A Comprehensive Overview of Spectral Geometry: Insights from the Laplace-Beltrami Operator on Riemannian Manifolds

Abstract

This study provides an overview of recent research concerning the geometric significance of the Laplace-Beltrami operator, inherently associated with a Riemannian manifold. Essentially, it offers an expanded examination of R. Brooks' expository paper [1], as well as closely related articles. Beginning with a basic exploration of the isospectrality of flat tori, the study progresses to elucidate Sunada's pioneering utilization of various theoretical concepts. It concludes with a concise portrayal of the fundamental aspects of isospectral deformations on Riemannian manifolds.

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Palestine Journal of Mathematics
Vol 13(4)(2024) , 1105–1137 © Palestine Polytechnic University-PPU 2024
A Comprehensive Overview of Spectral Geometry: Insights from
the Laplace-Beltrami Operator on Riemannian Manifolds
S. Ba¸saran and D. Amilo
Communicated by Ayman Badawi
MSC 2010 Classifications: Primary 58J53; 58J50; 53C20; Secondary 58J40; 11F72.
Keywords: Laplace-Beltrami operator, spectrum, isospectrality, isospectral deformation, nilmanifold.
The first author would like to acknowledge her late supervisor Prof. Dr. Cem Tezer.
Corresponding Author: D. Amilo
Abstract This study provides an overview of recent research concerning the geometric sig-
nificance of the Laplace-Beltrami operator, inherently associated with a Riemannian manifold.
Essentially, it offers an expanded examination of R. Brooks’ expository paper [1], as well as
closely related articles. Beginning with a basic exploration of the isospectrality of flat tori, the
study progresses to elucidate Sunada’s pioneering utilization of various theoretical concepts. It
concludes with a concise portrayal of the fundamental aspects of isospectral deformations on
Riemannian manifolds.
1 Introduction
The Laplace-Beltrami operator, a fundamental concept in differential geometry, plays a central
role in understanding the geometry and topology of Riemannian manifolds [2,3,4,5,6]. This
operator, canonically associated with a Riemannian manifold, encodes crucial geometric infor-
mation about the manifold, including its spectral properties [7,8,9,10,11,12]. The study of
these spectral properties and their geometric implications has been a topic of significant research
interest in recent years. This research aims to provide an overview of the key developments
in the field of geometric analysis related to the Laplace-Beltrami operator. It builds upon the
insights presented in an expository paper in [1], while also delving into closely related articles.
The primary focus of this study is to explore the geometric significance of the Laplace-Beltrami
operator, especially in the context of isospectrality and isospectral deformations. The journey of
this research begins with an elementary treatment of the isospectrality phenomenon observed in
flat tori. Understanding the isospectrality of these simple geometric objects serves as a founda-
tion for more intricate investigations. One of the highlights of this exploration is a presentation of
Sunada’s groundbreaking adaptation of a number-theoretical idea. Sunada’s work has profound
implications for our understanding of isospectral manifolds and the connections between geome-
try, topology, and number theory. The research culminates with an examination of the essentials
of isospectral deformations on Riemannian manifolds. Isospectral deformations are transforma-
tions that preserve the spectrum of the Laplace-Beltrami operator while altering the underlying
geometry. These deformations reveal the intricate interplay between the spectral properties of
the Laplace-Beltrami operator and the geometric structure of the manifold. In summary, this
study provides a comprehensive overview of recent research on the geometric implications of
the Laplace-Beltrami operator. It explores the isospectrality of flat tori, Sunada’s contributions,
and the broader context of isospectral deformations. By shedding light on these fundamental
aspects of Riemannian geometry, this research contributes to our understanding of the rich inter-
play between analysis and geometry on manifolds.

1106 S. Ba¸saran and D. Amilo
2 Preliminaries
Definition 2.1. A Riemannian manifold is an ordered pair (M , G)where Mis a smooth manifold,
Gis a smooth tensor field on Mof bidegree (0,2)such that for each m∈M,Gmis positive
definite.
Definition 2.2. The Levi-Civita connection ∇on (M, G)is the unique torsion-free connection
on Mwith respect to which Gis parallel that is ∇G=0.Given a smooth Riemannian manifold
(M, G), let D(M)be the set of all complex valued, smooth functions on M, and X(M)be the
set of smooth complexified vector fields on M. For each φ∈ D(M)The gradient of φ, denoted
by grad φis the vector field on Muniquely determined by the relation.
G(grad φ, Y ) = Y φ,
for all Y∈ X(M). For each Y∈ X(M)on M, the divergence div Yof Yis an element of
D(M), the value of which at each p∈Mis the trace of the linear map:
(u7−→ ∇uY):TpM7−→ TpM,
where ∇is the Levi - Civita connection on (M, G).
Definition 2.3. The Laplace - Beltrami operator
∆:D(M)7−→ D(M),
on (M, G)is defined by,
∆φ=−div ( grad φ).
If xis any chart on Mwith
G|dom(x)=Gij dxi⊗dxj,
then,
gradφ|dom(x)=Gij ∂φ
∂xj
∂
∂xj,
where,
Gij Gkj =δi
j,
and
div(Y)|dom(x)=∂Y i
∂xi+Γi
ikYk,
for any φ∈C(M)and any Y∈ X(M)where
Y|dom(x)=Yi∂
∂xi,
and Γi
jk are the Christoffel symbols of the second kind associated with the Levi-Civita connection
∇by
∇(∂
∂xj,∂
∂xk) = Γi
jk
∂
∂xi.
Thus,
∆φ|dom(x)=−div(grad φ)|dom(x)
=−∂
∂xi(Gim ∂φ
∂xm)−Γi
ikGk n ∂φ
∂xn,
for any φ∈C(M).λ∈Cis said to be an eigenvalue of ∆if there exists φ∈ D(M)such that
∆φ=λφ. A non-vanishing function φ∈ D(M)is called an eigenfunction of the Laplace -
Beltrami operator ∆if ∆φ=λφ for some λ∈C. Under these circumstances, we refer to φas an
eigenfunction of ∆with eigenvalue λ.

A Comprehensive Overview of Spectral Geometry 1107
Definition 2.4. Given a group Γand finite subgroups A, B ≤Γ,Aand Bare said to be equivalent
in the sense Gassmann.[22] if
#(c∩A) = #(c∩B),
for every conjugacy class cin ∇Γ.
Remark 2.5. (a)Subgroups equivalent in the sense of Gassmann have the same cardinality.
(b)If Aand Bare conjugate subgroups, then Aand Bare clearly equivalent in the sense of
Gassmann.
Definition 2.6. Given a Riemannian manifold (M, G)with the heat kernel H:M×M×R.>07−→
R,the theta function associated with (M, G)is a function
Θ=ΘM:R.>07−→ R,
defined by:
Θ(t) = ZM
H(m, m, t)dµ(m).
Let us illustrate these concepts on the basis of the following two simplest possible examples :
On the flat circle (R
LZ, dx⊗dx)of content L > 0, the Laplace - Beltrami operator reduces to
∆=−∂2
∂x2
It can routinely be checked that the eigenfunctions of ∆are of the form
φn(x) = e2πni
Lx
with eigenvalues
λn=4π2n2
L2.
for n∈Z, as depicted in Figures 1and 2.
Figure 1. Eigenfunctions of Laplace-Beltrami Operator in Example 2
On the real line (R,dx⊗dx), the Laplace-Beltrami operator is again of the form
∆=−∂2
∂x2.

