KOISO, NORIHITO

Osaka City University Repository

Koiso, N.
Osaka J. Math.
16 (1979), 413-421
ON THE SECOND DERIVATIVE OF THE TOTAL
SCALAR CURVATURE
NORIHITO KOISO
(Received March 6, 1978)
(Revised April 11, 1978)
0. Introduction
Let M be a compact connected C°°-manifold of dimension n^2. It is a
classical result due to D. Hubert [4] that a metric g on M is an Einstein metric
if and only if g is critical for the scalar curvature r, that is, g is such that
dt = 0
for any volume preserving deformation g(t) of £, where τg(t) is the scalar cuva-
ture of g(t) and vg(t) is the volume element of g(t). As for the derivative of
second order of the integral, Y. Muto [8] shows that there exist volume pre-
serving deformations which gives positive derivative and which gives negative
derivative.
In this paper we attempt to decide the sign of the derivative for given
volume preserving deformations. The results are as follows. Let (M,g) be an
Einstein manifold with certain condition (in Theorem 2.5). If (M,g) is not the
standard sphere, then any volume prserving deformation is decomposed to
a conformal deformation with positive derivative (Theorem 2.4), a trivial de-
formation with zero derivative and a deformation of constant scalar curvature
with negative derivative (Theorem 2.5).
The paper is organized as follows; after some preliminaries in 1, we prove
the above propositions in 2. Finally, in 3, we consider the case when M is a
complex manifold and g(t) are Ka'hler metrics.
1. Preliminaries
First, we introduce notation and definitions which will be used throughout
this paper. Let M be an rc-dimensional, connected and compact C°°-manifold,
and we always assume n^2. For a riemannian manifold (My g)y we consider
the riemannian connection and use the following notation;
vg the volume element defined by g,
R\ the curvature tensor defined by the riemannian connection,
414 N. Koiso
p the Ricci tensor
(For the standard sphere with orthonormal basis, Λ12i2<0 and Pn>0.)
τ; the scalar curvature,
( , ); the inner product in fibres of a tensor bundle defined by g,
< , >; the global inner product for sections of a tensor bundle over M, i.e.,
< , > = ( (.K,
J M
C°£(M)\ the vector space of all functions / such that \fvg=0y
C00(S2); the vector space of all symmetric covariant 2-tensor fields,
C7(*52); the vector space of all symmetric covariant 2-tensor fields h such
that<A,£>=0,
L; the operator operating on C°°(S2) defined by (Lh^—Rf fhkh
V; the covariant derivative defined by the riemannian connection,
δ; the formal adjoint of V with respect to <(,)>,
δ*; the formal adjoint of δ| C°°(52),
Δ=δί/; the Laplacian operating on the space C°°(Λf),
Δ=δV; the rough Laplacian operating on the vector space of tensor fields,
Hess = W; the Hessian on C°°(M) .
REMARK 1.1. Let (M, g) be an Einstein manifold. Then,
tr Lh = gliRf!hkl = ~pklhkl = - — tr h .
n
Therefore we see that L operates on tr'^O).
In this paper, we consider 1 -parameter families of riemannian metrics on
M. If £(0) =£, then we call such a family g(t) a deformation of g. The derivative
£'(0) of a deformation g(t) is called an infinitesimal deformation, or simply
i-deformation. Total of i-deformations forms the space C°°(*S2). Total of volume
preserving i-deformations consists with the space
2. The second derivative of the integral \ τv
a
First, we give a decomposition of C^(S2).
Proposition 2.1. Let g be a metric of constant scalar curvature such that
τg=Q or τg/(n— 1) is not an eigenvalue of Δ^. Then C~(S2) is decomposed as follows
C~(S2) = C7(M).^0Imδ*θδ-1(0)nα-1(0)nQ(52) , (2.1)
where a is an operator from C°°(S2) to C°°(M) which is defined by
a(h) = Δ{Δtr h+Sδh-(h, p)} .
