm{Ad}_{K}$-invariant decomposition

$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}.$

The subspace $\mathfrak{p}$ can be identified with the tangent space of $M$ at
the identity coset $eK\in G/K$. For an integer $p\geq 2$ we denote by
$s^{p}(\mathfrak{p}^{\ast})^{K}$ the $\mathrm{Ad}_{K}$-invariant symmetric
$p$-tensors (or equivalently the $\mathrm{Ad}_{K}$-invariant symmetric degree
$p$ polynomials over the space $\mathfrak{p}$). An element $q\in
s^{p}(\mathfrak{p}^{\ast})^{K}$ gives rise to a $G$-invariant symmetric
$p$-tensor field $\sigma(q)$ that coincides with $q$ at the identity coset.
Conversely, every such $G$-invariant symmetric tensor field arises this way.

We now assume we have a given $q\in s^{3}(\mathfrak{p}^{\ast})^{K}$ and a
Killing vector field $\eta\in\mathfrak{iso}(M)$, and we define a symmetric
$2$-tensor $h_{\eta}$ by

$h_{\eta}:=\iota_{\eta}\sigma(q).$ (2.6)

We will henceforth forget the dependence on $q$ and write $\sigma$ for
$\sigma(q)$. The following important result is due to Gasqui and Goldschmidt;
we give a new and direct proof.

######  Lemma 2.8 (Gasqui–Goldschmidt).

Let $M=G/K$ be a compact irreducible symmetric space with its canonical
Einstein metric $g$. Then the tensor field $h_{\eta}\in\varepsilon(g)$.

###### Proof.

To show that $h_{\eta}$ satisfies Equation (2.1) we consider the linear map
${\Phi:\mathfrak{p}\rightarrow\mathbb{R}}$ defined by

$\Phi(v):=\mathrm{tr}(\iota_{v}\sigma)|_{(e\cdot K)}.$

As $\Phi$ is $\mathrm{Ad}_{K}$-invariant, a non-trivial kernel would be an
$\mathrm{Ad}_{K}$-invariant subspace of $\mathfrak{p}$. Hence $\Phi$ must
vanish identically as the symmetric space $M$ is irreducible. For $a\in G$,
the trace of $h_{\eta}$ at a coset $(a\cdot K)$ is
${\Phi(a^{-1}\cdot\eta(a\cdot K))=0}$ (where we note that the non-vanishing
component of $a^{-1}\cdot\eta(a\cdot K)$ is in $\mathfrak{p}$).

To demonstrate Equation (2.2) we recall the result that any $G$-invariant
tensor field on a symmetric space is parallel. Thus

$(\nabla_{\cdot}h_{\eta})(\cdot,\cdot)=\sigma(\nabla_{\cdot}\eta,\cdot,\cdot).$

Taking the trace we see that

$\mathrm{div}(h_{\eta})(Y)=\sum_{j}\sigma(\nabla_{E_{j}}\eta,E_{j},Y),$

for an orthonormal frame $\\{E_{j}\\}$. However, after using the metric to
produce sections of $\mathrm{End}(TM)$, this is the trace of the composition
of the symmetric 2-tensor $\iota_{Y}\sigma$ and the skew-symmetric tensor
$\nabla_{\cdot}\eta$ (the skew-symmetry follows from the fact that $\eta$ is a
Killing field). Hence the trace vanishes and $h_{\eta}$ is divergence-free.

To demonstrate Equation (2.3) we use the fact that, as the metric is
homogeneous, we only need to check the equation

$\Delta(h_{\eta})(X,Y)+2\mathrm{Rm}(h_{\eta})(X,Y)=0,$

for Killing fields $X,Y$. We also consider a local orthonormal frame
$\\{E_{i}\\}$ with $\nabla E_{i}$ vanishing at the point at which we want to
compute the identity. To start with, we note

$\Delta
h_{\eta}(X,Y)=(\nabla^{2}_{E_{i},E_{i}}h_{\eta})(X,Y)=\sigma(\nabla^{2}_{E_{i},E_{i}}\eta,X,Y),$

and

$\mathrm{Rm}(h_{\eta})(X,Y)=\sigma(\eta,E_{i},R(X,E_{i})Y)=\sigma(\eta,E_{i},R(Y,E_{i})X).$

In general, given a Killing field $\xi$, we have the identity

$\sigma(\nabla_{W}\xi,Y,Z)+\sigma(W,\nabla_{Y}\xi,Z)+\sigma(W,Y,\nabla_{Z}\xi)=0,$
(2.7)

In particular

$\sigma(\nabla_{X}\eta,E_{i},Y)+\sigma(\nabla_{E_{i}}\eta,X,Y)+\sigma(\nabla_{Y}\eta,E_{i},X)=0.$

Taking the derivative with respect to $E_{i}$ and using the fact that we have
demonstrated tensors of the form $\iota_{\xi}\sigma$ are divergence free we
obtain,

$\sigma(\nabla_{E_{i}}\nabla_{X}\eta,E_{i},Y)+\sigma(\nabla_{E_{i}}\nabla_{E_{i}}\eta,X,Y)+\sigma(\nabla_{E_{i}}\eta,\nabla_{E_{i}}X,Y)\\\
+\sigma(\nabla_{E_{i}}\eta,X,\nabla_{E_{i}}Y)+\sigma(\nabla_{E_{i}}\nabla_{Y}\eta,E_{i},X)=0.$

Applying Equation (2.7) on the terms
$\sigma(\nabla_{E_{i}}\eta,\nabla_{E_{i}}X,Y)$ and
$\sigma(\nabla_{E_{i}}\eta,X,\nabla_{E_{i}}Y)$ yields

$\sigma(\nabla_{E_{i}}\eta,\nabla_{E_{i}}X,Y)=-\sigma(\nabla_{\nabla_{E_{i}}X}\eta,E_{i},Y).$

Hence

$\sigma(\nabla_{E_{i},X}^{2}\eta,E_{i},Y)+\sigma(\nabla_{E_{i},E_{i}}^{2}\eta,X,Y)+\sigma(\nabla_{E_{i},Y}^{2}\eta,E_{i},X)=0.$
(2.8)

We now expand

$\sigma(\eta,E_{i},\nabla^{2}_{X,E_{i}}Y)=X\cdot\sigma(\eta,E_{i},\nabla_{E_{i}}Y)-\sigma(\nabla_{X}\eta,E_{i},\nabla_{E_{i}}Y)=0$

which implies that Equation (2.8) becomes

$\sigma(R(X,E_{i})\eta,E_{i},Y)+\sigma(\nabla_{E_{i},E_{i}}^{2}\eta,X,Y)+\sigma(R(Y,E_{i})\eta,E_{i},X)=0.$
(2.9)

We note that we obtain a similar equation simply by swapping $\eta$ and $X$

$\sigma(R(\eta,E_{i})X,E_{i},Y)+\sigma(\nabla_{E_{i},E_{i}}^{2}X,\eta,Y)+\sigma(R(Y,E_{i})X,E_{i},\eta)=0.$
(2.10)

The next ingredient is an identity (see Lemma 33 in [24] but note the
differing sign convention) true for any Killing field $\xi$ on an arbitrary
Riemannian manifold:

$\nabla_{X,Y}^{2}\xi=R(\xi,X)Y.$

Hence

$\sigma(\nabla_{E_{i},E_{i}}^{2}X,\eta,Y)=\sigma(\nabla_{E_{i},E_{i}}^{2}\eta,X,Y)=-\lambda\sigma(\eta,X,Y).$

Thus subtracting Equation (2.9) from Equation (2.10) along with an application
of the first Bianchi identity yields

$\sigma(R(X,\eta)E_{i},E_{i},Y)=\sigma(R(Y,E_{i})\eta,E_{i},X)-\sigma(R(Y,E_{i})X,E_{i},\eta).$

The quantity $\sigma(R(X,\eta)E_{i},E_{i},Y)$ vanishes as it is the trace of
an antisymmetric map (as in the proof above that $h_{\eta}$ satisifes Equation
(2.2)). Hence we can reinterpret Equation (2.9) as

$\sigma(R(Y,E_{i})X,E_{i},\eta)+\sigma(\nabla_{E_{i},E_{i}}^{2}\eta,X,Y)+\sigma(R(X,E_{i})Y,E_{i},\eta)=0.$

This can be rewritten as

$(\Delta h_{\eta})(X,Y)+2\mathrm{Rm}(h_{\eta})(X,Y)=0,$

as required.

∎

Irreducible symmetric spaces with $G$-invariant symmetric 3-tensors form a
very short list.

######  Proposition 2.9 (Gasqui–Goldschmidt, [9] Proposition 2.1).

Let $M$ be an irreducible simply-connected symmetric space of compact type.
The space $s^{3}(\mathfrak{p})^{K}$ vanishes unless $M$ is one of the
following spaces:

  1. i

$SU_{n}$, with $n\geq 3$;

  2. ii

$SU_{n}/SO_{n}$, with $n\geq 3$;

  3. iii

$SU_{2n}/Sp_{n}$, where $n\geq 3$;

  4. iv

$E_{6}/F_{4}$.

For the cases (i)-(iv), then the space $s^{3}(\mathfrak{p})^{K}$ is one-
dimensional.

Note the slight difference in the spaces between the previous proposition and
Theorem 2.7. This is because the infinitesimal deformations of the
Grassmannians are not given by $\iota_{v}\sigma$, since the space
$s^{3}(\mathfrak{p})^{K}$ vanishes for the Grassmannians. We shall discuss
this in Section 6.

##  3\. Input from Representation Theory

Let $M=G/K$ be a symmetric space of type (i)-(v) from Theorem 2.7. If we
denote by $\psi(h,h)$ the projection of the quantity $E^{\prime\prime}(h,h)$
to the space of EIDs $\varepsilon(g)$ then $\psi$ can be seen to be a
symmetric bilinear $G$-equivariant map. In other words,
$\psi\in\mathrm{Hom}_{G}(s^{2}(\mathfrak{g}),\mathfrak{g})$. Lemma 2.5 can be
restated in the following manner:

$\textrm{an EID }h\in\varepsilon(g)\textrm{ is integrable to order 2 if and
only if }\psi(h,h)=0.$

The strategy we employ to show that infinitesimal deformations are not
integrable can by summarised as follows:

  1. (1)

Demonstrate that $\psi\in
V\subset\mathrm{Hom}_{G}(s^{2}(\mathfrak{g}),\mathfrak{g})$ where $V$ is a one
dimensional subspace of $\mathrm{Hom}_{G}(s^{2}(\mathfrak{g}),\mathfrak{g})$;
hence $\psi=C\Psi$ for some $C\in\mathbb{R}$ and $\Psi$ a generator of $V$.

  2. (2)

Compute the exact form of the map $\Psi$ and show that if $\Psi(x,x)=0$ then
$x=0$.

  3. (3)

Find some $h^{\ast}\in\varepsilon(g)$ so that the obstruction in Proposition
2.5

$\mathcal{I}(h^{\ast})=\langle\psi(h^{\ast},h^{\ast}),h^{\ast}\rangle_{L^{2}}\neq
0.$

  4. (4)

Conclude that $C\neq 0$ and so $\psi(h,h)\neq 0$ for any $h\in\varepsilon(g)$
with $h\neq 0$.