1108 S. Ba¸saran and D. Amilo
0
1000
2000
3000
4000
n-10
-5
0
5
10
n
-1
-0.5
0
0.5
1
0
500
1000
1500
2000
2500
3000
3500
Figure 2. Eigenvalues of Laplace-Beltrami Operator in Example 2
In this case every real number λ∈Ris an eigenvalue of ∆. Eigenfunctions with eigenvalue λ
are of the form
φ(x) =























Ae√−λx +Be−√−λx for A, B ∈Rif λ < 0
Cei√λx +De−i√λx for C, D ∈Rif λ > 0
Ex +Ffor E , F ∈Rλ=0
As it can be readily observed on the basis of the above examples, shown in Figues 3and 4the set
of eigenvalues of the Laplace-Beltrami operator on a compact manifold has a markedly different
nature from that on a non-compact manifold. We shall mostly concentrate on compact manifolds
on which the set of eigenvalues of the Laplace-Beltrami operator has a well-understood and tidy
structure: If (M, G)is a compact Riemannian manifold with the Laplace-Beltrami operator ∆,
then the eigenvalues of ∆constitute a countable, discrete and unbounded set of non-negative real
numbers. Furthermore the set of eigenfunctions corresponding to an eigenvalue λspan a finite
dimensional subspace Hλof D(M). The dimension of Hλis the multiplicity of λ. 0∈Ralways
occurs as an eigenvalue. However, H0consists of constant functions and thus dim H0=1. In
other words, the multiplicity of the eigenvalue 0 is always 1. Thus, in the case of the flat circle
of content Leach eigenvalue is of the form,
λn=4π2n2
L2,
and
Hλn=⟨e2πin
Lx, e −2πin
Lx⟩,
where n∈Z≥0.
We may now offer an elucidation of the expression “spectral geometry” [17]: Spectral geometry
done within the framework of Riemannian geometry is essentially the study of Riemannian man-
ifolds on the basis of the information consisting of magnitudes and multiplicities of the eigenval-
ues of the Laplace-Beltrami operator. It is very important to notice that the “information” in this

A Comprehensive Overview of Spectral Geometry 1109
Figure 3. φ(x)for λ < 0 in Example 2
Figure 4. φ(x)for λ > 0 in Example 2
question encompasses not only the magnitudes of the eigenvalues but the multiplicities thereof.
Thus, given a compact Riemannian manifold (M, G)we understand the spectrum of (M, G)to be
the set of eigenvalues of the Laplace-Beltrami operator on (M, G)with each eigenvalue tagged
by a number indicating its multiplicity. We shall denote the spectrum of (M, G)by Sp(M, G)or
Sp(M)unless confusion is likely. Observe that, in the case of a compact Riemannian manifold
(M, G), the spectrum Sp(M, G)may be identified with an increasing sequence
0=λ0< λ1≤. . . ≤λn≤. . . +∞.
Each eigenvalue occurs in the form of λnand is repeated as many times as its multiplicity. Rie-
mannian manifolds (M, G)and (M′,G′)are said to be isometric if there exists a diffeomorphism

1110 S. Ba¸saran and D. Amilo
φ:M7−→ M′satisfying:
Gp(u, v) = G′φ(p)(T φp(u), T φp(v)),
for p∈M,u,v∈TpM.
(M, G)and (M′,G′)are said to be isospectral if
Sp(M , G) = Sp(M′, G′).
It is clear that isometric Riemannian manifolds are isospectral. In spectral geometry of Rieman-
nian manifolds, the principal problem is to determine the extent to which the isometry class of a
Riemannian manifold is determined by its spectrum.
The broaching of the subject of “inverse spectral geometry” is popularly ascribed to M. Kac
[26] who raised now the famous question “Can one hear the shape of a drum?”. However, the
subject seems to have come up earlier in Riemannian geometry in [16]. A negative answer
to the question of whether “The isometry class of a Riemannian manifold is ‘audible’ ” was
provided by Milnor in a curt announcement [27]. Milnor’s pathbreaking work was followed by a
period of “sporadic counterexamples” during which diverse pairs of isospectral but non-isometric
Riemannian manifolds were produced. Good examples of matured products may be found in[36,
25,21]. Brilliantly constructed and certainly mathematically enriching as these counterexamples
were, they constituted only a collection of individual instances. A general method for producing
isospectrality was invented by Sunada who was inspired by the work of number theorists[35].
Sunada’s method proved to be fruitful not only in understanding isospectrality but was also
of fundamental importance in producing manifolds which “sounded the same while changing
shape” that is, manifolds which were isometrically deformable.[23].
3 The Case of Flat Tori
The extent to which the spectrum can determine the geometry on a Riemannian manifold can be
illustrated in the case of flat tori in a direct and elementary fashion.
For N⩾1, N ∈Z, consider RNwith its obvious additive group and vector space structure over
R. A lattice in RNis a discrete subgroup of RNwhich contains a basis of RN. Equivalently, a
lattice Λin RNis a subset of RNof the form
Λ=BZN,
where B∈RN×Nis a non - singular matrix. We note that lattices come in pairs: Given a lattice,
Λ=BZN⊆RN, the dual of Λis a lattice Λ∗⊆RNwhich is defined to be
Λ∗= (B−1)TZN.
The dual Λ∗of Λis also characterised by an important property of its elements : ℓ∗∈RNlies
in Λ∗if and only if ℓ∗Tℓis an integer for all ℓ∈Λ. A flat torus is known to be isometric to
TΛ={RN
Λ, δij dxi⊗dxj},
for some lattice Λin RN. We shall refer to TΛas the flat torus determined by the lattice Λ.
Notice that the content of TΛis exactly |detB|where Λ=BZN. It can be routinely checked
that the Laplace-Beltrami operator on TΛis of the form
∆=−δij
∂2
∂xi∂xj,
and the eigenfunctions of ∆are exactly the functions
φℓ∗=e2πiℓ∗Tx,
where ℓ∗∈Λ∗. Clearly the eigenvalue corresponding to φℓ∗is 4π2|ℓ∗|2=ℓ∗Tℓ∗. Consequently
the spectrum Sp(TΛ)is a sequence
0=λ0< λ1≤. . . ≤λn. . . ,

A Comprehensive Overview of Spectral Geometry 1111
in which, a quantity λoccurs as many times as the number of ℓ∗’s with λ=|ℓ∗|2.
The information contained in Sp(TΛ)can be converted into geometric information as follows :
(A)First we deposit Sp(TΛ)into an analytic function Fby writing
F(t) = ∞
X
n=0
e−λnt,
where the right hand side can be checked to be uniformly convergent on each compact subset of
R.>0. (The choice of Fwill be justified in the next section.) Notice that by thus squeezing the
spectrum of T∧into F“no information is lost”: Indeed, given F(t), we may read off Sp(Λ)
inductively : Clearly, λ0=0 and having inductively determined λ0, . . . , λnwe have,
λn+1=sup {m>o|lim
t→∞[F(t)−
n
X
k=0
e−λkt]emt =0}.
(B)Secondly, we remember that for each symmetric, positive definite S∈RN×Nthe Jacobi
theta function ΘSis the analytic function defined by:
ΘS(z) = X
n∈ZN
eπi(nTSn)z,
where the series on the right-hand side is uniformly convergent on each compact subset of {z∈
C|Im(z)>0}. Putting Λ=BZN, we easily obtain:
F(t) = X
ℓ∗∈Λ∗
e−4π2ℓ∗Tℓ∗t
=X
ℓ∗∈B−1TZN
e−4π2ℓ∗Tℓ∗t
=X
n∈ZN
e−4π2nTB−1(B−1)Tn t
=ΘB−1(B−1)T(4πit).
(C)Thirdly, we employ the Jacobi Inversion Formula
ΘS(z)=(z
i)−N
21
√detS
ΘS−1(−1
z),
with S= (BTB)−1=B−1(BT)−1and z=4πit to obtain:
F(t) = ΘB−1B−1T(4πit)
=
4πit
i
−N
21
p(det B−1)2ΘBTB(−1
4πit )
=|detB|
(4πt)N
2X
n∈ZN
e−πi(nTBTB n)1
4πit
=|detB|
(4πt)N
2X
ℓ∈Λ
e−ℓTℓ
4t.
(D)Finally, we construct again for purposes of simple and clear exposition the sequence,
0=µ0< µ1≤µ2≤. . . ≤µn≤. . .
in which a quantity µoccurs as many times as the number of ℓ∈Λwith |ℓ|2=µ. We have found
in (C)that
F(t) = |detB |
(4πt)N
2
∞
X
n=0
e−µn
4t,

