C OMPOSITIO M ATHEMATICA
R UISHI K UWABARA
On isospectral deformations of riemannian metrics
Compositio Mathematica, tome 40, n
o3 (1980), p. 319-324
<http://www.numdam.org/item?id=CM_1980__40_3_319_0>
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319
ON ISOSPECTRAL DEFORMATIONS OF RIEMANNIAN METRICS
Ruishi Kuwabara
@ 1980 Sijthoff & Noordhoff International Publishers - Alphen aan den Rijn
Printed in the Netherlands
1. Introduction
Let M be an n-dimensional
compact
orientable Coomanifold, and g
be a C°° Riemannian metric on M. It is known that the
Laplace-
Beltrami
operator àg
=-gij~i~j acting
on C~ functions on M has aninfinite sequence of
eigenvalues (denoted by Spec(M, g))
each
eigenvalue being repeated
as many as itsmultiplicity.
Consider the
following problem [1,
p.233]:
Letg(t) (- E
t E, E >0)
be a1-parameter
C~ deformation of a Riemannian metric on M.Then,
is there a deformationg(t)
such thatSpec(M, g(t))=
Spec(M, g(o))
for every t? Such a deformation is called anisospectral deformation.
First,
wegive
some definitions to state the results of this article.The deformation
g(t)
is called trivial if for each t there is adiffeomorphism q(t)
such thatg(t) = n(t)*g(0) (the pull-back
ofg(O) by ~(t)).
For a deformationg(t)
thesymmetric
covariant 2-tensorh - g’(0)
is called theinfinitesimal deformation (i-deformation,
forshort) [2]. By Berger-Ebin [3] h
isdecomposed
aswhere
~ihij
= 0(V being
the connection inducedby g(0))
andLx
is the Lie derivative withrespect
to X. The i-deformation h is called trivial ifh
= 0.0010-437X/80/03/0319-06$00.20/0
320
The main result of this article is the
following.
THEOREM A: There is no non-trivial
isospectral i-deformation of
ametric
of flat
torus.REMARK:
Concerning
theisospectral
deformation of a metric of constant curvature, we caneasily get
thefollowing:
There is nonon-trivial
isospectral deformation of
a metricof
constant curvatureK if
2~ dim M~5,
or dim M = 6 and K > 0. This result is obtainedby combining
the results ofspectral geometry [ 1 ], [4]
and thoseconcerning
thenon-deformability
of a metric of constant curvature, the latterbeing directly
derived from the results ofBerger-Ebin [3],
Mostow
[5]
and Koiso[2]. (See
Tanaka[6],
for the case dim M = 2and K
0.)
In the case of flat metrics Sunada[7]
showed that thereare
only finitely
manyisometry
classes of flat manifolds with agiven spectrum.
The author wishes to thank Professor M. Ikeda for
carefully reading
themanuscript
andoffering
valuable comments. Thanks arealso due to Mr. N.
Koiso,
OsakaUniversity,
forhelpful
discussions.2. Proof of Theorem A
The main
part
of theproof
is tostudy
the variation of the coefficients of Minakshisundaram’sexpansion:
If
g(t)
is anisospectral deformation,
the coefficientsa;(t)
must beconstants.
For a Riemannian
metric g
onM,
dV(g), R’,,, Rij
and T denote thevolume
element,
the curvature tensor, the Ricci tensor and the scalar curvature,respectively.
Thecoefficients aj (j
=0, 1, 2)
aregiven by
where
|R|2 = RijkmRijkm
and|03C1|2 = RijRij.
Let
g(t)
be a deformation andh(t)
~g’(t)
the i-deformation ofg(t)
at t. Let
aj(t)
be the Minakshisundaram’s coefficients for(M, g(t)).
Then the
following
formulas are obtainedby straightforward
cal-culation.
(2.1) a’(t) 0 2f, hss dV(g(t)),
(2.2) a,(t) = 6 fm (lrh 2 s s - Riihii) dV(g(t)),
(2.3) a’2(t) = 1/360~M 2 [12(~j~i03C4)hji - 6(~k~kRji)hji+ 8RikRfhij
-
4RkjmiRkmhji - 4RjkmsRikms h j’
+9(àT) h s s -1003C4Rjihji+ |R|2hss _ |03C1|2 hss + 5/203C42hss ] d V(g(t)).
Let
(M, go)
be a flat manifold. Then we haveLEMMA 2.1:
If g(t)
is anisospectral deformation
withg(O)
= go, andVihij
= 0 holds at t =0,
then(2.4a, b) h ss = 0, V khji
= 0hold at t = 0.
PROOF:
Starting
from(2.3),
we haveby
tediouscalculation,
(Note
that~ihij = 0
andRjkm
= 0 at t =0.)