In this paper we will consider this strategy for $G=SU_{n}\times SU_{n}$ and
$G=SU_{n}$. In this section we carry out steps (1) and (2) for these groups.
We expect this material is standard and considerations such as these are
implicit in much of the works of Koiso, and Gasqui and Goldschmidt; we include
the proofs for completeness.

######  Lemma 3.1.

Let $G=SU_{n}\times SU_{n}$ with $n\geq 2$, identify
${\mathfrak{g}:=\mathrm{Lie}(G)}$ with
${\mathfrak{su}_{n}\oplus\mathfrak{su}_{n}}$, and consider $\mathfrak{g}$ as a
representation of $G$ via the adjoint representation. Then the dimension of
real vector space
$\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)$ is given by:

$\mathrm{dim}\;\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)=\begin{cases}0&\;n=2\\\
2&\;n\geq 3.\end{cases}$

###### Proof.

Let $G_{\mathbb{C}}=SL_{n}\left(\mathbb{C}\right)\times
SL_{n}\left(\mathbb{C}\right)$ and identify
$\mathfrak{g}_{\mathbb{C}}:=\mathrm{Lie}\;G_{\mathbb{C}}$ with
$\mathfrak{sl}_{n}\oplus\mathfrak{sl}_{n}$. Consider
$\mathfrak{g}_{\mathbb{C}}$ as a representation of $G_{\mathbb{C}}$ and
$\mathfrak{g}_{\mathbb{C}}$ via the adjoint representation. It follows from
the fact that $G$ is maximal compact subgroup in $G_{\mathbb{C}}$ that

$\displaystyle\mathrm{dim}_{\mathbb{R}}\;\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)$
$\displaystyle=\mathrm{dim}_{\mathbb{C}}\;\mathrm{Hom}_{G_{\mathbb{C}}}\left(s^{2}(\mathfrak{g}_{\mathbb{C}}),\mathfrak{g}_{\mathbb{C}}\right).$

If $T$ is a maximal torus in $G_{\mathbb{C}}$, $W=N_{G_{\mathbb{C}}}/T$ is the
associated Weyl group, and $\mathfrak{t}=\mathrm{Lie}\;T$ then by Theorem 1 in
[2] there is an isomorphism of graded vector spaces (in fact of graded
$\mathbb{C}[\mathfrak{g}]^{G_{\mathbb{C}}}\cong\mathbb{C}[\mathfrak{t}]^{W}$-modules)

$\mathrm{Hom}_{G_{\mathbb{C}}}\left(\mathrm{S}\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}\right)\cong\mathrm{Hom}_{W}\left(\mathrm{S}\mathfrak{t},\mathfrak{t}\right)$

and in particular an isomorphism of complex vector spaces

$\mathrm{Hom}_{G_{\mathbb{C}}}\left(s^{2}(\mathfrak{g}_{\mathbb{C}}),\mathfrak{g}_{\mathbb{C}}\right)\cong\mathrm{Hom}_{W}\left(s^{2}(\mathfrak{t}),\mathfrak{t}\right).$

This discussion shows that it suffices to prove that

$\mathrm{dim}_{\mathbb{C}}\;\mathrm{Hom}_{W}\left(s^{2}(\mathfrak{t}),\mathfrak{t}\right)=\begin{cases}0&\;n=2\\\
2&\;n\geq 3.\end{cases}$ (3.1)

Fix $T=T^{\prime}\times T^{\prime}$ where $T^{\prime}$ is the standard maximal
torus in $SL_{n}\left(\mathbb{C}\right)$. Then $W$ is naturally identified
with $S_{n}\times S_{n}$. Denote by $\mathbb{C}$ the trivial representation of
$S_{n}$, by $U$ the one dimensional alternating representation, by $V$ the
standard $n-1$ dimensional irreducible representation, and for $n\geq 3$, by
$V_{\left(d-2,2\right)}$ the Specht module corresponding to the partition
$\left(d-2,2\right)$. For $M_{1}$ and $M_{2}$ representations of $S_{n}$ we
use $M_{1}\boxtimes M_{2}$ to denote the representation $M_{1}\otimes M_{2}$
of $S_{n}\times S_{n}$ in which the first (respectively the second) copy of
$S_{n}$ acts on the first (respectively the second) tensor factor. See for
example Chapter 4 of [7] for further details about representations of
symmetric groups.

The $S_{n}\times S_{n}$ representation $\mathfrak{t}$ decomposes into
irreducible $S_{n}\times S_{n}$ representations as

$\mathfrak{t}\cong\left(V\boxtimes\mathbb{C}\right)\oplus\left(\mathbb{C}\boxtimes
V\right).$ (3.2)

There is the decomposition of the representation $s^{2}(\mathfrak{t})$ as

$\displaystyle s^{2}(\mathfrak{t})$ $\displaystyle\cong
s^{2}\left(\left(V\boxtimes\mathbb{C}\right)\oplus\left(\mathbb{C}\boxtimes
V\right)\right)$ $\displaystyle\cong
s^{2}\left(V\boxtimes\mathbb{C}\right)\oplus\mathrm{s}^{2}\left(\mathbb{C}\boxtimes
V\right)\oplus\left(V\boxtimes\mathbb{C}\right)\otimes\left(\mathbb{C}\boxtimes
V\right)$
$\displaystyle\cong\left(s^{2}(V)\boxtimes\mathbb{C}\right)\oplus\left(\mathbb{C}\boxtimes
s^{2}(V)\right)\oplus\left(V\boxtimes V\right).$ (3.3)

If $d=2$ then $V=U$, $s^{2}(V)=V^{\otimes 2}=\mathbb{C}$, and (3.3) decomposes
into irreducible $S_{n}\times S_{n}$ representations as

$\left(\mathbb{C}\boxtimes\mathbb{C}\right)\oplus\left(\mathbb{C}\boxtimes\mathbb{C}\right)\oplus\left(V\boxtimes
V\right).$ (3.4)

If $d\geq 3$ then by Exercise 4.19 in [7] there is an isomorphism of $S_{n}$
representations $s^{2}(V)\cong U\oplus V\oplus V_{\left(d-2,2\right)}$, and so
(3.3) decomposes into irreducible $S_{n}\times S_{n}$ representations as

$\displaystyle\left(U\boxtimes\mathbb{C}\right)\oplus\left(V\boxtimes\mathbb{C}\right)\oplus\left(V_{\left(d-2,2\right)}\boxtimes\mathbb{C}\right)\oplus\left(\mathbb{C}\boxtimes
U\right)$ $\displaystyle\oplus\left(\mathbb{C}\boxtimes
V\right)\oplus\left(\mathbb{C}\boxtimes
V_{\left(d-2,2\right)}\right)\oplus\left(V\boxtimes V\right).$ (3.5)

Equation (3.1) then follows from Schur’s Lemma and comparing (3.2) with (3.4)
and (3.2) with (3.5). ∎

The map $\psi$ can be given explicitly.

######  Lemma 3.2.

Let $G$ and $\mathfrak{g}$ be as in Lemma 3.1. For $n\geq 3$ the vector space
$\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)$ has a basis
consisting of the following two functions:

$\displaystyle\psi_{1}\left(\left(X,Y\right),\left(X^{\prime},Y^{\prime}\right)\right)$
$\displaystyle:=\sqrt{-1}\left(\left(XX^{\prime}+X^{\prime}X\right)-\frac{1}{n}\mathrm{tr}\left(XX^{\prime}+X^{\prime}X\right)\mathrm{Id},0\right)$
$\displaystyle\psi_{2}\left(\left(X,Y\right),\left(X^{\prime},Y^{\prime}\right)\right)$
$\displaystyle:=\sqrt{-1}\left(0,\left(YY^{\prime}+Y^{\prime}Y\right)-\frac{1}{n}\mathrm{tr}\left(YY^{\prime}+Y^{\prime}Y\right)\mathrm{Id}\right)$

Furthermore, the subspace of
$\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)$ consisting of
maps invariant under the involution on
${\mathfrak{su}_{n}\oplus\mathfrak{su}_{n}}$ defined by
$(x,y)\rightarrow(y,x)$ is one dimensional and is spanned by the map
$\Psi:=\psi_{1}+\psi_{2}$.

###### Proof.

It is straightforward to check that $\psi_{1}$ and $\psi_{2}$ are linearly
independent symmetric bilinear maps from $\mathfrak{g}\times\mathfrak{g}$ to
$\mathfrak{g}$. The result then follows from Lemma 3.1. The second assertion
is easily checked directly. ∎

######  Lemma 3.3.

Let $\Psi$ be as in Lemma 3.2. Then, for $n$ odd, the zero set of $\Psi$ is
$\left\\{0\right\\}$; for $n$ even, the zero set of $\Psi$ is the $G$-orbit of
the plane spanned by the elements $\left(\Lambda,0\right)$ and
$\left(0,\Lambda\right)$ of
$\mathfrak{g}=\mathfrak{su}_{n}\oplus\mathfrak{su}_{n}$, where $\Lambda$ is
the block diagonal sum of $\frac{n}{2}$ copies of the matrix

$\left(\begin{array}[]{cc}\sqrt{-1}&0\\\ 0&-\sqrt{-1}\end{array}\right).$

###### Proof.

Suppose that $\Psi\left(x\right)=0$. Write $x=\left(X,Y\right)$ where
$X,Y\in\mathfrak{su}_{n}$. Then

$\Psi\left(x\right)=2\sqrt{-1}\;\left(X^{2}-\frac{1}{n}\mathrm{tr}(X^{2})\mathrm{Id},Y^{2}-\frac{1}{n}\mathrm{tr}(Y^{2})\mathrm{Id}\right).$

As $\Psi$ is $G$-equivariant we can assume that $X$ and $Y$ are diagonal. The
kernel of the linear map
$T:\mathrm{Mat}_{n}\left(\mathbb{C}\right)\to\mathrm{Mat}_{n}\left(\mathbb{C}\right)$
given by

$T(X)=X-\frac{1}{n}\mathrm{tr}(X)\mathrm{Id},$

is the set of multiples of the identity matrix $\mathrm{Id}$. It follows that
$\Psi\left(x\right)=0$ if and only if each of $X$ and $Y$ satisfy the
condition that the squares of their eigenvalues are all equal. For an element
$Z$ of $\mathfrak{su}_{n}$ (the elements of which are traceless with imaginary
eigenvalues) this condition is only satisfied if the eigenvalues are either
all zero, or if $n$ is even and the eigenvalues are of the form $ai$ and $-ai$
for some $a\in\mathbb{R}$, with each occurring with multiplicity
$\frac{n}{2}$, in which case $Z$ is $SU_{n}$-conjugate to a multiple of
$\Lambda$. ∎

Finally in this section, we note that a very similar proof to that in Lemma
3.1 yields the following result.