1112 S. Ba¸saran and D. Amilo
from which we can now read off |detB |and the sequence (µk)∞
k=0as follows:
Clearly,
|detB |=lim
t→0+(4πt)N
2F(t).
On the other hand µ0=0 and having inductively determined µ0, . . . , µnalong with |det B|we
can compute µn+1by (2)
µn+1=sup {m > 0|lim
t→0+[F(t)
|detB |(4πt)N
2−
n
X
k=0
e−µk
4t]em
4t=0}.
This shows us that the function F(t), which is presently on our hands still without any motiva-
tion, has the peculiarity of translating spectral information consisting of the sequence λ0, λ1, . . .,
into geometric information consisting of |det B|and µ0, µ1, . . . , µn, . . .
Theorem 3.1. Two 2-dimensional flat tori are isometric iff they are isospectral.
Proof. Clearly flat tori TΛ1,TΛ1of the same dimension, say N, are isometric iff there exists an
isometry of RNwhich sends Λ1to Λ2, that is iff Λ1and Λ2are isometric subsets of RNwith its
ordinary Euclidean structure. Suppose N=2 and Sp(TΛ1) = Sp(TΛ2). This means that if e1, f1
and e2, f2are the vectors of smallest length generating Λ1and Λ2respectively then,
|e1|=|e2|,
|f1|=|f2|.
Moreover the areas of the parallelograms spanned by e1, f1and e2, f2are equal. Therefore
these parallelograms are congruent in R2with its ordinary Euclidean structure. Therefore Λ1
is isometric to Λ2, hence TΛ1is isometric to TΛ2. Milnor’s historical example [27] consisted in
pointing out that on R16 there were lattices Λ1,Λ2, which were well known among number the-
orists to be non-isometric but representing integers the same number of times, which, translated
into Riemannian geometry means that TΛ1is isospectral to TΛ2.[37]. Although there is consider-
able amount of confusion, it seems to be well-established by now that the above theorem is not
valid for N > 2.It is however known that each flat toral isospectral class contains finitely many
isometry classes [31,18].
4 The Sunada Concept
Given a Riemannian manifold (M, G)with the Laplace-Beltrami operator ∆, the heat equation
associated with (M, G)is the partial differential equation
∆U+∂U
∂t =0
where,
U:M×R.>07−→ R.
The importance of the heat equation lies in its close relationship with the structure of the spec-
trum of the Laplace-Beltrami operator. For any continuous function f:M7−→ R, the heat
equation has a unique solution Usubject to the initial condition,
lim
t→0+U(x, t) = f(x),
for all x∈M.
This existence and uniqueness result can be brought into a form which is independent of the
choice of “initial conditions” by means of the concept of the heat kernel. Given a Riemannian
manifold (M, G), the heat kernel is a function,
H:M×M×R.>07−→ R
such that,
∆|xH+∂
∂t H=0

A Comprehensive Overview of Spectral Geometry 1113
and
lim
t→0+ZM
H(x, y, t)f(y)dµ(y) = f(x),
for any continuous f:M7−→ Rwhere µstands for the Lebesque measure on Minduced by the
Riemannian metric G. Given a continuous f:M7−→ R, it is clear that
U(x) = ZM
H(x, y, t)f(y)dµ(y)
is the unique solution of the heat equation subject to limt→0+U(x, t) = f(x).
Quite generally, given a compact Riemannian manifold (M, G)where the Laplace-Beltrami op-
erator has the spectrum,
0=λ0< λ1≤λ2≤. . . λn≤. . .
which correspond to the respective normalized eigenfunctions,
φ0, φ1, φ2, . . . , φn, . . .
When it exists the heat kernel is unique and can be verified by direct computation that when
(M, G)has a heat kernel HM
HM(x, y, t) = ∞
X
n=0
φn(x)φn(y)e−λnt.
Consider the Euclidean space:
RN, δij dxi⊗dxj.
It can be verified by direct computation that,
HRN(x, y, t) = 1
4πt
N
2
e−|x−y|2
4t.
is the heat kernel for
RN, δij dxi⊗dxj.
Consider the case of the flat torus
TΛ= (Rn/Λ, δij dxi⊗dxj). In view of the remark preceding the above example.
H(x, y, t) = X
ℓ∗∈Λ∗
e2πℓ∗Txe−2πℓ∗Tye−4π|ℓ|∗2t.
Of basic importance for the theory which is to be introduced in this section there is another
technique for constructing heat kernels in quotient manifolds. Consider a smooth manifold M
acted upon by a group Γproperly discontinuously. Let M=M /Γ. Clearly, Mis a smooth
manifold and the quotient map p:M7−→ Mis a covering projection. We put Riemannian
metrics G,Gon M,Mrespectively, so that p:M7−→ Mbecomes a local isometry. It is possible
to do this by choosing Gto be invariant under the action of Γand by defining Γto be the
quotient Riemannian tensor. Equivalently we may take a Riemannian metric Gon Mand lift it
to Mvia pto attain Gwhich is then automatically invariant under Γ. We shall also make the
provision that Gbe sufficiently generic to exclude non-trivial isometries between open subsets of
M. Such a Riemannian metric is folklorically referred to as “bumpy”: The fact that on smooth
paracompact manifolds bumpy metrics exist (in fact abundantly) has been formulated and proven
by Sunada.[35].
We state the theorem and omit the proof which is rather technical.
Theorem 4.1. Given a smooth paracompact manifold M, there exists a Riemannian metric Gon
Msuch that for any disjoint open subsets U,V⊆M, no map φ:U7−→ Vcan be an isometry
with respect to G.

1114 S. Ba¸saran and D. Amilo
Figure 5. Heat Kernel HR2(x, y, t)for N=2 and t=1 in Example 3
Figure 6. Heat Kernel H(x, y, t)in Example 3
Proposition 4.2. His invariant under the action of Γin the sense that
H(gm1, gm2, t) = H(m1, m2, t)
for any m1,m2∈Mand g∈Γ.
Proof. This is a direct consequence of the uniqueness clause for the heat kernel. Indeed, given
g∈Γif we define
H
′:M×M×R.>07−→ R,
to be,
H
′
(m1, m2, t) = H(gm1, gm2, t),