SECOND DERIVATIVE OF THE TOTAL SCALAR CURVATURE 415
REMARK 2.2. By [7, p. 135], we know that the first positive eigenvalue of
- on an Einstein manifold. Moreover by [9, Theorem 5], the equality
n — 1
holds if and only if the Einstein manifold is isometric to the standard sphere.
Therefore, the condition of the scalar curvature is satisfied if (M,g) is an Einstein
manifold but not the standard sphere.
REMARK 2.3. Moreover we can get the following decomposition [6, Corol-
lary 2.9]. Let g ( t ) be a deformation of g. Then g(t) is decomposed into
/(Z)γ(£)*£(ί), where f(t) is a 1-parameter family of positive functions, 7(2) is a
1 -parameter family of diffeomorphisms and g(t) is a volume preservnig defor-
mation of constant scalar curvature such that δ '^(O)— 0. The decomposition
(2.1) is the differential of this decomposition.
Proof . First we show the decomposition
Q(S2) = Cϊ(M) g®cΓ\0) Π C~(S2) .
IffgGa~l(Q), then a(fg)=Q, which implies
By the condition of g,f is constant, which implies /=0 because of/eCj(M),
hence C;(M)
 <?nα~
1(0)n C~(S2)=Q. If h eC~(S2) is orthogonal to C~(M) g+
cΓ^O) n C7(S2), then tr λ=0 and <A, cr^O) Π C7(S2)>=0. But here, the formal
adjoint of a is given by
α*(/) = Δ2/ £+HessΔ/-Δ/ /> ,
and has injective symbol. Therefore, by [3, Corollary 6.9], C°°(S2) = Imα*0
αΓ^O) (orthogonal direct sum), which implies that there aie a function / and an
element ψ of a'^O) such that λ=α*/+ ψ . We easily see that α*/eC7(ιS2),
and so ψeα'XO) Π Cj(52). Therefore
0 - <α*/+ψ, ψ> - <ψ, x/τ>
and ψ»=0. Since tr A— 0, we see (n— 1)Δ2/— τΔ/=0. Therefore / is a con-
stant, which implies h=a*f=0. Thus we get the above decomposition.
By [2, (3.1)], we know C°°(52)- Im δ*θδ'1(0). Since δ£=0, Imδ*c
C7(S2). Therefore we get
= Im δ*θδ'1(0) Π C;(S2) .
Moreover, αδ*f=Δ{Δtr(δ*f)+δδ(8*f)-(δ*f, p)}, and
416 N. Koiso
~
-0 .
Therefore we see Im δ*Cα~1(0), which implies
α-^O) Π C~(S2) = Im δ*θcT1(0) Π δ'^O) Π C°
Now, we decide the sign of the second derivative according to the decom-
position (2.1). Recall that any element of Im δ* is an i-deformation of a tri-
vial deformation 7(ΐ)*g ([2, Lemma 6.2]). Therefore <τ, 1>" = 0 for any
element of Im δ*.
Theorem 2.4. Let (M, g) be an Einstein manifold but not be the standard
sphere. If h=fg is a conformal and volume preserving non-zero i-deformationy
i.e., <A, £>=0, then any volume preserving deformation g(t) of g such that g'(0)=h
satisfies
- \dt20l
Proof. We recall the formula [8, 2]
Hence we get <τ, l>//=/t^<(w— 1)Δ/- τ/,/>. But here, by Remark 2.2, we
know that the first eigenvalue of Δ > . Thus, since /e C%(M), we see
n— 1
Theorem 2.5. Let (My g) be an Einstein manifold. We denote by α0
the minimum eigenvalue of the operator L: tr~1(0)->tr~1(0). We assume that
in<— , — — 1. Then for any volume preserving deformation g(ί) of g such
I n 2n)
andg'(Q)Φθ, we get
Proof. We set g'(ϋ)=h. Recall the formula [8, (1.5)]
SECOND DERIVATIVE OF THE TOTAL SCALAR CURVATURE 417
+4-<v<v» w>+-<Λ,,,AJy>-f <*Λ v>Z n Δn
But here SA=0, ah = Q and <trA, 1>=0, which implies Δ t r Λ — —trh = 0.