######  Lemma 3.4.

Let $G=SU_{n}$ with $n\geq 2$, identify ${\mathfrak{g}:=\mathrm{Lie}(G)}$ with
${\mathfrak{su}_{n}}$, and consider $\mathfrak{g}$ as a representation of $G$
via the adjoint representation. Then the dimension of real vector space
$\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)$ is given by:

$\mathrm{dim}\;\mathrm{Hom}_{G}\left(s^{2}(\mathfrak{g}),\mathfrak{g}\right)=\begin{cases}0&\;n=2\\\
1&\;n\geq 3.\end{cases}$

##  4\. Calculations for $SU_{n}$

In this section we give the proof of the Theorem A using the strategy outlined
in the previous section; in this section we focus on step (3) — demonstrating
that the obstruction map does not vanish identically. It is important to fix
conventions at this stage to ensure the correct calculation of various
constants. The Lie algebra $\mathfrak{su}_{n}$ is the space of skew-Hermitian
$n\times n$ matrices. We endow $\mathfrak{su}_{n}$ with the Euclidean inner
product

$\langle A,B\rangle:=-\mathrm{tr}(AB).$

This inner product is $(1/2n)$ times the one induced by the Killing form and
so the resulting bi-invariant Einstein metric has Einstein constant
$\lambda=n/2$ (as the Killing form induces an Einstein metric with Einstein
constant $1/4$).

To construct the Einstein infinitesimal variations given by the
Gasqui–Goldschmidt construction (2.6) we require a tensor $\sigma\in
s^{3}(\mathfrak{su}(n))^{SU_{n}}$. Up to scale, there is only one such tensor
given by

$\sigma(X,Y,Z):=\sqrt{-1}\left(\mathrm{tr}(XYZ)+\mathrm{tr}(XZY)\right),$
(4.1)

where $X,Y,Z\in\mathfrak{su}_{n}.$

The following result allows us to reduce computing Koiso’s obstruction (2.5)
to a pointwise calculation (rather than having to integrate over the whole
manifold).

######  Lemma 4.1.

Let $\eta\in\mathfrak{su}_{n}$ and denote by $\tilde{\eta}$ the corresponding
left-invariant (resp. right-invariant) extension to a vector field on
$SU_{n}$. Then the tensor $h_{\eta}$ is left-invariant (resp. right
invariant).

###### Proof.

We shall demonstrate the left-invariant statement. Let $g,k\in SU_{n}$ and let
$X,Y\in\mathfrak{X}(SU_{n})$. We compute

$L_{g}^{\ast}(h_{\eta})(X,Y)(k)=h_{\eta}(L_{g\ast}X,L_{g\ast}Y)(gk)=\sigma(\tilde{\eta}(gk),L_{g\ast}X,L_{g\ast}Y)(gk),$

and thus as $\tilde{\eta}$ is left-invariant we obtain

$L_{g}^{\ast}(h_{\eta})(X,Y)(k)=L_{g}^{\ast}(\sigma)(\tilde{\eta},X,Y)(k)=\sigma(\tilde{\eta},X,Y)(k)=h_{\eta}(X,Y)(k).$

∎

###  4.1. Results in $\mathfrak{su}_{n}$

If we use the previous lemma to produce a left-invariant tensor $h_{\eta}$ and
choose the identity as the point in $SU_{n}$ at which to perform calculations,
the algebra we need to do takes place in $\mathfrak{su}(n)$. In this
subsection we collect the relevant results about the Euclidean vector space
${(\mathfrak{su}_{n},\langle\cdot,\cdot\rangle)}$. All of the following
lemmata are proved directly using straightforward (if tedious) matrix algebra;
most of the proofs are omitted.

In order to perform calculations, we fix an orthonormal basis of
$\mathfrak{su}_{n}$.

######  Lemma 4.2.

For $1\leq k\leq n-1$, define matrices

$T_{k}=\frac{\sqrt{-1}}{\sqrt{k(k+1)}}\mathrm{Diag}\left(\underbrace{1,1,\dots,1}_{k\
\mathrm{terms}},-k,0,\dots,0\right),$

and, for $1\leq k<l\leq n$, define matrices $E^{r}(k,l)$ and $E^{c}(k,l)$ by

$(E^{r}(k,l))_{pq}=\frac{1}{\sqrt{2}}\left(\delta_{kp}\delta_{lq}-\delta_{kq}\delta_{lp}\right),$

and

$(E^{c}(k,l))_{pq}=\frac{\sqrt{-1}}{\sqrt{2}}\left(\delta_{kp}\delta_{lq}+\delta_{kq}\delta_{lp}\right).$

Then the matrices

$\mathcal{B}=\\{T_{1},T_{2},\dots,T_{n-1}\\}\cup\\{E^{r}(k,l),E^{c}(k,l)\\}_{1\leq
k<l\leq n},$

form an orthonormal basis of $(\mathfrak{su}(n),\langle\cdot,\cdot\rangle)$
with the $T_{k}$ spanning to the Lie subalgebra of a maximal torus generated
by the diagonal matrices in $SU_{n}$.

We see from the form of the tensor $\sigma$ in equation (4.1), it will be
useful to consider various quantities of the form $XY+YX$ with
$X,Y\in\mathfrak{su}(n)$. To do this, we introduce a small variation on the
notation for the basis given in Lemma 4.2. Given an unordered pair
$\\{k,l\\}$, if $a=\min\\{k,l\\}$ and $b=\max\\{k,l\\}$, then we define

$E^{r}(\\{k,l\\})=E^{r}(a,b)\qquad\mathrm{and}\qquad
E^{c}(\\{k,l\\})=E^{c}(a,b).$

######  Lemma 4.3.

Let $T_{k}$, $E^{r}(k,l)$, and $E^{c}(k,l)$ be the orthonormal basis of
$(\mathfrak{su}_{n},\langle\cdot,\cdot\rangle)$ defined in Lemma 4.2. Then

  1. (i)

If $p=q$, then

$T_{p}T_{q}+T_{q}T_{p}=\dfrac{2}{p(p+1)}\mathrm{Diag}\left(\underbrace{-1,-1,\dots,-1}_{p\
\mathrm{terms}},-p^{2},0,\dots,0\right)$

  2. (ii)

If $p>q$, then

$T_{p}T_{q}+T_{q}T_{p}=\dfrac{2}{\sqrt{pq(p+1)(q+1)}}\mathrm{Diag}\left(\underbrace{-1,-1,\dots,-1}_{q\
\mathrm{terms}},q,0,\dots,0\right)$

  3. (iii)

$T_{p}E^{r}(k,l)+E^{r}(k,l)T_{p}=((T_{p})_{kk}+(T_{p})_{ll})E^{r}(k,l)$

  4. (iv)

$T_{p}E^{c}(k,l)+E^{c}(k,l)T_{p}=((T_{p})_{kk}+(T_{p})_{ll})E^{c}(k,l)$

  5. (v)

If $\\{k,l\\}\neq\\{p,q\\}$, then $E^{r}(k,l)E^{r}(p,q)+E^{r}(p,q)E^{r}(k,l)=$

$\dfrac{\sqrt{-1}}{\sqrt{2}}\left(\delta_{ql}E^{c}(\\{p,k\\})-\delta_{qk}E^{c}(p,l)-\delta_{lp}E^{c}(k,q)+\delta_{pk}E^{c}(\\{l,q\\})\right).$

  6. (vi)

If $k=p$ and $l=q$, then $E^{r}(k,l)E^{r}(p,q)+E^{r}(p,q)E^{r}(k,l)=$

$\mathrm{Diag}\left(\underbrace{0,0,\dots,0}_{k-1\
\mathrm{terms}},-1,\underbrace{0,0,\dots,0}_{l-k-1\
\mathrm{terms}},-1,0,0,\dots,0\right)$

  7. (vii)

If $\\{k,l\\}\neq\\{p,q\\}$, $E^{r}(k,l)E^{c}(p,q)+E^{c}(p,q)E^{r}(k,l)=$

$\dfrac{\sqrt{-1}}{\sqrt{2}}\left(\pm\delta_{ql}E^{r}(\\{p,k\\})+\delta_{pl}E^{r}(k,q)+\delta_{kq}E^{r}(p,l)\pm\delta_{pk}E^{r}(\\{q,l\\})\right).$

  8. (viii)

If $k=p\ \mathrm{and}\ l=q$, then
$E^{r}(k,l)E^{c}(p,q)+E^{c}(p,q)E^{r}(k,l)=0$.

  9. (ix)

If $\\{k,l\\}\neq\\{p,q\\},$, then
$E^{c}(k,l)E^{c}(p,q)+E^{c}(p,q)E^{c}(k,l)=$

$\dfrac{\sqrt{-1}}{\sqrt{2}}\left(\delta_{ql}E^{c}(\\{p,k\\})+\delta_{pl}E^{c}(k,q)+\delta_{kq}E^{c}(p,l)+\delta_{pk}E^{c}(\\{q,l\\})\right)$

  10. (x)

If $k=p$ and $l=q$, then $E^{c}(k,l)E^{c}(p,q)+E^{c}(p,q)E^{c}(k,l)=$

$\mathrm{Diag}\left(\underbrace{0,0,\dots,0}_{k-1\
\mathrm{tems}},-1,\underbrace{0,0,\dots,0}_{l-k-1\
\mathrm{terms}},-1,0,0,\dots 0\right)$

We will see that the precise signs of the terms in (vii) will not matter; the
real utility of Lemma 4.3 is that, if we choose basis elements
$X,Y,\in\mathcal{B}$ with $X\neq Y$, then

$\sqrt{-1}(XY+YX)\perp T_{n-1}.$

Thus choosing $\eta$ to be a convenient multiple of $T_{n-1}$ turns out to
greatly simplify calculations.

######  Lemma 4.4.

Let $\eta=\sqrt{-1}\mathrm{Diag}(\underbrace{1,1,\dots,1}_{n-1},-(n-1))$ and
let $H\in s^{2}(\mathfrak{g}^{\ast})$ be defined by

$H(X,Y)=\sqrt{-1}\hskip 1.0pt\mathrm{tr}(\eta(XY+YX)).$

Then $H$ is diagonal with respect to the orthonormal basis $\mathcal{B}$
defined in Lemma 4.2. Furthermore

$\displaystyle H(T_{p},T_{p})=2\qquad\mathrm{for}\quad p<n-1,$ (4.2)
$\displaystyle H(T_{n-1},T_{n-1})=-2(n-2),$ (4.3) $\displaystyle
H(E^{r}(k,l),E^{r}(k,l))=H(E^{c}(k,l),E^{c}(k,l))=2\qquad\mathrm{for}\quad
l<n,$ (4.4) $\displaystyle
H(E^{r}(k,n),E^{r}(k,n))=H(E^{c}(k,n),E^{c}(k,n))=-(n-2).$ (4.5)

###### Proof.

We begin by noting that $\eta=\sqrt{n(n-1)}T_{n-1}$ and

$H(X,Y)=\langle\eta,\sqrt{-1}(XY+YX)\rangle.$

Using Lemma 4.3 we see that all the non-diagonal terms of $H$ vanish. The
remaining diagonal terms are easily calculated. ∎

In order to compute the terms of Koiso’s obstruction (2.5) that contain
covariant derivatives we will need information about Lie brackets
$[\cdot,\eta]$. The following is again straightforward algebra.