A Comprehensive Overview of Spectral Geometry 1115
for m1, m2∈M,H
′satisfies all the conditions for the heat kernel:
∆|m1H
′
+∂H
′
∂t =0.
It is obvious by the Γ−invariance of Gand hence that of ∆. Trivially
H
′
(m1, m2, t) = H
′
(m2, m1, t).
Finally, given any continuous f:M7−→ Rwe have:
lim
t→0+ZM
H
′
(m1, m2, t)f(m2)dµ(m2) = lim
t→0+ZM
H(gm1, gm2, t)f(m2)dµ(m2)
=lim
t→0+ZM
H(gm1, m2, t)f(g−1m2)dµ(g−1m2)
=lim
t→0+ZM
H(gm1, m2, t)f(g−1m2)dµ(m2)
=f(g−1gm1) = f(m1).
By the uniqueness of heat kernels, we conclude that, H
′
=H.
Proposition 4.3. The heat kernel Hon Msatisfies
H(m1, m2, t) = X
g∈Γ
H(gm1, m2, t),
where m1∈p−1(m1),m2∈p−1(m2)provided that the sum at the right hand side is meaningful.
Proof. It should be noted that the right hand side quantity is well-defined in the sense that it is
independent of the choice of m1∈p−1(m1). Indeed for any m1
′∈p−1(m1), m2
′∈p−1(m2)
there exist g1,g2∈Γsuch that m2
′=g2m2,m1
′=g1m1and
H(m1, m2, t) = X
g∈Γ
H(gm1
′, m2
′, t)
=X
g∈Γ
H(gg1m1, g2m2, t)
=X
g∈Γ
H(g2−1gg1m1, m2, t)
=X
g∈Γ
H(gm1, m2, t).
Consider TΛ= (RN/Λ, δij dxi⊗dxj)where Λ=BZN. Since the heat kernel of RNis
HR2(x, y, t) = 1
4πt e−|x−y|2
4t,
we obtain:
HTΛ(x, y, t) = 1
4πt X
l∈Λ
e−|x−y+l|2
4t.
, as shown in Figure 7
For the flat torus TΛthe theta function is easily computed to be:
Θ(t) = X
ℓ∗∈Λ∗
e−4π|ℓ∗|2t,
which, at long last, justifies our choice of the function Fin section 5. The fundamental observa-
tion of Sunada [35] is that, when Γis a finite group,the relationship between Hand Hallows us
to obtain the theta function Θ=ΘMof Min terms of Hand Γ. For any set K, let #(K)denote
the cardinality of K:

1116 S. Ba¸saran and D. Amilo
Figure 7. Heat kernel in Example 3
Lemma 4.4.
Θ(t) = ZM
H(m, m, t)dµ(m) = X
g∈Γ
1
#(Γ)Ig(t)
where,
Ig(t) = ZM
H(gm, m, t)dµ(m).
Proof. Obvious. It is important to notice that Ig(t)is constant across conjugacy classes in Γ:
That is, given g, g′∈Γwith g′=h−1gh for some h∈Γ, we have,
Ig′(t) = ZM
H(g′m, m, t)dµ(m)
=ZM
H(h−1ghm, m, t)dµ(m)
=ZM
H(h−1ghm, m, t)dµ(m)
=ZM
H(ghm, hm, t)dµ(m)
=ZM
H(ghm, hm, t)dµ(hm)
=ZM
H(gm, m, t)dµ(m)
=Ig(t)
Let’s denote the set of conjugacy classes in Γby Con(Γ)and put
I[c](t) = Ig(t)
for c∈Con(Γ)where g∈c. The left hand side is well-defined owing to the constancy of Ig
across conjugacy classes.

A Comprehensive Overview of Spectral Geometry 1117
Lemma 4.5.
Θ(t) = X
c∈Con(Γ)
#(c)
#(Γ)I[c](t).
Proof. This is quite simple in view of the previous lemma and the above observations:
Θ(t) = 1
#(Γ)X
g∈GIg(t)
=1
#(Γ)X
c∈Con(Γ)
#(c)I[c](t)
=X
c∈Con(Γ)
#(c)
#(Γ)I[c](t).
Given a subgroup A≤Γ, let MA=M/A, with its canonical Riemannian tensor GAobtained
either by lifting Gon M=M/Γor by lowering Gon M.
We can now obtain the theta function ΘA(t)of (MA,GA)by employing similar arguments :
Lemma 4.6.
ΘA(t) = X
c∈Con(A)
#(c∩A)
#(A)I[c](t).
Proof. Clearly the heat kernel HAof MA=M/A satisfies,
HA(m1, m2, t) = X
a∈A
H(am1, m2, t),
with m1∈pA−1(m1), m1∈pA−1(m2)where pA:M7−→ MA=M/A is the obvious covering
projection. Let µAdenote the Lebesque measure induced on MAby GA. We have,
ΘA(t) = ZMA
H(m, m, t)dµA(m)
=1
#(A)ZMX
a∈A
H(am, am, t)dµ(m).
=X
a∈A
1
#(A)Ia(t).
Once again we notice that Ia(t)is constant across conjugacy classes and consequently,
ΘA(t) = X
c∈Con(Γ)
#(c)
#(A)I[c](t),
hence,
ΘA(t) = X
c∈Con(Γ)
#(c∩A)
#(A)I[c](t).
Theorem 4.7. Let Gbe a “bumpy” Riemannian metric on M=M/Γ, A, B ≤Γbe subgroups
of Γ, let MA=M /A, MB=M /B with respective natural Riemannian metrics.
(i) MAis isometric to MBiff Aand Bare conjugate subgroups of Γ.
(ii) ΘA(t) = ΘB(t)if Aand Bare equivalent in the sense of Gassmann.

1118 S. Ba¸saran and D. Amilo
Proof. ii is now clear by the observations preceding this theorem. As for i: Each element of
M=M/Γis an equivalence class under the equivalence relation ∼Γwhere m1∼Γm2if ∃g∈Γ
such that m2=gm1. Let’s denote the ∼Γthe equivalence class containing mby [m]Γ. Similarly,
we introduce the notations [m]A,[m]B. Suppose first that Aand Bare conjugate groups.There
exists γ∈Γsuch that
B=γAγ−1.
Define Fγ:MA7−→ MBby Fγ([m]A)=[γ m]BFirst check that this is well defined. Indeed if
m1∼Am2then there exists a∈Msuch that m2=am1. Hence,
γm2=γa m2=γaγ−1γ m1.
As γ aγ−1∈γ Aγ−1=B, we conclude γm2∼Bγm1.Therefore Fγis well-defined. As
m7−→ γ m is an isometry, so is Fγ:MA7−→ MBan isometry. Suppose, conversely, that
MA,MBare isometric, that is there exists an isometry F:MA7−→ MB. Let ˆpA:MA=
M/A 7−→ M=MΓ=M /Γbe the obvious covering projection sending [m]Ainto [m]Γ.
Similarly define ˆpB:MB7−→ M. Take [m]A∈MAand assume that F([m]A)=[m′]B.
If m=ˆpA([m]A)=[m]Γ= [m′]Γ=ˆpB([m]B) = m′then we may choose open disjoint
neighbourhoods U,Vof m,m′and open neighbourhoods ˆ
U,ˆ
Vof [m]A,[m]Γsuch that:
ˆpA|ˆ
U:ˆ
U7−→ U,
F|ˆ
U:ˆ
U7−→ ˆ
V ,
ˆpB|ˆ
V:ˆ
V7−→ V,
are isometries. Consequently,
ˆpB|ˆ
V◦F|ˆ
U◦(ˆpA|ˆ
U)−1:U7−→ V
is an isometry. This is impossible since Gis a “bumpy” metric. We conclude m=m′that is
[m]Γ=ˆpA([m]A) = ˆpB([m′]B) = [m′]Γ.
This being true for any [m]A∈MA,we conclude that Fhas to be a lifting of the identity map on
M, that is
ˆpB◦F=ˆpA
Thus for a given [m]A∈MA, if F([m]A)=[m′]Bthen [m]Γ= [m′]Γand there exists γ∈Γ
with
m′=γm.
By the uniqueness of liftings with a given action on a single point we conclude that,
F=Fγ.
and
B=γAγ−1.
For any commutative ring Rwith 1, let U(R)denote the multiplicative group of units in R.
Given groups G, H where Gacts upon Hon the left by φ:G7−→ Aut(H), let us write for
simplicity gh instead of φ(g)(h). The semi-direct product G⋉φH( or simply G⋉H)of Gand
Hconsists of pairs (g, h)∈G×Hwith the binary operation.
(g, h)(g′, h′)=(gg′, h(gh′)).
The group structure can be routinely checked.
Consider now the group Z/8Z={0,1,2,3,4,5,6,7}of integers modulo 8. U(Z/8Z) = {1,
3,5,7}acts on Z/8Zby multiplication. Consider the group Γ=U(Z/8Z)⋉ Z/8Z.In plain
language, Γis the group consisting of pairs (32 in all!),
(x, y)∈ {1,3,5,7}×{0,1,2,3,4,5,6,7},