From(2.1)
and(2.5),
a0’(0)
=a2(0)
= 0 holds if andonly
if(2.4a, b)
holdgood. Q.E.D.
LEMMA 2.2: Let
g(t)
be as in Lemma 2.1. Thenholds
for
eacheigenfunction 0 of L1g(O).
PROOF: For the
eigenvalue Àk(t)
ofL1g(t),
thefollowing
was obtainedby Berger [8]:
322
where 0
is theeigenfunction
forÀk(O). Therefore, (2.6)
follows from(2.4). Q.E.D.
Now,
let us prove Theorem A. Let(M, go)
be a flat torusgiven by R"/L,
where L is alattice, i.e.,
a discrete abeliansubgroup
of thegroup of Euclidean motions in Rn. Let L* denote the dual
lattice, consisting
of all x E R" such that(x, y) = 1 ?= 1xiyi
is aninteger
for ally E L. Then the sets of
eigenfunctions
andeigenvalues
aregiven by {~x(y)
= cos203C0(x, y), 03C8x(y)
= sin203C0(x, y) ; x OE L*)
and403C02(x, x) ;
x EL*}, respectively. Recalling (2.4b),
we see thathij
are constants in thecoordinates induced from R".
Therefore,
from(2.6)
we havehence
hijxixj = (hx, x)
= 0 for x E L*. This leads to h =0,
because the set{x/~x~;
x EL*}
isobviously
dense inf x
ER" ; IIXII
=1}. Q.E.D.
3. Conformal deformations
In this section we restrict our
study
to the conformaldeformation,
Set
a(t)
=p’(t),
andstraightforward
calculationgives
LEMMA 3.1 : Assume that
(M, go)
has a constant scalar curvature.If g(t)
is avolume-preserving deformation,
we havePROOF: It is easy to see that
a"(0)
= 0 is written asBy
thisequation
and T = const.,(3.2)
is led to(3.3). Q.E.D.
From now on, we assume n = dim M > 2.
By
virtue of the above lemma we have thefollowing
theorem.THEOREM B: Let
(M, go)
have a constant scalar curvature andAI
be the non-zerofirst eigenvalue of L1g(O). If
holds,
there is noconformal isospectral deformation of
go.PROOF: Let
g(t)
be a conformalisospectral
deformation of go.Since
a0’(0)
=0,
we havef Ma-
dV(go)
= 0.Therefore,
holds
(see [1,
p.186],
forexample). Accordingly,
if(3.4) holds,
wehave
a1"(0)
> 0 unless=0, i.e.,
h = 0.Q.E.D.
The condition
(3.4)
isobviously
satisfied if T s 0.Further,
as shownby
Obata[9],
it is also satisfied for an Einstein space not isometric with asphere.
In the case of asphere, 03BB1
=rl(n - 1),
hencea’’1(0) ~ 0
holds. Theequality
holdsonly
when is theeigenfunction
for03BB1,
which isequivalent
to~i~j03C3 + {03C4/n(n - 1)}
ugij = 0(see [9]).
There-fore,
Thus h is trivial.
As a consequence we have the
following
theorem.THEOREM C:
Suppose (M, go)
is an Einstein space, or a spaceof non-positive
constant scalar curvature. Then there is no non-trivialconformal isospectral i-deformation of
go.REMARK: In the case of dim M =
2,
it follows from the Gauss- Bonnet formula that the coefficiental(t)
is invariant.324
REFERENCES
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[2] N. KOISO: Non-deformability of Einstein metrics, Osaka J. Math. 15 (1978) 419-433.
[3] M. BERGER and D. EBIN: Some decompositions of the space of symmetric tensors
on a Riemannian manifold. J. Diff. Geom. 3 (1969) 379-392.
[4] S. TANNO: Eigenvalues of the Laplacian of Riemannian manifolds. Tôhoku Math.
Journ. 25 (1973) 391-403.
[5] G.D. MOSTOW: Strong rigidity of locally symmetric spaces. Princeton, 1973.
[6] S. TANAKA: Selberg’s trace formula and spectrum. Osaka J. Math. 3 (1966) 205-216.
[7] T. SUNADA: Spectrum of a compact flat manifold (preprint).
[8] M. BERGER: Sur les premières valeurs propres des variétés riemanniennes. Com- positio Math. 26 (1973) 129-149.
[9] M. OBATA: Certain conditions for a Riemannian manifold to be isometric with a
sphere. J. Math. Soc. Japan 14 (1962) 333-340.
(Oblatum 21-111-1978 & 12-XII-1978) Department of Applied Mathematics and Physics
Faculty of Engineering Kyoto University Kyoto 606, Japan