######  Lemma 4.5.

Let $\eta$ be as in Lemma 4.4. Then for any
${X\in\mathrm{Mat}(\mathbb{C})^{n\times n}}$ we have

$[X,\eta]=n\sqrt{-1}\left(\begin{array}[]{ccccc}0&0&\dots&0&-X_{1n}\\\
0&0&\dots&0&-X_{2n}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\dots&0&-X_{(n-1)n}\\\
X_{n1}&X_{n2}&\dots&X_{n(n-1)}&0\end{array}\right).$

The fact that $H$ is diagonal will mean we need only compute very few Lie-
bracket-type terms. The following collects what we will need.

######  Lemma 4.6.

Let $\eta$ be as in Lemma 4.4. Then

  1. (i)

for $0<k<l<n$

$[E^{r}(k,l),[E^{r}(k,l),\eta]]=[E^{c}(k,l),[E^{c}(k,l),\eta]]=0,$

  2. (ii)

$[E^{r}(k,n),[E^{r}(k,n),\eta]]=n\sqrt{-1}\mathrm{Diag}\left(\underbrace{0,0,\dots,0}_{k-1\
\mathrm{terms}},-1,0,0,\dots,0,1\right),$

  3. (iii)

$[E^{c}(k,n),[E^{c}(k,n),\eta]]=n\sqrt{-1}\mathrm{Diag}\left(\underbrace{0,0,\dots,0}_{k-1\
\mathrm{terms}},-1,0,0,\dots,0,1\right).$

###### Proof.

Part (i) follows immediately from Lemma 4.5 as, for any $0\leq j\leq n$,

${(E^{r}(k,l))_{jn}=(E^{c}(k,l))_{jn}=0},$

if $l<n$. Parts (ii) and (iii) are also straightforward after using Lemma 4.5
to compute the first Lie bracket. ∎

###  4.2. Computing $\mathcal{I}$

In this section we consider the symmetric 2-tensor $h_{\eta}$ that is
generated by the inclusion of the left-invariant extension of $\eta$ in
$\sigma$ defined by Equation (4.1). We proceed by computing each of the terms
of the obstruction $\mathcal{I}$ given in Equation (2.5).

######  Lemma 4.7.

Let $\eta$ be as in Lemma 4.4 and let $h_{\eta}=\iota_{\tilde{\eta}}\sigma$,
where $\tilde{\eta}$ is the left-invariant extension of $\eta$ and $\sigma$ is
the symmetric 3-tensor defined in Equation (4.1). Then

$\langle(h_{\eta})^{k}_{i}(h_{\eta})_{kj},(h_{\eta})_{ij}\rangle_{L^{2}}=2(n-1)(n-2)(12-n^{2})\mathrm{Vol}(SU_{n}),$

where $\mathrm{Vol}(SU_{n})>0$ is the volume of the bi–invariant metric on
$SU_{n}$ induced the inner product $\langle\cdot,\cdot\rangle$ on
$\mathfrak{su}(n)$.

###### Proof.

The tensor $h_{\eta}$ is left-invariant by Lemma 4.1. Hence we can compute at
the identity and so

$\langle(h_{\eta})^{k}_{i}(h_{\eta})_{kj},(h_{\eta})_{ij}\rangle_{L^{2}}=\langle
H_{i}^{k}H_{kj},H_{ij}\rangle\mathrm{Vol}(SU_{n}),$

where $H$ is the symmetric bilinear form defined in Lemma 4.4 (we also use
$\langle\cdot,\cdot\rangle$ to denote the inner product on
$s^{2}(\mathfrak{su}(n))$ induced by the one on $\mathfrak{su}(n)$). As $H$ is
diagonal with respect to the orthonormal basis $\mathcal{B}$ defined in Lemma
4.2 we can compute

$\langle H_{i}^{k}H_{kj},H_{ij}\rangle=\mathrm{tr}(H^{3}),$

where the trace on the right-hand side is the usual trace of a matrix. Using
equations (4.2)-(4.5) we can compute

$\mathrm{tr}(H^{3})=((n-2)\times 8)-(8(n-2)^{3})+((n-2)(n-1)\times
8)-((n-1)\times(n-2)^{3}).$

The result follows after simplification. ∎

######  Lemma 4.8.

Let $h_{\eta}$ be as in Lemma 4.7. Then

${\langle\nabla_{i}\nabla_{j}(h_{\eta})_{kl},(h_{\eta})_{ij}(h_{\eta})_{kl}\rangle_{L^{2}}=[(n-2)^{2}(n-1)n(n+2)]\mathrm{Vol}(SU_{n}).}$

###### Proof.

As $\sigma$ is parallel we have (computing at the identity)

$\nabla_{i}\nabla_{j}(h_{\eta})_{kl}=\sigma(\nabla_{i}\nabla_{j}\tilde{\eta},E_{k},E_{l}),$

where $\\{E_{i}\\}$ is any basis of $\mathfrak{g}$. We are using bi-invariant
metric on $SU_{n}$ so, using left-invariant extensions of the basis
$\\{E_{i}\\}$, we have

$\nabla_{i}\nabla_{j}\tilde{\eta}=\frac{1}{4}[E_{i},[E_{j},\eta]].$

Thus

$\langle\nabla_{i}\nabla_{j}(h_{\eta})_{kl},(h_{\eta})_{ij}(h_{\eta})_{kl}\rangle_{L^{2}}=\frac{1}{4}\sigma([E_{i},[E_{j},\eta]],E_{k},E_{l})H_{ij}H_{kl}\mathrm{Vol}(SU_{n}).$

If we take $\\{E_{i}\\}=\mathcal{B}$, and as $H$ is diagonal with respect to
the basis $\mathcal{B}$, we need only consider $E_{i}=E_{j}$ and $E_{k}=E_{l}$
and so

$\langle\nabla_{i}\nabla_{j}(h_{\eta})_{kl},(h_{\eta})_{ij}(h_{\eta})_{kl}\rangle_{L^{2}}=\frac{1}{4}\sigma([E_{i},[E_{i},\eta]],E_{k},E_{k})H_{ii}H_{kk}.$

From Lemma 4.6 we see the right-hand side of the previous equation is non-zero
only when $E_{i}=E^{r}(k,n)$ or $E_{i}=E^{c}(k,n)$; in either case,
${H_{ii}=-2(n-2)}$ and we obtain using Lemma 4.6

$\frac{1}{4}\langle[E_{i},[E_{j},\eta]],H_{ij}H_{kl}\rangle=-\left(\frac{n(n-2)}{2}\right)\sum_{k=1}^{n-1}\langle\sigma(v_{k},\cdot,\cdot),H(\cdot,\cdot)\rangle,$

where

$v_{k}=\sqrt{-1}\mathrm{Diag}\left(\underbrace{0,0,\dots,0}_{k-1\
\mathrm{terms}},-1,0,0,\dots,0,1\right).$

We split the sum up as follows:

$\displaystyle\langle\sigma(v_{k},\cdot,\cdot),H(\cdot,\cdot)\rangle$
$\displaystyle=\sum_{i=1}^{n-1}\sigma(v_{k},T_{i},T_{i})H(T_{i},T_{i})$
$\displaystyle+2\sum_{1\leq i<j<n}\sigma(v_{k},E^{r}(i,j),E^{r}(i,j))$
$\displaystyle+2\sum_{1\leq i<j<n}\sigma(v_{k},E^{c}(i,j),E^{c}(i,j))$
$\displaystyle-(n-2)\sum_{i=1}^{n-1}\sigma(v_{k},E^{r}(i,n),E^{r}(i,n))$
$\displaystyle-(n-2)\sum_{i=1}^{n-1}\sigma(v_{k},E^{c}(i,n),E^{c}(i,n)),$

where we have used Equations (4.4) and (4.5) to simplify the final two
summations. Using Lemma 4.3 and the definition of $\sigma$ we can compute

$\langle\sigma(v_{k},\cdot,\cdot),H(\cdot,\cdot)\rangle=\sum_{i=1}^{n-1}\sigma(v_{k},T_{i},T_{i})H(T_{i},T_{i})\
-4(n-2)-2(n-2)^{2}.$

Summing over the possible values of $k$ yields

$\displaystyle\sum_{k=1}^{n-1}\langle\sigma(v_{k},\cdot,\cdot),H(\cdot,\cdot)\rangle$
$\displaystyle=\left(\sum_{k=1}^{n-1}\sum_{i=1}^{n-1}\sigma(v_{k},T_{i},T_{i})H(T_{i},T_{i})\right)$
$\displaystyle-4(n-1)(n-2)-2(n-1)(n-2)^{2}.$

Switching the order of summation we can compute

$\sum_{k=1}^{n-1}\sigma(v_{k},T_{i},T_{i})H(T_{i},T_{i})=\left\\{\begin{array}[]{cc}-4&\mathrm{if}\
i<n-1,\\\ -4(n-2)^{2}&\mathrm{if}\ i=n-1.\end{array}\right.$

Hence

$\displaystyle\sum_{k=1}^{n-1}\langle\sigma(v_{k},\cdot,\cdot),H(\cdot,\cdot)\rangle$
$\displaystyle=-4(n-2)(n-1)-4(n-1)(n-2)-2(n-1)(n-2)^{2}$
$\displaystyle=-2(n-2)(n-1)(n+2).$

The result now follows. ∎

######  Lemma 4.9.

Let $h_{\eta}$ be as in Lemma 4.7. Then

$\langle\nabla_{i}\nabla_{j}h_{kl},h_{ik}h_{jl}\rangle_{L^{2}}=-\dfrac{1}{2}[(n-2)^{2}(n-1)n(n+2)]\mathrm{Vol}(SU_{n}).$

###### Proof.

As in the proof of Lemma 4.8, the fact that $\sigma$ is parallel means that,
for any local frame $\\{E_{i}\\}$,

$\nabla_{i}\nabla_{j}h_{kl}=\sigma(\nabla_{i}\nabla_{j}\eta,E_{k},E_{l}).$

Computing at the identity in the orthonormal basis $\mathcal{B}$, Lemma 4.4
yields $h_{\eta}=H$ is diagonal and so, to compute
${\langle\nabla_{i}\nabla_{j}h_{kl},h_{ik}h_{jl}\rangle}$, we need only
consider terms of the form

$\sigma(\nabla_{i}\nabla_{j}\eta,E_{i},E_{j})=\dfrac{1}{4}\sigma([E_{i},[E_{j},\eta]],E_{i},E_{j}).$

Noting that $\sigma$ is Ad-invariant and we can write

$\sigma(\nabla_{i}\nabla_{j}\eta,E_{i},E_{j})=-\dfrac{1}{4}\sigma([E_{j},\eta],E_{i},[E_{i},E_{j}]).$