A Comprehensive Overview of Spectral Geometry 1119
with a binary operation,
(x, y)(x′, y′)=(xx′, y +xy′).
It can be readily checked that (1,0)∈Γis the identity element, that is,
(1,0)(x, y)=(1x, 0+1y)=(x, y).
(x, y)(1,0)=(x1, y +x0)=(x, y)
and the inverse element,
(x, y)−1= (x−1,−x−1y).
Now we consider A, B ≤Γwhere,
A={1,3,5,7}×{0}
B={(1,0),(3,4),(5,4),(7,0)}.
A routine tabulation of the conjugacy classes in Γshows that Aand Bare equivalent in the sense
of Gassmann.
Let Snbe the symmetric group of degree nwhich consists of the permutations of nobjects.
Every finite group Gof order ncan be embedded as a subgroup of the group of permutations
SG≃Snof the carrier set of Gby means of the so-called Cayley representation which is a group
monomorphism.
Gi
7−→ SG,
where i(g)is defined to be:
i(g)(x) = g(x),
for all x∈G.
Let pbe a prime number and consider the groups:
A=Z/pZ⊕Z/pZ⊕Z/pZ.
B={




1x y
0 1 z
0 0 1




|x, y, z ∈Z/pZ}.
which are both finite groups of order p3which we consider to be subgroups of Sp3by means of
the above described procedure.
Clearly the order of each non-zero element of Ais p. The same holds for the elements of B
since,





1a b
0 1 c
0 0 1





n
=






1na nb +n(n−1)
2ac
0 1 nb
0 0 1






,
which can be obtained by a simple induction as,






1na nb +n(n−1)
2ac
0 1 nb
0 0 1











1a b
0 1 c
0 0 1





n
=






1(n+1)a(n+1)b+n(n+1)
2ac
0 1 (n+1)c
0 0 1






.

1120 S. Ba¸saran and D. Amilo
This means that all non-identity elements of A, B ≤Sp3are products p2disjoint cycles of length
p. Consequently, given g∈Sp3, either g=eand
#([g]∩A) = #([g]∩B) = 1
or g=eand
#([g]∩A) = #([g]∩B) = p3−1.
Therefore A, B are subgroups of Sp3which are equivalent in the sense of Gassmann. On the
other hand, let alone being conjugate, A, B are not even isomorphic since Ais Abelian but Bis
not.
Let, n≥3G=SL(n, Z/pZ)and consider:
A={[aij ]1≤i,j≤n∈SL(n, Z/pZ)|ai1=0 for i≥2}
B={[bij ]1≤i,j≤n∈SL(n, Z/pZ)|b1j=0 for j≥2}.
Let
a=












1
0
0
.
.
.
0












∈(Z/pZ)n.
Clearly x∈Aiff xa =λa for some λ∈Z/pZ− {0}. In other words, x∈Ais a (right)
eigenvector of x. Similarly y∈Biff aTy=µaTfor some µ∈Z/pZ− {0}. Again, this means
that y∈Biff aToccurs as a (left) eigenvector of y. Since, for any g∈SL(n, Z/pZ)a occurs
as a right eigenvector of xgx−1if and only if aToccurs as a left eigenvector of (x−1)Tg. These
show that Aand Bare not conjugate but equivalent in the sense of Gassmann. Since any finite
group is known to arise as the fundamental group of a compact, smooth manifold of dimension
4 ( in fact as the fundamental group of a compact complex projective algebraic surface) [34], in
view of theorem 4.7, examples 3,3and 3above provides us with isospectral but non-isometric
Riemannian manifolds.
5 Isospectral Deformations on Nilmanifolds
Having produced isospectral Riemannian manifolds which are not isometric, it is natural to ask
whether it is possible to find continuous families of isospectral manifolds which are naturally
isometrically distinct. Rephrased in the manner of Kac, we ask, whether it is possible for a Rie-
mannian manifold to change its shape continuously while sounding the same. To be precise, the
problem is to find a continuously parametrised family Gtof Riemannian metrics on a manifold
Mfor t∈[0,1]such that (M, Gt)and (M , Gt′)are isospectral for any t, t′∈[0,1], isometric
only when t=t′. The manifolds (M , Gt)are said to constitute isospectral deformations from
G0to G1. To emphasize the requirement that (M , Gt)and (M, Gt′)are not isometric for t=t′,
one may talk about non-trivial isospectral deformations. Non-trivial isospectral deformations
are known not to exist on flat tori[31] and the so called Heisenberg manifolds[32] and compact
manifolds of negative sectional curvature [24,28] In this section we present a method allied to
that of Sunada by means of which it is possible to produce non-trivial isospectral deformations
on a special but large class of Riemannian manifolds. Given a Lie group Gwith Lie algebra
G, the group Aut(G)of Lie group automorphisms of Gis a Lie group. If Gis connected then
Aut(G)can be naturally immersed in Aut[G]. If Gis connected and simply connected then
Aut(G)may be naturally identified with Aut[G]. The Lie algebra of Aut[G]and hence that of
Aut(G)can be identified with the algebra Der[G]of derivations of G, that is, of vector space
endomorphisms.
φ:G 7−→ G

A Comprehensive Overview of Spectral Geometry 1121
such that:
φ([X, Y ]) = [θX, Y ]+[X, θY ].
φ∈Aut(G)is said to be an inner automorphism if there exists g∈ G such that
φ(x) = gxg−1,
for all x∈G. We shall denote such φ∈Aut(G)by ig. It can be readily checked that inner
automorphisms of Gconstitute a normal Lie subgroup Inn(G)of G. The Lie algebra of Inn(G)
can be identified with the ideal ad[G]of Der[G]which consists of derivations of the form ad[X],
for X∈ G where ad[X]:G 7−→ G is defined by:
ad[X](Y)=[X, Y ].
The fact that ad[X]is a derivation is tantamount to the Jacobi identity for Lie brackets. If Z, z
denote the centers of G, Grespectively, that is,
Z={x∈G|xg =gx ∀g∈G},
z={X∈ G | [X, Y ] = 0∀Y∈ G},
then it can be easily checked that,
Inn(G)≃G/Z
ad[G]≃ G/z.
Let’s abbreviate the matrix





1x y
0 1 z
0 0 1





,
by 




x
y
z





and consider the Heisenberg group of degree 1 defined by:
G={




x
y
z




|x, y, z ∈R}.
It can be easily checked that the Lie algebra Gof Gis generated by the left-invariant vector fields,
X=∂
∂x ,
Y=∂
∂y +x∂
∂z ,
Z=∂
∂z ,
with
[X, Y ] = Z,
[Y, Z ] = [Z, X] = 0.
Clearly,
Z={




0
0
z




|z∈R} ≤ G.