n
Therefore
-<v,V, v»A»,>+-L<v,A/, vA*>-f <*Λ V>
Z Z/ί
A),
=— <Δtr A, tr A>— — <trA, tr A > = 0 .
z z/x
Moreover we see
and
Thus we get
= — -<Δh+2Lh, A>.
Z
We remark that the equation Δ t r A — — tr A— 0 implies tr A=0. In fact,
by Remark 2.2, the first positive eigenvalue of Δ^-  . Therefore if T Φ O
n— 1
then tr A=0. Even if τ=0, tr h is constant. But here h is volume preserving,
i.e., <A, £>=(), and so trA=0. Now we define the operator <5V: C°°(*S2)-*
C°°(Γ30), 5V: C-ίS^-^C^ΓJ) by
, y, z) -
where w, ϋ, w^R, u2+v2+w2= 1. Set p=uv -\-vw+wu. Then the minimum
and maximum of p is — — and 1, respectively. By simple computations we have
and
418 N. Koiso
Therefore we get <A^—2pL^— — rp^+2pV8^ ψ >^0. Thus, since δ&=0,
we see
<Δh+2Lh, hy ^  2<((l+p)Lh+—ph,
If α0>-, we assume ρ=~. Then (l+p)a+ί.p=λ.aί>~>Q. Ifn 2 n 2 2n
aQ>—^-> we assume p=l. Then (l+p)a0+^-p=2a0+^->0, which com-2n n n
pletes the proof.
REMARK 2.6. We have many examples of Einstein metrics which satisfy
the condition of the operator L in Theorem 2.5. (See [5].)
i) ([5, Proposition 3.4]) An Einstein metric which is decomposed locally
to the riemannian manifolds with negative sectional curvature.
ii) ([5, Corollary 3.7]) An Einstein metric whose sectional curvature
ranges in the interval ί — - , 1 .
iii) ([5, Corollary 3.5]) An Einstein metric which is decomposed locally to
the irreducible symmetric spaces of non-comabt type of dimension >2.
iv) ([5, Table 1, Table 2]) The irreducible symmetric spaces of the follow-
ing types.
AΠI St/φ+iyS^xE/O
BDI SO(p+q)/SO(p)χSO(q)
CII Sp(p+l)ISp(p)xSp(l)
Dili SO(2p)IU(p)
EVII Ei/EsXT1
3. The case of Kahler deformation
Let (M,J) be a compact complex manifold of dimension m=—. We
consider 1 -parameter families of Kahler metrics on (M,J), which we call Kahler
deformations. For a Kahler metric g on My we denote by ω and p the Kahler
form and Ricci form of g, respectively, i.e., ωij=Jiί9 Pij=pikjkj If g(t) is a
Kahler deformation, then ω' is a closed real 2-form and p' is a 0-cohomologous
closed real 2-form. First we will show some formulae. For the integral of
closed forms, recall that the exterior product of a closed form and 0-cohomolo-
gous form is 0-cohomologous, and the integral of a 0-cohomologous 2w-form
vanishes. By easy computation we see
SECOND DERIVATIVE OF THE TOTAL SCALAR CURVATURE 419
(3.1)
, ω)ω", (3.2)
{(φ, ω)(ψ, ω)-2(φ, ψ)}ωM , (3.3)
-
Zm
2
 *
where φ and i/r are 2-forms on M. By intergrating both sides of (3.1), we get
»". (3 4)
and so - — α / Λ ω - 1 , (3.5)
(/W — 1)1
(ί°*)//= π^ ( ω//Λω""1+Γ^Ϊ ( ω/Λω'Λω-2 . (3.6)\J / w— 1 ! J w— 2 ! J) ( )
Since τ=(/δ, ω), we see, by the formula (3.2),
(ί- )'-
We assume the deformation is a Kahler deformation, hence β' is 0-cohomolo-
gous, which implies
Moreover if /5 is cohomologous to £ω for some real number £, then
By the formulae (3.4) and (3.7), if gl and g2 are Kahler metrics such that
their Kahler forms ω
γ
 and ω2 are cohomologous to each other, then \ vκ =
JM lS vσr) and I T vff =\ τ v f f 9 . Therefore we can consider "critical classes'*M g2 JM gl l JM g2 8Z
for the scalar curvature.