Using the definition of $\sigma$ given by Equation(4.1) we have

$\displaystyle-\dfrac{1}{4}\sigma([E_{j},\eta],E_{i},[E_{i},E_{j}])$
$\displaystyle=-\dfrac{\sqrt{-1}}{4}\mathrm{tr}([E_{j},\eta](E_{i}[E_{i},E_{j}]+[E_{i},E_{j}]E_{i}))$
$\displaystyle=-\dfrac{\sqrt{-1}}{4}\mathrm{tr}([E_{j},\eta](E_{i}^{2}E_{j}-E_{j}E_{i}^{2}))$
$\displaystyle=-\dfrac{\sqrt{-1}}{4}\langle[E_{j},\eta],[E_{j},E_{i}^{2}]\rangle$
$\displaystyle=\dfrac{\sqrt{-1}}{4}\langle[E_{j},[E_{j},\eta]],E_{i}^{2}\rangle,$

where the final equality follows as the inner product
$\langle\cdot,\cdot\rangle$ is also Ad-invariant. Thus we have (relabelling
indices as needed)

$\displaystyle\langle\nabla_{i}\nabla_{j}h_{kl},h_{ik}h_{jl}\rangle_{L^{2}}$
$\displaystyle=-\dfrac{\sqrt{-1}}{4}\mathrm{tr}([E_{i},[E_{i},\eta]]E_{k}^{2})H_{ii}H_{kk}\mathrm{Vol}(SU_{n})$
$\displaystyle=-\frac{1}{8}\sigma([E_{i},[E_{i},\eta]],E_{k},E_{k})H_{ii}H_{kk}\mathrm{Vol}(SU_{n}).$
$\displaystyle=-\frac{1}{2}{\langle\nabla_{i}\nabla_{j}(h_{\eta})_{kl},(h_{\eta})_{ij}(h_{\eta})_{kl}\rangle_{L^{2}}},$
$\displaystyle=-\dfrac{1}{2}[(n-2)^{2}(n-1)n(n+2)]\mathrm{Vol}(SU_{n}),$

where Lemma 4.8 was applied to deduce the final equality. ∎

We now have everything needed to prove the main theorem in this paper.

###### Proof of Theorem A.

As $2n+1$ is odd, the obstruction to integrability to second order $\psi$ has
no non-trivial zeroes by Lemma 3.3. Thus, to show that none of the elements of
$\varepsilon(g)$ are integrable to second order we simply need to find
$h^{\ast}\in\varepsilon(g)$ such that $\mathcal{I}(h^{\ast})\neq 0$. We take
$h^{\ast}=h_{\eta}$ where $\eta$ is as in Lemma 4.4, and then, using Lemmas
4.7, 4.8, and 4.9, and the fact that the Einstein constant
$\lambda=\dfrac{n}{2}$ we compute

$\displaystyle\mathcal{I}(h^{\ast})=$ $\displaystyle 2\lambda\langle
h_{i}^{k}h_{kj},h_{ij}\rangle+3\langle\nabla_{i}\nabla_{j}h_{kl},h_{ij}h_{kl}\rangle-6\langle\nabla_{i}\nabla_{j}h_{kl},h_{ik}h_{jl}\rangle$
$\displaystyle=$
$\displaystyle\left(2\cdot\frac{n}{2}\left[2(n-1)(n-2)(12-n^{2})\right]\right.$
$\displaystyle\left.+3\left[(n-2)^{2}(n-1)n(n+2)\right]\right.$
$\displaystyle\left.+6\dfrac{1}{2}[(n-2)^{2}(n-1)n(n+2)]\right)\mathrm{Vol}(SU_{n}).$

Simplifying, this yields

$\mathcal{I}(h^{\ast})=4n^{3}(n-1)(n-2)\mathrm{Vol}(SU_{n})\neq 0.$

Hence all the infinitesimal variations are non-integrable and so the Einstein
metric is isolated in its moduli space. ∎

##  5\. Applications to Solitonic Rigidity

###  5.1. Ricci Solitons

Einstein metrics are fixed points (up to diffeomorphism and homothethic
scaling) of the Ricci flow

$\frac{\partial g}{\partial t}=-2\mathrm{Ric}(g).$ (5.1)

There are also non-Einstein fixed points of Equation 5.1 known as Ricci
solitons which are metrics that solve

$\mathrm{Ric}(g)+\frac{1}{2}L_{X}g=\lambda g,$ (5.2)

for a vector field $X\in\Gamma(TM)$ and constant $\lambda\in\mathbb{R}$. A
soliton is known as expanding, steady, or shrinking depending upon whether
$\lambda<0,\lambda=0,$ or $\lambda>0$. If one can find a function $f$ such
that $X=\nabla f$ then Equation (5.2) becomes

$\mathrm{Ric}(g)+\mathrm{Hess}(f)=\lambda g,$ (5.3)

and the metric $g$ is known as a gradient Ricci soliton with potential
function $f$. Perelman [23] demonstrated that, on compact manifolds, any non-
Einstein Ricci soliton is a shrinking gradient Ricci soliton. Therefore we
only consider shrinking gradient Ricci solitons on compact manifolds in the
sequel.

###  5.2. Ricci soliton moduli

The study of moduli spaces for Ricci solitons was initiated by Podestà and
Spiro in [25]. It is clear from Equation (5.3) that, given a gradient Ricci
soliton $g$ with potential function $f$, adding a constant to $f$ yields
admissible potential function. Hence, in order to study moduli of solitons,
the potential functions should be normalised.

If $\mathcal{M}$ denotes the space of all Riemannian metrics on a manifold
$M^{m}$, Podestà and Spiro introduce the set
${\mathcal{P}\subset\mathcal{M}\times C^{\infty}(M)}$ defined by

$\mathcal{P}=\left\\{(g,f)\ \bigg{|}\
\int_{M}e^{-f}d\mathrm{Vol}_{g}=(2\pi)^{m/2}\right\\}.$

They define the set $\mathcal{S}ol\subset\mathcal{P}$ to be the pairs solving
Equation (5.3) after fixing $\lambda=1$. Given a Ricci soliton
${(g,f)\in\mathcal{S}ol}$ they define an ‘$f$-twisted’ slice inside
$\mathcal{M}$ for the action of the diffeomorphism group and the premoduli
space about $(g,f)$ is the intersection of the slice (or rather the product of
the slice and $C^{\infty}(M)$) and the set $\mathcal{S}ol$. We will not
directly need the theory developed and so we refer the reader to the paper
[25] for further details. There is also a solitonic notion of infinitesimal
variation generalising that of EIDs given in Definition 2.2. Podestà and Spiro
use a different definition of the space of essential infinitesimal soliton
deformations and then show the following is equivalent (see Theorem 3.3 in
[25]).

######  Definition 5.1 (Essential infinitesimal solitonic variation).

Let $(g,f)\in\mathcal{S}ol$ be a shrinking Ricci soliton and let ${h\in
s^{2}(T^{\ast}M)}$ satisfy

${}\mathrm{tr}(h)=\alpha,$ (5.4)
${}\mathrm{div}_{f}(h):=\mathrm{div}(h)-\iota_{\nabla f}h=0,$ (5.5)

and

${}\Delta_{f}h+2\mathrm{Rm}(h)=0,$ (5.6)

where $\alpha\in C^{\infty}(M)$ and
${\Delta_{f}(\cdot):=\Delta(\cdot)-\nabla_{\nabla f}(\cdot)}$. Then the pair
$(h,a)$ is referred to as an (essential)  infinitesimal solitonic deformation
(ISD). We denote the space of such $(h,\alpha)$ by $\sigma(g,f)$.

In [25], Podestà and Spiro prove the solitonic analogue of Koiso’s Theorem
2.3: given $(g,f)\in\mathcal{S}ol$, there is a real analytic submanifold
$\mathcal{Z}$ of the product of the $f$-twisted slice and $C^{\infty}(M)$ such
that the tangent space to $Z$ at $(g,f)$ is $\sigma(g,f)$ and the premoduli
space for $(g,f)$ is a real-analytic subset of $\mathcal{Z}$.

If the Ricci soliton is an Einstein metric then the potential function $f$ is
constant and

$f=\log\left(\mathrm{Vol}(M,g)\right)-\frac{m}{2}\log\left(\frac{2\pi}{\lambda}\right).$

In this case the space of infinitesimal solitonic deformations decomposes as
the direct sum of the infinitesimal Einstein deformations and an eigenspace
for a particular eigenvalue of the Laplacian on functions.

######  Lemma 5.2 (Podesta–Spiro, Proposition 4.4 in [25] \- see also Kröncke
Lemma 6.2 in [17]).

Let $(M,g)$ be an Einstein manifold with Einstein constant $\lambda$. Let EID
be the space of infinitesimal Einstein deformations and ISD the space of
infinitesimal solitonic deformations. Then ISD has the following
decomposition:

$\mathrm{ISD}=\mathrm{EID}\oplus\left\\{\lambda v\cdot
g+\mathrm{Hess}(v)\bigg{|}v\in C^{\infty}(M),\Delta v=-2\lambda v\right\\}.$

We will denote the $-2\lambda$-eigenspace of the Laplacian by $V_{-2\lambda}$.
As with infinitesimal Einstein deformations, a natural question is whether an
element of ISD is the tangent to a genuine family of Ricci solitons. Kröncke
proved the following analogy to Lemma 2.5.

######  Theorem 5.3 (Kröncke, Theorem 5.7 in [18]).

Let $(M,g)$ be an Einstein metric with Einstein constant $\lambda>0$ and let
$v\in V_{-2\lambda}$.
Then ${h=\lambda v\cdot g+\nabla^{2}v\in\mathrm{ISD}}$ is not integrable to
second order if there exists a function $w\in V_{-2\lambda}$ such that

$\int_{M}v^{2}w\ d\mathrm{Vol}_{g}\neq 0.$

An equivalent formulation of this theorem is to say that, if $v\in
V_{-2\lambda}$, then the corresponding ISD $h$ is integrable to second order
if and only if the $L^{2}$-projection of $v^{2}$ to $V_{-2\lambda}$ vanishes.

Let $M=G/K$ be a compact irreducible Hermitian symmetric space; a classical
result due to Matsushima [21] gives a $G$-equivariant isomorphism
$\Phi:\mathfrak{g}\rightarrow V_{-2\lambda}$. The methods of Section 4 thus
give a proof of the following theorem of Kröncke.

###### Proof of Theorem C.

The space of infinitesimal Einstein deformations $\varepsilon(g)$ vanishes
[15] and so $\mathrm{ISD}\cong\mathfrak{su}_{2n+1}$.

The obstruction given by Theorem 5.3 is an element of
$\mathrm{Hom}_{SU_{2n+1}}(s^{2}(\mathfrak{g}),\mathfrak{g})$, where
${\mathfrak{g}=\mathfrak{su}_{2n+1}}$ which is one dimensional by Lemma 3.4.
The obstruction can be understood to be a multiple of the map

$\Psi(X,Y)=\sqrt{-1}\left(XY+YX-\frac{\mathrm{tr}(XY+YX)}{2n+1}\mathrm{Id_{2n+1}}\right),$

and as $2n+1$ is odd, this map does not have any non-trivial zeroes. It is
possible to find an eigenfunction $v\in V_{-2\lambda}$ such that

$\int_{\mathbb{CP}^{2n}}v^{3}d\mathrm{Vol}_{g}\neq 0,$

(see [11], [13], or [20]). Hence the obstruction is a non-zero multiple of
$\Psi$ and so does not vanish for any perturbation induced by $v\in
V_{-2\lambda}$. ∎

Podestà and Spiro also considered the following notion of rigidity for an
Einstein metric.