1122 S. Ba¸saran and D. Amilo
z=< Z > ≤ G
It can be readily checked that each θ∈Der[G]is of the form
θ(X) = aX +bY +cZ
θ(Y) = a′X+b′Y+c′Z
θ(Z)=(a+b′)Z.
Therefore
dim(Aut(G)) = dimAut[G] = dim(Der[G]) = 6
whereas
dim(Inn(G)) = dim(Ad[G])
=dim(G)−dim(Z) = dim(G)−dim(z)
=3−1=2.
Proceeding more directly we can easily check that each φ∈Aut[G]is of the form:
(∗)























φ(X) = aX +bY +cZ
φ(Y) = a′X+b′Y+c′Z
φ(Z)=(ab′−a′b)Z
,
where ab′−a′b=0, from which we observe once again that,
dim(Aut[G]) = dim(Aut(G)) = dim(Der[G]) = 6.
In this particular example where Gis connected, simply connected and nilpotent, the relationship
between Gand Gis very simple. The maps:
exp :G 7−→ G
log :G7−→ G
are diffeomorphisms and can be explicitly given by:
exp(aX +bY +cZ) = 




a
b
c+1
2ab





,
log(




x
y
z





) = xX +yY + (z−xy
2)Z.
This allows us to write down the general form of the automorphisms of G: Indeed each F∈
Aut(G)is of the form F=Fφ∈Aut(G)where,
Fφ=exp ◦φ◦log,
for some φ∈Aut(G). Explicitly, we employ the general form of φas given in (∗)and put
F(




x
y
z





) = exp ◦φ◦log(




x
y
z





),

A Comprehensive Overview of Spectral Geometry 1123
=exp ◦φ[xX +yY + (z−xy
2)Z]
=exp[x(aX +bY +cZ) + y(a′X+b′Y+c′Z)+(z−xy
2)(ab′−a′b)Z]
=exp{(ax +a′y)X+ (bx +b′y)Y+ [cx +c′y+∆(z−xy
2)]Z},
where ∆=ab′−a′b. Thus,
F(




x
y
z





) = 




ax +a′y
bx +b′y
cx +c′y+∆z−(ab′−a′b)xy
2+1
2abx2+1
2ab′xy +1
2a′bxy +1
2a′b′y2





=




ax +a′y
bx +b′y
cx +c′y+∆z+1
2(abx2+2a′bxy +a′b′y2)





.
[23,20]
Let’s abbreviate the matrix,


















1x1x2z10 0 0
0 1 0 y10 0 0
0 0 1 y20 0 0
0 0 0 1 0 0 0
0 0 0 0 1 x1z2
0 0 0 0 0 1 y2
0 0 0 0 0 0 1


















,
by















x1
x2
y1
y2
z1
z2















,
and consider the Lie group,
G={















x1
x2
y1
y2
z1
z2















|x1, x2, y1, y2, z1, z2∈R}.

1124 S. Ba¸saran and D. Amilo
It can be readily checked that,















x1
x2
y1
y2
z1
z2






























x1
′
x2
′
y1
′
y2
′
z1
′
z2
′















=















x1+x1
′
x2+x2
′
y1+y1
′
y2+y2
′
z1+z1
′+x1y1
′+x2y2
′
z2+z2
′+x1y2
′















and















x1
x2
y1
y2
z1
z2















−1
=















−x1
−x2
−y1
−y2
−z1+x1y1+x2y2
−z2+x1y2















.
The Lie algebra Gof Gis generated by the left invariant vector fields:
X1=∂
∂x1,
X2=∂
∂x2,
Y1=∂
∂y1+x1∂
∂z1,
Y2=∂
∂y2+x2∂
∂z1+x1∂
∂z2,
Z1=∂
∂z1,
Z2=∂
∂z2,
which obey:
[X1, Y1] = Z1,
[X2, Y2] = Z1,
[X1, Y2] = Z2,
all the remaining brackets being zero. It can be readily checked that,
Z={















0
0
0
0
z1
z2















|z1, z2∈R},

A Comprehensive Overview of Spectral Geometry 1125
and
z=< Z1, Z2> .
Consequently
dim Inn(G) = dim Ad[G] = dimG/Z =dim G/z=6−2=4.
It is not easy to express the elements of Aut(G)explicitly. On the other hand, it is relatively
easier to investigate the elements of Aut(G)and to conclude that,
dim(Aut(G)) = dim(Aut[G]) = dim(Aut[G]) = 22 .
An inner automorphism igof Gis of the form:
ig(















x1
x2
y1
y2
z1
z2















) =















x1
x2
y1
y2
z1+a1y1+a2y2−b1x1−b2x2
z2+a1y2−b2x1















,
where,
g=















a1
a2
b1
b2
c1
c2















.
The logarithm and the exponential can be computed routinely :
exp[a1X1+a2X2+b1Y1+b2Y2+c1Z1+c2Z2]
=















a1
a2
b1
b2
c1+1
2(a1b1+a2b2)
c2+1
2a1b2















.
log(















x1
x2
y1
y2
z1
z2















) = x1X1+x2X2+y1Y1+y2Y2+ (z1−x1y1+x2y2
2)Z1+ (z2−x1y2
2)Z2.

1126 S. Ba¸saran and D. Amilo
Intuitively each derivation θ∈Der[G]of Gis an“infinitesimal” automorphism of G, hence of
G. Similarly elements of Ad[G]may be regarded as “infinitesimal” inner automorphisms. Now,
let us revisit the case of finite G. Let φ:G7−→ Gbe an automorphism with the property that
for each g∈G, there exists x=x(g)∈Gwith φ(g) = xgx−1. It can be easily checked that
for any subgroup A≤G,Ais equivalent in the sense of Gassmann to φ(A)≤G. An obvious
analogue of the above situation can be formulated as follows: [23] Given a Lie group Gwith Lie
algebra G, φ ∈Aut(G)is said to be an almost inner automorphism if for each g∈G, there
exists x=x(g)∈Gsuch that φ(g) = xgx−1. A derivative ξ∈Der[G]is called an almost inner
derivative if for each Y∈ G, there exists X=X(Y)∈ G such that
ξ(Y)=[X, Y ].
We denote the set of almost inner automorphisms of Gby AIA(G), the set of almost inner
derivatives of Gby AID(G)Clearly
Inn(G)⊆AIA(G)⊆Aut(G),
Ad[G]⊆AID[G]⊆Der[G].
The Heisenberg group admits no non-trivial almost inner automorphism. To be precise, each
almost inner automorphism in the Heisenberg group is an inner automorphism. To see this, note
that for any
g=




α
β
γ





,
ig(




x
y
z





) = 




α
β
γ










x
y
z










α
β
γ





−1
=




α
β
γ










x
y
z










−α
−β
−γ+αβ





=




α+x
β+y
γ+z+αy










−α
−β
−γ+αβ





=




x
y
z+αy +αβ + (α+x)(−β)





=




x
y
z+αy −βx





.
Given any automorphism φof Gdefined by:
φ(




x
y
z





) = 




ax +a′y
bx +b′y
cx +c′y+∆z+1
2[abx2+2a′bxy +a′b′y2]





.

A Comprehensive Overview of Spectral Geometry 1127
In order for φto be an almost inner automorphism it is clear that a=b′=1, a′=b=0 in
which case
φ=ig,
where g may be chosen to be of the form:
g=




c′
−c
γ





.
for arbitrary γ∈R.
In the six-dimensional nilpotent Lie group of Example 2it is fairly easy to work out the
general form of an almost inner derivative : Since all Lie brackets take their values in z=<
Z1, Z2>, we conclude that for an almost inner derivative δ∈AID[z], δ(X)∈zfor all X∈ G.
On the other hand δ|z≡0. Therefore an almost inner derivative δmust have the form:
δ:





























































X17−→ a11Z1+a12 Z2
X27−→ a21Z1+a22 Z2
Y17−→ b11Z1+b12 Z2
Y27−→ b21Z1+b22 Z2
Z17−→ 0
Z27−→ 0
However, as Lie brackets containing X2or Y1can take values only in < Z1, Z2>we conclude
that a22 =b22 =0.It is easy to check now that an endomorphism δof Gis an almost inner
derivative iff δis of the form:
δ:





























































X17−→ a11Z1+a12 Z2
X27−→ a21Z1
Y17−→ b11Z1
Y27−→ b21Z1+b22 Z2
Z17−→ 0
Z27−→ 0

1128 S. Ba¸saran and D. Amilo
This shows that AID[G]is a six dimensional subalgebra of Der[G]. In view of the four linearly
independent inner derivatives,
Ad[X1]:





























