Theorem 3.1. Let (M, g) be a Kahler manifold. Then
420 N. Koiso
for any volume preserving deformation g(t) of gy if and only if there exists a real
number 8 such that p is cohomologous to £ω.
Proof. If p is cohomologous to £ω, then the formula (3.8) implies
( \τv
σ
) = -- \ ω'/\ωm~l. On the other hand, the deformation is volume
\J V (iw— 2)1 J
preserving, hence the formula (3.5) implies \ω'/\ωm~1 = Q. Thus we see
,
If ί \ vrg j =0 for any volume preserving Kahler deformation, \ p/\φ/\ωm~2
=0 for any closed real 2-form φ such that \ φ/\ωm~1=Q. Thus if we define 1-
formsp and q on the space of all real closed 2-forms by p(φ)= \ p/\φ/\ωm~2 and
q(φ) = I φ/\ωm~l then there is a real number c such that p=cq, i.e.,
for all real closed 2-forms φ. Let ψ be the harmonic part of p—cω. Then
, ω)— (Δ v/r, ω)=0 and so ( v/r, ω) is constant. Moreover, by the formula (3.3),
and j {(φ, ω)(ψ, ω)-2(φ,»}ω" - 0 .
We set φ=(λjr, ω)ω— 2ψ, then
I ((Λ/Γ, ω)ω — 2ψ, (Λ/Γ, ω)ω — 2 v/r)ωw = 0,
which implies (Λ/Γ, ω)ω— 2-v/r=0 and so p is cohomologous to |c+ — (i/r, ω) >ω.
Theorem 3.2. JΓ/" ίA^r^ exists a positive (resp. negative) number 6 such that
d2p is cohomologous to 8ω} then
at
fI τgωvg(t) ύ positive (resp. negative) for allJ
volume preserving Kάhler deformations g(t) such that ω'(0) is not 0-cohomologous.
Proof. Since g(t) is volume preserving, the formulae (3.5) and (3.6)
implies ί ω'Λω^-^O and ( ω/f /\ωm~l + (m— 1) ( ω' /\ω' /\ωm~2= 0. Then, by
the formula (3.9),
SECOND DERIVATIVE OF THE TOTAL SCALAR CURVATURE 421
Let ψ be the harmonic part of ω'. Then, by the formula (3.3),
But here ( ψ , ω) is constant and \ ψ Λω"1"1 =0, hence (ψ, ω)=0. Thus we see
'= έ ί
 (
* *>""
Since ω' is not 0-cohomologous, we see -ψ* is a non-zero real 2-form, which
completes the proof.
OSAKA UNIVERSITY
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a riemannian manifold, J. Differential Geometry 3 (1969), 379-392.
[3] D.G. Ebin: The manifold of riemannian metrics, Global Analysis (Proc. Sympos.
Pure Math.) 15 (1968), 11-40.
[4] D. Hubert: Die Grundlagen der Physic, Nachr. Akad. Wiss. Gottingen (1915), 395-
407.
[5] N. Koiso: Non-deformability of Einstein metrics, Osaka J. Math. 15 (1978), 419-
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On the second derivative of the total scalar curvature

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