######  Definition 5.4 (Weak solitonic rigidity - Definition 5.1 in [25]).

A compact Einstein metric $g$ is said to be weakly solitonic rigid if there is
a neighbourhood $\mathcal{U}\subset\mathcal{M}$ of $g$ such that every Ricci
soliton in $\mathcal{U}$ is Einstein.

We can now prove Theorem B which is a generalisation of Theorem C in this
weaker setting.

###### Proof of Theorem B.

The structure theorem of Podestà–Spiro for the solitonic premoduli space
guarantees that in a small enough neighbourhood of $g$, all solitons come from
integrable deformations of $g$.

For the symmetric space metric $g$ on $Gr_{k}(\mathbb{C}^{2n+1})$ the space of
soliton deformations splits as in Lemma 5.2. As in the proof of Theorem C, the
eigenspace $V_{-2\lambda}$ is $SU_{2n+1}$-equivariantly isomorphic to
$\mathfrak{su}_{2n+1}$. In [11], the authors show that there exists an
eigenfunction $f\in V_{-2\lambda}$ with

$\int_{Gr_{k}(\mathbb{C}^{2n+1})}f^{3}d\mathrm{Vol}_{g}\neq 0.$

Hence Kröncke’s obstruction in Theorem 5.3 does not vanish and is given by a
multiple of the map $\Psi$ from the proof of Theorem B which has no non-
trivial zeroes on $\mathfrak{su}_{2n+1}$. Hence the only possibly integrable
definitions are EIDs. ∎

##  6\. Applications to stability of the Ricci flow and to other rigidity
problems

###  6.1. Applications to dynamical stability

For a general compact Riemannian manifold $(M^{m},g)$, Perelman’s
$\nu$-entropy [23] is given by

$\nu(g)=\inf\left\\{\mathcal{W}(g,f,\tau):f\in C^{\infty}(M),\ \tau>0,\
(4\pi\tau)^{-\frac{m}{2}}\int_{M}e^{-f}d\mathrm{Vol}_{g}=1\right\\},$

where

$\mathcal{W}(g,f,\tau)=\int_{M}[\tau(\mathrm{S}(g)+|\nabla
f|^{2})+f-m]e^{-f}d\mathrm{Vol}_{g}.$

For any metric $g$, the infimum is always achieved by a pair $(f,\tau)$.
Perelman demonstrated the following formula for the first variation of $\nu$;
let ${g(s):=g+sh}$ for some $h\in s^{2}(T^{\ast}M)$ and
$s\in(-\varepsilon,\varepsilon)$ then

$\dfrac{d}{ds}\bigg{|}_{s=0}\nu(s)=-\tau(4\pi\tau)^{-\frac{m}{2}}\int_{M}\bigg{\langle}h,\
\mathrm{Ric}(g)+\mathrm{Hess}(f)-\frac{1}{2\tau}g\bigg{\rangle}e^{-f}d\mathrm{Vol}_{g}.$
(6.1)

What Equation (6.1) demonstrates is that Ricci solitons are critical points of
$\nu$. Furthermore, from Equation (6.1), it can be easily deduced that the
$\nu$-entropy is increasing along the Ricci flow except at these critical
points; hence, variations that increase the entropy are destabilising.

We return to the case where $g$ is an Einstein metric but, in order to keep
our formulae in line with others in the literature, we will write the Einstein
constant $\lambda=\frac{1}{2\tau}$. The second variation variation of $\nu$ at
an Einstein metric was computed by Cao, Hamilton and Ilmanen [3] (see [5] for
a proof in the general case of a Ricci soliton). They showed that the second
variation on TT tensors is controlled by the spectrum of the Lichnerowicz
Laplacian and that the second variation vanishes if $h$ is an EID. Thus, to
determine whether an EID $h$ is destabilising, we must compute the third
derivative of $\nu(g(s))$ at $s=0$. To do this we will follow the method used
by Knopf and Sesum [13] to compute the third variation in conformal directions
and write

$\displaystyle\dfrac{d\nu(s)}{ds}\bigg{|}_{s=0}$
$\displaystyle=A\int_{M}\bigg{\langle}h,\
\mathrm{Ric}(g)+\mathrm{Hess}(f)-\frac{1}{2\tau}g\bigg{\rangle}e^{-f}d\mathrm{Vol}_{g}$
$\displaystyle=A\int_{M}\underbrace{\bigg{(}g^{ip}g^{jq}h_{pq}\bigg{)}}_{B}\underbrace{\bigg{(}\mathrm{Ric}_{ij}+\mathrm{Hess}(f)_{ij}-\frac{1}{2\tau}g_{ij}\bigg{)}}_{C}\underbrace{\bigg{(}e^{-f}d\mathrm{Vol}_{g}\bigg{)}}_{D},$

where $A=-\tau(4\pi\tau)^{-\frac{m}{2}}$. In the following proof we will also
use the convention of [13] and write
$\mathcal{A}^{\prime},\mathcal{A}^{\prime\prime},etc.$ for the derivative with
respect to $s$ and evaluated at $s=0$ of any quantity $\mathcal{A}$.

###### Proof of Theorem D.

Clearly, when $s=0$ we have that $C=0$. Hence Lemma 2.4 of [5] implies that
$\tau^{\prime}=0$ and so $A^{\prime}=0$. Using the formulae for
$\mathrm{Ric}^{\prime}$ (see [31] for example), the fact that $h$ is an EID
and that $f(0)$ is constant, we have

$C^{\prime}=-\mathrm{Hess}(f^{\prime}).$

Cao and Zhu give a charactersiation of $f^{\prime}$ for variations at general
Ricci soliton (see the proof of Lemma 2.4 in [5]). It is clear that at an
Einstein metric, if $h$ is TT then $f^{\prime}=0$ and so $C^{\prime}=0$.
Elementary calculus then yields

$\displaystyle\frac{d^{3}}{ds^{3}}\nu(g(s))\bigg{|}_{s=0}$
$\displaystyle=A\int_{M}BC^{\prime\prime}D$
$\displaystyle=-\tau(4\pi\tau)^{-\frac{m}{2}}\int_{M}\bigg{\langle}h,\
\mathrm{Ric}(g)^{\prime\prime}+(\mathrm{Hess}(f))^{\prime\prime}\bigg{\rangle}e^{-f}d\mathrm{Vol}_{g}.$

As $f^{\prime}=0$,

$(\mathrm{Hess}(f))^{\prime\prime}=\mathrm{Hess}(f^{\prime\prime}).$

The TT tensors are $L^{2}$-orthogonal to Lie derivatives of the metric and so,
as $\mathrm{Hess}(f^{\prime\prime})=\frac{1}{2}L_{\nabla f^{\prime\prime}}g$,
we have

$\frac{d^{3}}{ds^{3}}\nu(g(s))\bigg{|}_{s=0}=-\tau(4\pi\tau)^{-\frac{m}{2}}\int_{M}\bigg{\langle}h,\
\mathrm{Ric}(g)^{\prime\prime}\bigg{\rangle}e^{-f}d\mathrm{Vol}_{g}.$

The result then follows from Proposition 4.2 and Lemma 4.3 in [15]. ∎

###### Proof of Theorem E.

For any $n>2$ we have demonstrated an EID $h$ such that $\mathcal{I}(h)\neq
0$. For the family of metrics $g(s):=g+sh$, the first and second derivatives
of $\nu(g(s))$ but, by Theorem D, the third derivative of $\nu(g(s))$ does not
vanish. Hence $g$ is not a local maximum of the $\nu$-entropy and so $g$ is
dynamically unstable. ∎

###  6.2. Applications to other rigidity problems

For the symmetric spaces $SU_{2n+1}/SO_{2n+1}$, if one can demonstrate that,
for some $h_{\eta}$, the quantity $\mathcal{I}$ does not vanish then none of
the EID on these spaces is integrable. It would be particularly interesting to
compute $\mathcal{I}$ given Theorem D, as the dynamical stability of the five-
dimensional space $SU_{3}/SO_{3}$ is currently unknown.

Our methods show almost all the EIDs on $SU_{2n}$ are not integrable. Higher
order obstructions need to be considered for the remaining cases.

Koiso demonstrated in [15] that
$\mathrm{Hom}_{E_{6}}(s^{2}(\mathfrak{e}_{6}),\mathfrak{e}_{6})=0$ and so all
the EIDs on this space are integrable to second order.

The Grassmannians have EIDs, but these do not arise from the construction
outlined in Section 2. For Grassmannians $SU_{p+q}/S(U_{p}\times U_{q})$,
where

$K=S(U_{p}\times
U_{q})\qquad\mathrm{and}\qquad\mathfrak{p}=\left(\begin{array}[]{cc}0_{p\times
p}&Z\vspace{2pt}\\\ -\overline{Z}^{t}&0_{q\times q}\end{array}\right),$

for $Z\in M^{p\times q}(\mathbb{C})$. As the product of any three matrices in
$\mathfrak{p}$ is trace-free, we see directly that the space
$s^{3}(\mathfrak{p})^{K}$ vanishes in this case. The construction of
infinitesimal variations for the Grassmannians is detailed in Chapter VIII of
the book [8]. The method of generating EIDs uses some of the complex
differential geometry specific to the case of the Grassmannians and in
particular, the generalised Euler sequence. Again, for the spaces
$SU_{2n+1}/S(U_{n-k}\times U_{k})$, if one such deformation is obstructed then
they are all obstructed. We hope to investigate this construction in a future
work.

Finally, we note that the bi-invariant metric on the compact Lie group $G_{2}$
is known to admit infinitesimal solitonic deformations [4]. In [10], the
second author demonstrated that there exist deformations that are not
integrable to second order and hence the Einstein metric is dynamically
unstable. It would be interesting to extend this analysis to characterise
precisely which solitonic deformations are not integrable.