X17−→ 0
X27−→ 0
Y17−→ Z1
Y27−→ Z2
Z17−→ 0
Z27−→ 0
Ad[X2]:





























































X17−→ 0
X27−→ 0
Y17−→ 0
Y27−→ Z1
Z17−→ 0
Z27−→ 0
Ad[Y1]:





























































X17−→ −Z1
X27−→ 0
Y17−→ 0
Y27−→ 0
Z17−→ 0
Z27−→ 0

A Comprehensive Overview of Spectral Geometry 1129
Ad[Y2]:





























































X17−→ −Z2
X27−→ −Z1
Y17−→ 0
Y27−→ 0
Z17−→ 0
Z27−→ 0
it is clear that,
AID[G] =< Ad[X1], Ad[X2], Ad[Y1], Ad[Y2], δ, ε >
where,
δ:





























































X17−→ Z2
X27−→ 0
Y17−→ 0
Y27−→ 0
Z17−→ 0
Z27−→ 0
ε:





























































X17−→ 0
X27−→ 0
Y17−→ 0
Y27−→ Z2
Z17−→ 0
Z27−→ 0

1130 S. Ba¸saran and D. Amilo
It can be directly checked that automorphisms of Gobtained by exponentiation from elements
of AID[G]are almost inner automorphisms. At this stage, we content ourselves by noticing that
the maps
φ=exp ◦expG(δ)◦log =exp ◦(I+δ)◦log
ψ=exp ◦expG(ε)◦log =exp ◦(I+ε)◦log
are both almost inner automorphisms. Indeed,
φ(















x1
x2
y1
y2
z1
z2















) =















x1
x2
y1
y2
z1
z2+x1















=ig(















x1
x2
y1
y2
z1
z2















),
where,
g=















0
0
x2/x1
−1
0
0















,
if x1=0, otherwise g=eG.
Again
ψ(















x1
x2
y1
y2
z1
z2















) =















x1
x2
y1
y2
z1
z2+y2















=ih(















x1
x2
y1
y2
z1
z2















),
where
h=















1
−y1/y2
0
0
0
0















,
if y2=0, otherwise h=eG.
Theorem 5.1. Almost inner automorphisms of Gconstitute a normal subgroup of Aut(G).

A Comprehensive Overview of Spectral Geometry 1131
Proof. Clearly IdG∈AIA(G). Suppose φ, ψ ∈Aut(G).Given x∈G, ∃g=g(x)and
h=h(x)such that
φ(x) = gxg−1
ψ(x) = hxh−1.
hence
ψ◦φ(x) = ψ(φ(x))
=ψ(gxg−1)
=ψ(g)ψ(x)ψ(g)−1
=ψ(g)hx(ψ(g)h)−1.
This being true for arbitrary x∈G, it is seen that ψ◦φ∈AIA(G). Therefore AI A(G)≤
Aut(G).
As for normality:
Given φ∈AIA(G)and α∈Aut(G), consider x∈Gand choose g=g(α−1(x)) Thus
α◦φ◦α−1(x) = α◦φ(α−1(x))
=α(gα−1(x)g−1)
=α(g)x α(g)−1.
This being true for each x∈G, we conclude that α◦φ◦α−1∈AIA(G). Consequently
AIA(G)Aut(G)
Theorem 5.2. Let G∗stands for the dual of G. Given a simply connected Lie group Gand
φ∈Aut(G),the following are equivalent:
(i)φis an almost inner automorphism.
(ii)For each X∈ G there exists g=g(x)∈Gsuch that
φ∗(X)=(ig)∗(X).
(iii)For each ξ∈ G∗there exists g=g(ξ)∈Gsuch that
ξ◦φ∗=ζ◦(ig)∗
Proof. (i) =⇒(ii): Given X∈ G
φ∗(X)(e) = Teφ(X(e)) = d
dt|t=0φ(γ(t)),
where γ:(−ε, ε)−→ Gis any smooth path with γ(0) = eand ˙γ(0) = X(e).We choose
γ(t), t ∈Rto be the one-parameter subgroup of Ggenerated by X∈ G. There exists g∈Gsuch
that
φ(γ(1)) = ig(γ(1)).
It can be checked that
φ(γ(1)) = ig(γ(t)),
for all t∈R. Consequently
φ(X)=(ig)∗(X).
(ii) =⇒(iii): Choose a left-invariant Riemannian metric Gon G. Equivalently Gmay be
understood to be an innerproduct on G. Given ξ∈ G∗, there exists unique X∈ G such that
ξ=G(X, .).
Therefore, for any Y∈ G,
ξ◦φ∗(Y) = G(X, φ∗(Y))
=G(φ∗(X), Y )
=G((ig)∗(X), Y )

1132 S. Ba¸saran and D. Amilo
for some g=g(X)∈G. This being true for arbitrary Y∈ G, we find
ξ◦φ∗=ξ◦(ig)∗.
It is obvious that the dual of this argument allows us to conclude that
(iii) =⇒(ii).
(ii) =⇒(i): Consider x∈G. Choose X∈ G such that γ(1) = xwhere γ(t)is the
one-parameter subgroup generated by X∈ G. Choose g=g(X)∈Gwith
φ∗(X)=(ig)∗(X).
It can now be checked that φ(γ(t)) is the one parameter subgroup generated by (ig)∗(X)and
φ(x) = φ(γ(1)) = gγ(1)g−1=gxg−1.
Theorem 5.3. In a connected, simply connected Lie group G,AIA(G)is a Lie subgroup of
Aut(G).
Proof. By a standard theorem of E. Cartan a closed subgroup of a real Lie group is a Lie sub-
group [29]. We have already shown that AIA(G)is a subgroup (in fact a normal subgroup)
of Aut(G). Therefore, it will be sufficient to show that AIA(G)is closed in Aut(G). To this
end notice that Gis nilpotent and there exists m∈Nsuch that for all X∈ G , Ad[X]m≡0.
Consequently exp(Ad[X]) is a polynomial in Ad[X]which has an order independent of X∈ G.
As a result, the orbits of Inn(G)on Gare closed in G. Since the orbits of AIA(G)on Gcoincide
with those of Inn(G)we conclude that the orbits of AIA(G)on Gare closed in G. Therefore
AIA(G)is closed in Aut(G)≃Aut[G].
Theorem 5.4. Given a connected, simply connected nilpotent Lie group G, the Lie subalgebra
corresponding to the Lie subgroup AIA(G)of almost inner derivations.
Proof. Assume Gis of nilpotence length m. Let {Gk}m
k=0be the central series of Gconsisting of
the iterated derived subalgebras Gkdefined inductively by
G0=G. . . Gk+1= [G,Gk].
Consider any φ∈AIA(G). Clearly,
(φ∗−I)Gk⊆ Gk+1,
for each k≥0. In other words, φ∗−Iis nilpotent on G. If φ=expGDfor some D∈Der[G].
Again we have,
DGk⊆ Gk+1,
for each k≥0.Given X∈ G, there exists Y∈ G such that,
φ∗X=expG(Ad[Y])X.
Consequently,
expG(−Ad[Y])expG(D)X=X. (∗),
Now, remember that Ad[G]is an ideal in Der[G],in fact [D, Ad[Y]]G=Ad[D(y)].Employing
the Hausdorff-Campbell formula on the left hand side of (∗)we obtain
expG(D−Ad[Z])X=X, (∗∗),
where Z=Y−1
2D(Y) + ··· which terminates after finitely many terms involving powers of
D. But, D−Ad[Z]is again nilpotent on Gand f(D−Ad[Z]) is invertible where f:CC is the
entire function defined by
f(z) = 








ez−1
zfor z=0
1 for z = 0 .