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11institutetext: School of Computer Science, University of Sydney, NSW 2008,
Australia 22institutetext: Brain and Mind Centre, University of Sydney, NSW
2050, Australia 33institutetext: School of Mathematics and Statistics,
University of Sydney, NSW 2050, Australia 44institutetext: I-MED Radiology
Network, NSW 2050, Australia 55institutetext: Sydney Neuroimaging Analysis
Centre, NSW 2050, Australia
55email<EMAIL_ADDRESS>
# Symmetry Awareness Encoded Deep Learning Framework for Brain Imaging
Analysis

Yang Ma 1122 Dongang Wang 2255 Peilin Liu 2233 Lynette Masters 44 Michael
Barnett 2255 Weidong Cai 11 Chenyu Wang 2255

###### Abstract

The heterogeneity of neurological conditions, ranging from structural
anomalies to functional impairments, presents a significant challenge in
medical imaging analysis tasks. Moreover, the limited availability of well-
annotated datasets constrains the development of robust analysis models.
Against this backdrop, this study introduces a novel approach leveraging the
inherent anatomical symmetrical features of the human brain to enhance the
subsequent detection and segmentation analysis for brain diseases. A novel
Symmetry-Aware Cross-Attention (SACA) module is proposed to encode symmetrical
features of left and right hemispheres, and a proxy task to detect symmetrical
features as the Symmetry-Aware Head (SAH) is proposed, which guides the
pretraining of the whole network on a vast 3D brain imaging dataset comprising
both healthy and diseased brain images across various MRI and CT. Through
meticulous experimentation on downstream tasks, including both classification
and segmentation for brain diseases, our model demonstrates superior
performance over state-of-the-art methodologies, particularly highlighting the
significance of symmetry-aware learning. Our findings advocate for the
effectiveness of incorporating symmetry awareness into pretraining and set a
new benchmark for medical imaging analysis, promising significant strides
toward accurate and efficient diagnostic processes. Code is available at
https://github.com/bitMyron/sa-swin.

###### Keywords:

Symmetry-Aware Cross-Attention (SACA) Self-Supervised Learning Neuroimaging
Diagnosis

##  1 Introduction

In many neurological and psychiatric conditions, the symmetry nature of the
left and right brain hemispheres plays a pivotal role in brain disease
diagnosis and monitoring, often acting as a precursor to various disorders.
The measurement of pathological structural and functional asymmetries could
also be used in managing disease progression and the effectiveness of
treatment [18]. The significance of these asymmetrical alterations is
especially marked in diseases like Alzheimer’s disease (AD) [14] and
schizophrenia [22], where hemispheric symmetry is also considered as one of
the most indicative biomarkers by radiologists. Figure 1 showcases instances
of pathologies affecting the brain’s symmetry.

Figure 1: Brain T1 images: (a) represents a healthy case, displaying
symmetrical shape, structure, and image intensity. (b)-(d) depict patients
with a brain tumor, Alzheimer’s Disease, and focal epilepsy, respectively, and
the noticeable structural asymmetry parts are circled out.

Leveraging the clinical experience and identifying such asymmetrical features
in neuroimaging analysis tasks can further enhance the efficacy of deep
learning tools used in triage, annotation, and diagnosis. Certain approaches
attempt to identify asymmetries by subtracting the L/R flipped image from the
original and using the resulting subtraction as input for deep neural networks
[12]. Other strategies involve segmenting the regions of interest on the
original and flipped images separately, employing the pair of segmentation
masks to compute an additional loss to improve model performance [13].
However, hemispheric symmetrical features were only used for pre-processing or
data augmentation rather than being encoded within the network structure,
which limits the generalization ability to extend to the general neuroimaging
analysis process.

To better leverage symmetrical features, we introduce an innovative Symmetry-
Aware Cross-Attention (SACA) module, advancing the attention mechanism’s
application in neuroimaging analysis. Our module enhances the transformer
blocks’ capabilities by implementing cross-attention mechanisms between the
original input and its symmetrical counterpart, compelling the network to
focus on symmetrical comparisons and semantic feature encoding. This approach
mimics the expert diagnostic strategy, leveraging self-supervised and
supervised training processes to improve the model’s understanding of brain
anatomy and its inherent symmetries.

The SACA module can be pretrained with a substantial amount of data based on a
self-supervised learning framework to be further used for downstream tasks,
including disease diagnosis, brain structure, and lesion segmentation. The
advancement of self-supervised learning has been illustrated in previous work
[1], which involve generating pre-text tasks, including inpainting [10],
rotation [9] and contrastive coding [2], to embed intrinsic features. Besides
these tasks widely used in self-supervised learning methods, we introduce a
Symmetry-Aware Head (SAH) to enhance the pretraining process by focusing on
the contrastive symmetrical features of the left and right hemispheres. Since
they are entangled with the patient status (i.e., healthy or disease), the
hemispheric symmetrical features are utilized in computing the symmetry-aware
loss.

To fulfill the pretraining, we have compiled an extensive dataset comprising
brain MRI and CT images from both healthy subjects (assumed to be symmetrical)
and diseased subjects (assumed with asymmetrical features). Empirical
experiments of our approach, through extensive pretraining on brain CT and MRI
datasets and subsequent evaluations on various classification and segmentation
tasks, underscore a notable improvement over existing baseline models.

In general, our primary contributions are outlined as follows:

  * $\bullet$

We introduce a novel Symmetry-Aware Cross-Attention (SACA) module, which
leverages a cross-attention mechanism to analyze the relationship between an
image and its symmetrical counterpart, facilitating a deeper understanding of
brain anatomy and its inherent symmetries.

  * $\bullet$

The proposed network is pretrained on a symmetry-aware self-supervised process
and can be further applied in real-world clinical datasets and analysis tasks,
including classification and segmentation for multiple modalities.

  * $\bullet$

A vast dataset spanning MRI and CT modalities has been curated, and our
approach has undergone extensive testing and demonstrated state-of-the-art
performance on diagnosis and segmentation tasks.

##  2 Methodology

In this study, we leverage a substantial corpus of 3D brain imaging data,
denoted as $\mathcal{I}^{0}=\\{\mathbf{I}_{n}\\}$. Each image is associated
with a binary label $\mathcal{Y}^{0}=\\{0,1\\}$ indicating the health status
of the subject (healthy or diseased), where $y_{n}^{0}\in\mathcal{Y}^{0}$. Our
objective is to train a model to encapsulate structural insights into the
human brain. The comprehensive pipeline is depicted in Figure 2.

###  2.1 Preprocssing

A midway registration pipeline was introduced to standardize the input image
to position the brain within a standardized 3D coordinate framework, ensuring
that flipping the image volume along its sagittal axis could generally map the
left and right cerebral hemispheres correctly. Given an original input image
$\mathbf{I}$, we first generate its mirror image $\mathbf{I}^{\prime}$ by
flipping $\mathbf{I}$ across the vertical plane of the image. We then obtain
the affine transformation
$\mathcal{T}:{\mathbf{I}}\rightarrow{\mathbf{I}}^{\prime}$ with the widely
used linear registration tool FMRIB’s Linear Image Registration Tool (FLIRT)
[17]. Based on the generated transformation matrix, the midway points of all
transformation operations can be further calculated and formed the midway
transformation $\mathcal{T}_{1/2}$. The midway transformation is applied to
$\mathbf{I}$, resulting in $\bar{\mathbf{I}}$, whose sagittal plane is
perfectly aligned with respect to the vertical plane of the image volume. By
the midway registration, the simple flip of the input image ensures the
mirroring of the left and right hemispheres, assisting the model in learning
and utilizing the inherent hemispheric symmetry of the human brain.

After the midway registration, random sub-volumes, denoted as
$\mathcal{X}=\\{\mathbf{x}_{n}\\}$, are extracted as inputs to networks from
the collection of all aligned images
$\mathcal{\bar{I}}=\\{\bar{\mathbf{I}}_{n}\\}$. Since the input images are
standardized by the midway registration, for each patch
$\mathbf{x}\in\mathbb{R}^{H\times W\times D}$, the mirrored counterpart
$\mathbf{x}^{\prime}$ with respect to the sagittal plane $\Phi_{s}$ can be
identified and can be flipped to obtain $\tilde{\mathbf{x}}$. Note that the
generated $\tilde{\mathbf{x}}$ is expected to closely resemble the original
patch $\mathbf{x}$, barring minor structural discrepancies inherent to the
intrinsic hemispheric symmetry of the human brain. Both original patch
$\mathbf{x}$’s and their flipped symmetrical counterparts
$\tilde{\mathbf{x}}$’s are used as the input to the encoder network as sub-
volumes.

As illustrated in Figure 2, two distinct rotations $R=\\{r_{1},r_{2}\\}$ are
conducted on both sub-volumes and two random inner cutouts
$C=\\{c_{1},c_{2}\\}$ are conducted on original sub-volume $\mathbf{x}$,
resulting in augmented sub-volumes
$\\{\mathbf{x}_{1},\mathbf{x}_{2},\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2}\\}$.
The augmented sub-volumes are segmented into patches of the size $(h,w,d)$,
ending up with a sequence of the length
$\frac{H}{h}\times\frac{W}{w}\times\frac{D}{d}$. These patches are processed
through an embedding network and subsequently passed into a 3D Sliding-window
(Swin) Transformer architecture [20], which consists of four stages and eight
transformer layers. This results in a reduced sequence of the length
$\frac{H}{16h}\times\frac{W}{16w}\times\frac{D}{16d}$ with the embedding
feature dimension $768$.

Figure 2: (a) The process of midway registration to align input images; (b)
The framework applied to input sub-volumes and their sagittal symmetrical
counterparts. This framework is augmented by the Symmetry-Aware Cross-
Attention (SACA) module and is optimized through a composite loss function
that integrates inpainting, rotation, contrastive, and symmetry losses.

###  2.2 Symmetry-Aware Cross-Attention (SACA) Module

Following the Swin Transformer Encoder [11], we introduce a Symmetry-Aware
Cross-Attention (SACA) module to foster deep symmetrical feature alignment and
referencing. The SACA module processes input embedding sequences $\mathbf{f}$,
applying self-attention mechanisms to $\mathbf{f}$ and cross-attention to the
flipped symmetrical counterpart $\tilde{\mathbf{f}}$. This dual attention
strategy, as depicted in Figure 3, is intended to exchange information between
the raw sub-volume and the flipped symmetrical counterpart.

The cross-attention mechanism within this module is represented as follows:

$\operatorname{SACA}(\mathbf{f},\tilde{\mathbf{f}})=\operatorname{Softmax}\left(\frac{Q\tilde{K}^{T}}{\sqrt{D}}\right)\tilde{V},\text{where
}\Bigg{\\{}\begin{aligned} Q&=\mathbf{f}W_{Q}\\\
\tilde{K}&=\tilde{\mathbf{f}}W_{K}\\\
\tilde{V}&=\tilde{\mathbf{f}}W_{V}\end{aligned}\ \ \ .$ (1)

$Q$ represents the queries generated from the features of the original patch
$\mathbf{f}$, and $\tilde{K}$ and $\tilde{V}$ are keys and values generated
from the counterpart patch $\tilde{\mathbf{f}}$, with $D$ denoting the
dimensionality of the query and key vectors. The matrices $W$ are learnable
weights for the query, key, and value transformations within the SACA module.
With our novel SACA module, the contrastive information from the original
patch and its flipped symmetrical counterpart could be utilized, and the level
of symmetry of the patch could be learned along the process, enabling the
network to recognize symmetrical features of input brain images.

Figure 3: Symmetry-Aware Cross-Attention (SACA) Module: (a) The computation of
attention for each token within the SACA module; (b) Cross-attention
calculation of SACA module structure.

###  2.3 Symmetry-Aware Pretraining Proxies

Upon obtaining the encoded features from the SACA-based encoder, they are
directed to four distinct proxy heads, each responsible for a specific
pretraining task. The proxies include one symmetry-aware head that identifies
the symmetrical features of the input sub-volumes and three heads that deal
with intrinsic features of the input sub-volume (i.e., inpainting, rotation,
and contrastive coding) as shown in Figure 2.

The Symmetry-Aware Head (SAH) consists of a max pooling layer followed by a
multi-layer perceptron (MLP), which distills the features for an original
input patch and the symmetrical counterpart as $\mathbf{f}^{s}$ and
$\tilde{\mathbf{f}}^{s}$. These features serve as the foundation of computing
the conditional symmetry loss, expressed as:

$\mathcal{L}_{Symmetry}=-\log\frac{\sum\exp(\operatorname{sim}(\mathbf{f}^{s},\tilde{\mathbf{f}}^{s})/\tau)\cdot
y^{0}}{\sum\exp(\operatorname{sim}(\mathbf{f}^{s},\tilde{\mathbf{f}}^{s})/\tau)},$
(2)

where $y^{0}$ indicates the subject’s health status, with 1 for healthy
control and 0 for patients. The parameter $\tau$ is a temperature factor
adjusting the logits’ scaling. The cosine similarity
$\operatorname{sim}(\mathbf{a},\mathbf{b})$ is used to measure the similarity
between embedded features. The core premise of this formulation is that for
subjects with healthy brains, a given patch and its symmetrical counterpart
should exhibit similar visual and structural characteristics, whereas
deviations and asymmetries are expected in the presence of pathological
conditions, reflecting the brain’s asymmetrical response to disease.

The heads for intrinsic feature awareness follow [27], including the
Inpainting Head (IH), defined as a decoder to reconstruct the occluded voxels;
the Rotation Head (RH), defined by an MLP classifier to identify the rotation
angles; and the Contrastive Coding Head (CH), defined by two MLP layers to
distinguish the present sub-volumes from other sub-volumes. The losses are
defined as:

$\displaystyle\mathcal{L}_{Inpainting}=\left\|\mathbf{f}^{p}-\mathbf{x}\right\|_{1}$
(3)
$\displaystyle\mathcal{L}_{Rotation}=-r\log(\operatorname{Softmax}(\mathbf{f}^{r})),$
$\displaystyle\mathcal{L}_{Contrastive}=-\log\frac{\exp(\operatorname{sim}(\mathbf{f}^{c}_{i},\mathbf{f}^{c}_{j})/\tau)}{\sum\mathbbm{1}_{k\neq
i}\exp(\operatorname{sim}(\mathbf{f}_{i}^{c},\mathbf{f}_{k}^{c})/\tau)},$

where $\mathbf{f}^{p},\mathbf{f}^{r},\mathbf{f}^{c}$ represent the
concatenated output from each head, and $\mathbf{x}$ is the input sub-volume.
Through comprehensive grid search experimentation, the four losses are
configured to be evenly summed up for pretraining, balancing the contribution
of each task effectively. The pretrained symmetry-aware network can be further
applied for downstream tasks, as shown in Section 3.

##  3 Experiments and Results

We evaluated our proposed approach by pretraining on large neuroimaging
datasets and downstream classification and segmentation tasks.

###  3.1 Datasets and Downstream Tasks

We curated two large datasets for brain image pretraining using head CT and
brain MRI images separately. The MRI dataset comprises 3D T1 images of 3,509
healthy adults and 2,917 patients sourced from a variety of publicly available
datasets [4, 29, 24, 23, 19, 16, 3, 16] (more details in supplementary). The
CT dataset is derived from our proprietary in-house collection, including 2025
images from healthy individuals and 4693 images from patients afflicted with a
range of brain diseases, with images from three brain diseases (i.e.,
hemorrhage, fracture, midline shift) reserved for downstream tasks.

For MRI-based evaluations, we assess the zero-shot and few-shot classification
capabilities of our model across several disease identification tasks,
including Focal Epilepsy [25], ADHD [5], and Schizophrenia [26]. Furthermore,
we explore the model’s transfer learning proficiency in segmentation tasks
using images with different modalities from the MRI pretraining. This includes
the BraTS2021 dataset [3] and the MSSEG2016 dataset [8].

For CT imaging, we conducted few-shot training using reserved in-house data of
three brain diseases, including intracranial hemorrhage, bone fracture, and
midline shift, and evaluated the performance on the public dataset CQ500 [7]
to demonstrate the model’s versatility across varied clinical conditions.

###  3.2 Implementation Details

The proposed model undergoes pretraining with distinct settings for image
classification and segmentation tasks. Specifically, the whole image
classification task pretraining is conducted with a batch size of 2 and images
resized to $160\times 160\times 160$, and a batch size of 4 and sub-volume
dimensions of $96\times 96\times 96$ was used for segmentation-related
pretraining. The pretraining phase is executed with a learning rate of
$6\times 10^{-6}$ over 300 epochs. For downstream tasks, the learning rate is
set to $1\times 10^{-5}$, spanning 100 epochs. All models are trained
utilizing NVIDIA Tesla V100 GPUs.

The pretrained models are further experimented with downstream tasks under a
5-fold cross-validation approach across all experiments, with the exception of
the multiple sclerosis lesion segmentation on MSSEG, which has a predefined
validation set [21]. The average performance scores, including Area under the
ROC Curve (AUC), F1-measure, and Dice Similarity Coefficient (DSC), are
reported.

###  3.3 Results

The classification outcomes are delineated in Tables 1 and 2. In the
supervised fine-tuning scenario (checked under “+SFT”), our model consistently
surpassed the performance of the Swin MLP [27] across all evaluated downstream
tasks, especially in some diseases that display significant asymmetrical
features, such as ADHD and Midline Shift. Besides, our method has achieved
comparable performance in CT disease diagnosis as that of [6], but only
hundreds of labeled cases were used for fine-tuning, much less than in [6]
where 313,318 scans were collected and manually labeled. This underscores the
significant advantage of incorporating symmetrical awareness during the
pretraining phase.

Table 1: Benchmark methods on the MRI T1-only classification tasks. Experiments |  |  |  | Focal Epilepsy | Schizophrenia | ADHD
---|---|---|---|---|---|---
+Pretrain | +SACA | +SFT | AUC | F1 | AUC | F1 | AUC | F1
Swin MLP [27] | ✗ | ✗ | ✓ | 0.678 | 0.683 | 0.656 | 0.810 | 0.685 | 0.625
✓ | ✗ | ✗ | 0.522 | 0.651 | 0.594 | 0.749 | 0.514 | 0.609
✓ | ✗ | ✓ | 0.713 | 0.695 | 0.627 | 0.803 | 0.642 | 0.645
Ours | ✓ | ✓ | ✗ | 0.631 | 0.683 | 0.663 | 0.760 | 0.515 | 0.625
✓ | ✓ | ✓ | 0.778 | 0.788 | 0.719 | 0.836 | 0.792 | 0.727

Furthermore, we assessed the impact of symmetrical awareness in a zero-shot
learning context. Remarkably, our pretrained model exhibited superior zero-
shot capabilities in almost all tasks. This indicates that our pretraining
strategy effectively captures critical pathological features based on symmetry
characteristics, facilitating generalization to novel tasks without requiring
task-specific data.

Table 2: Benchmark methods on the CT classification tasks evaluated on the CQ500 dataset [6].  Experiments |  |  |  | Hemorrhage | Fracture | Midline Shift
---|---|---|---|---|---|---
+Pretrain | +SACA | +SFT | AUC | F1 | AUC | F1 | AUC | F1
Qure.ai [6] | ✗ | ✗ | ✓ | 0.942 | 0.761 | 0.962 | 0.508 | 0.970 | 0.704
Swin MLP [27] | ✗ | ✗ | ✓ | 0.872 | 0.776 | 0.918 | 0.460 | 0.926 | 0.640
✓ | ✗ | ✗ | 0.751 | 0.685 | 0.781 | 0.353 | 0.788 | 0.413
✓ | ✗ | ✓ | 0.911 | 0.748 | 0.952 | 0.469 | 0.942 | 0.620
Ours | ✓ | ✓ | ✗ | 0.793 | 0.722 | 0.803 | 0.267 | 0.866 | 0.508
✓ | ✓ | ✓ | 0.958 | 0.781 | 0.969 | 0.513 | 0.986 | 0.728

The assessment of models’ proficiency in adapting to segmentation tasks is
summarized in Table 3, which manifests the superior performance of our method.
Additionally, we conducted an ablation study to discern the relative
contribution of our proposed SACA module and SAH proxy. Both the SAH only
pretraining process and the SACA module could improve the performance of
segmentation steadily, and the combination of all pretraining proxies and the
novel SACA module could further yield the most substantial benefits,
underscoring the synergetic effect of combining these diverse learning
strategies. This comprehensive evaluation demonstrates our method’s superior
capability in leveraging pretraining proxies to enhance model performance on
segmentation tasks, highlighting its potential for advancing medical image
analysis.

Table 3: Benchmark methods and ablation study of our method on the MRI segmentation tasks. “IRC” represents commonly used pretrain heads for inpainting (I), rotation (R), and contrastive (C). Experiments |  |  | BraTS [3] | MSSEG [8] |
---|---|---|---|---|---
+IRC | +SAH | +SACA | $\operatorname{Dice}_{ET}$ | $\operatorname{Dice}_{TC}$ | $\operatorname{Dice}_{WT}$ | $\operatorname{Dice}_{Avg}$ | Dice
nnU-Net [15] | ✗ | ✗ | ✗ | 0.883 | 0.927 | 0.913 | 0.908 | -
TransBTS [28] | ✗ | ✗ | ✗ | 0.868 | 0.911 | 0.898 | 0.891 | -
SwinUNETR [11] | ✗ | ✗ | ✗ | 0.891 | 0.933 | 0.917 | 0.913 | 0.594
Ours | ✗ | ✓ | ✗ | 0.897 | 0.935 | 0.920 | 0.917 | 0.615
✗ | ✗ | ✓ | 0.901 | 0.937 | 0.928 | 0.922 | 0.640
✗ | ✓ | ✓ | 0.909 | 0.940 | 0.935 | 0.928 | 0.654
| ✓ | ✓ | ✓ | 0.912 | 0.946 | 0.940 | 0.932 | 0.680

##  4 Conclusion

Our study introduces an innovative framework utilizing 3D brain imaging data
of various modalities, significantly enhancing neuroimage analysis by encoding
symmetry-aware features in deep neural networks. The integration of the
Symmetry-Aware Cross-Attention (SACA) module and the symmetry-aware
pretraining proxy has shown remarkable improvements in classification and
segmentation tasks, setting new benchmarks over existing models. This work
highlights the critical role of anatomical symmetry in neuroimaging analysis
and suggests a promising direction for developing more accurate, robust, and
generalized AI models in healthcare leveraging intrinsic anatomical features.

####  4.0.1 Acknowledgements

The authors would like to express their gratitude to the BCR Scholarship for
Computational Neuroscience and the Faculty of Engineering Research Stipend
Scholarship at the University of Sydney for their financial support, which
enabled this research and its contributions to medical imaging and healthcare.

####  4.0.2

The authors have no competing interests to declare that are relevant to the
content of this article.

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