A Comprehensive Overview of Spectral Geometry 1133
By (∗∗)we have,
f(D−Ad[Z])(D−Ad[Z])X
=expG(D−Ad[Z])X=X,
from which we conclude by the invertibility of f(D−Ad[Z]) that,
(D−Ad[Z])X=0,
or equivalently
DX =Ad[Z]X,
which, being true for arbitrary Ximplies that,
D∈AID[G].
We conclude that,
AIA(G)⊆exp(AID[G]).
A similar argument can be employed to reverse this inclusion.
Remark 5.5. The proof of the above theorem, rephrased in slightly greater detail will allow us
to observe that for a nilpotent Lie group Gof nilpotence length m, AIA(G)is a nilpotent group
of nilpotence length at most m−1. A similar result is valid for AID[G].
Remark 5.6. In view of the fact that AIA(G)is normal, we conclude that AID[G]is a Lie ideal
in Der[G]. A nilmanifold Mis a quotient manifold of right cosets M=Γ\Gwhere Gis a
simply connected nilpotent Lie group, Γis a discrete cocompact subgroup of G.
Notice that a right coset of Γin Gis nothing but an orbit of the action of Γon Gby multiplication
on the left. Consequently a tangent vector of M=Γ\Gcan be regarded as an equivalence class
of tangent vectors on Gwhere u∈TxGand v∈TyGare are understood to be equivalent if there
exists γ∈Γsuch that
y=γx,
and
v=TxLg(u).
The equivalence class containing u∈TxGwhich we denote by [u]Γcan be identified with
a tangent vector at Γx∈M=Γ\G. Thus, a left-invariant Riemannian metric Gon Gwill
naturally induce a Riemannian metric GΓon M=Γ\Gdefined by
GΓ([u]Γ,[v]Γ) = G(u, v).
By an obvious and convenient use of notation we shall simply write u,Ginstead of [u]Γ,GΓin
the sequel.
ARiemannian nilmanifold is a pair (Γ\G, G)where Gis a Riemannian metric which is induced
by a left invariant one on Gin the manner described above.
Lemma 5.7. On a Lie group Gwith left invariant metric G, the Laplacian ∆=∆Gis of the form
∆f=
n
X
i=1{−EiEif+ (∇EiEi)f},
for each f∈ D(G), where {Ei}1≤i≤nis any basis for Gthat is orthonormal with respect to G
and ∇is the Levi-Civita connection attached to G.
Proof. Notice that for any smooth f:G7−→ Rwe have
grad(f) =
n
X
i=1
(Eif)Ei.

1134 S. Ba¸saran and D. Amilo
Now, given A=AiEi∈ X(G),
div(A) = trace{Er∇Er(AiEi)}
=trace{Er(ErAi)Ei+ (AiΓri
m)Em},
where we put Γri
mEmfor ∇Er(Ei).Thus we conclude,
div(A) = ErAr+AiΓri
r.
The result follows from the observation Γijk=−Γik jfor any 1 ≤j, k ≤n, which can be
derived by noticing that ∇G=0,Gis left invariant and hence
G(∇EiEj, Ek) + G(Ej,∇EiEk) = 0.
The following theorems are best derived by means of standard but heavy techniques. We offer
sketches of proofs.
Theorem 5.8. If Gis a bi-invariant Riemannian metric on the nilpotent Lie group G, then the
Riemannian nilmanifolds (Γ\G, G)and (Γ\G, φ∗G)are isospectral for each φ∈AIA(G).
Proof. When Gon the nilpotent Lie group Gis biinvariant( such tensors exist on Gsince G
admits cocompact subgroups!) then the Levi-Civita connection ∇obeys ∇XY=0 for all
X, Y ∈ G and ∆reduces to the form
∆f=−
n
X
i=1
EiEif,
for each f∈C(G)where {Ei}1≤i≤nis any basis for Gwhich is orthonormal with respect to
G. By Kirillov’s theory on the unitary representations of nilpotent Lie groups [33], each unitary
irreducible representation of Gon the Hilbert space of functions which are square integrable
by the Lebesque measure induced by G, are parametrised by elements of G,any almost inner
automorphism induces well-behaved unitary transformations between these. Since, in the pres-
ence of a biinvariant metric, the effect of each X∈ G on D(G)can be expressed in terms of
irreducible representations, the same applies to the above mentioned Ei’s and to ∆.
Theorem 5.9. Riemannian nilmanifolds (Γ\G, G1)and (Γ\G, G2)are isometric iff there exists
g∈Gand φ∈Aut(G)with φ(Γ) = Γsuch that
G2= (ig◦φ)∗G1.
Proof. Suppose G2= (ig◦φ)∗G1where g∈Gand φ∈Aut(G)with φ(Γ) = Γ. Clearly
ig◦φ=Lg◦Rg−1◦φ,
where Lxand Ryrepresent multiplications in Gby xand yon the left and right respectively. G1
being left invariant we find
G2= (Rg−1◦φ)∗G1.
But Rg−1◦φ=fis the lifting of a map f:Γ\G7−→ Γ\Gwhich is an isometry from (Γ\G, G1)
to (Γ\G, G2).
Conversely suppose
f:(Γ\G, G1)7−→ (Γ\G, G2)
is an isometry. flifts to an isometry
f:(Γ\G, G1)7−→ (Γ\G, G2).

A Comprehensive Overview of Spectral Geometry 1135
By a standard result of [13]G1=Ψ∗G2,for some Ψ∈Aut(G)with Ψ(Γ) = Γ. Thus f◦Ψ−1
is an isometry of (Γ\G, G2).Let
σ=Lg◦f◦Ψ−1,
where g−1=f◦Ψ−1(e)∈Γ. Therefore σis an isometry of (Γ\G, G2). with σ(e) = e. Again
from [14,15], we conclude that σ∈Aut(G). Consequently
σ◦Ψ=Lg◦Rg−1(Rg◦f).
=ig◦(Rg◦f).
But Rg◦f(Γ) = Γ.
Remark 5.10. It is possible to give a direct but not quite as natural a proof for theorem 5.9 in the
case where Gis of nilpotence length 2[20].
Remark 5.11. The theorems 5.8 and 5.9 indicate that on nilmanifolds like that in Example 5on
which inner automorphisms have sufficiently large codimension inside the group of almost inner
automorphisms, one can trivially obtain non-trivial isospectral deformations.
6 Conclusion
This study highlighted that the spectrum gives fairly detailed and complete information on large
families of Riemannian manifolds such as flat tori, Heisenberg spaces and manifolds of nega-
tive sectional curvature. This is what should perhaps be called the realm of “spectral rigidity”.
Moreover, away from spectral rigidity, (which we consider to be “generic”) that is where the
possibility of spectral deformations take over, a meagre category of highly structured manifolds
await exploration. It would be interesting to develop criteria distinguishing these situations more
clearly.
7 Competing interests
The authors declare no competing interests regarding the publication of this manuscript.
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Author information
S. Ba¸saran, Department of Computer Information Systems, Near East University TRNC, Mersin 10, Nicosia
99138, Turkey. https://orcid.org/0000-0001-9983-1442,.
E-mail:
seren.basaran@neu.edu.tr

A Comprehensive Overview of Spectral Geometry 1137
D. Amilo, Mathematics Research Center, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey.
Department of Mathematics, Near East University, Nicosia, 99010, Cyprus. Faculty of Art and Science,
University of Kyrenia, Kyrenia, TRNC, Mersin 10, Turkey. https://orcid.org/0000-0003-0206-2689,.
E-mail:
amilodavid.ikechukwu@neu.edu.tr
Received: 2023-11-28
Accepted: 2024-05-13