KOISO, NORIHITO

Koiso, Norihito

Osaka University Knowledge Archive

A decomposition of the space Μ of Riemannian metrics on a manifold

Osaka City University Repository

Koiso, N.
Osaka J. Math.
16 (1979), 423-429
A DECOMPOSITION OF THE SPACE Ά OF
RIEMANNIAN METRICS ON A MANIFOLD
NORIHITO KOISO
(Received March 6, 1978)
0. Introduction
Let M be a compact C°°-manifold. We denote by c ί^, S) and £F the space of
all riemannian metrics on M, the diffeomorphism group of M, and the space of
all positive functions on M, respectively. Then the group 3) and £F acts on
<3tt by pull back and multiplication, respectively. D. Ebin and N. Koiso
establish Slice theorem [4, Theorem 2.2] on the action of 3).
In this paper, we shall give a decomposition theorem on the action of £F
(Theorem 2.5). That is, there is a local diffeomorphism from ΞF X C into JΛ
where if is a subspace of <3H of riemannian metrics with volume 1 and of con-
stant scalar curvature τg such that ^=0 or τg\(n—1) is not an eigenvalue of
Δ^. Combining the above theorems, we get the following decomposition of a
deformation (Corollary 2.9). Let g^C and g(t) be a deformation of g. Then
there are a curve f(t) in £F, a curve j(t) in S) and a curve g(t) in C such that
δ£'(0)=0, which satisfy the equation g(t)=f(t)*f(t)*g(t). (For the operator δ,
see 1.)
The author wishes to express his thanks to the referee.
1. Preliminaries
First, we introduce notation and definitions which will be used throughout
this paper. Let M be an w-dimensional, connected and compact C°°-manifold,
and we always assmue n^2. For a vector bundle T over M, we denote by Hr(T)
the space of all /Γ-sections, where Hr means an object which has derivatives
defined almost everywhere up to order r and such that each partial derivative
is square integrable. Then Hr(T) is isomorphic to a Hubert space and the
space C°°(T) of all C°°-sections becomes an inverse limit of {Hr(T)}
r
^
 2....
Therefore such a space is said to be an ILH-space. If a topological space 3£ is
isomorphic to an ILH-space locally, 3£ is said to be an ILH-manifold. For
details, see [5].
424 N. Koiso
Let g be an /Γ-metric on M. We consider the riemannian connection and
use the following notations:
vg\ the volume element with respect to g,
R the curvature tensor,
p; the Ricci tensor,
(For the standard sphere with orthnormal basis, jRi2i2==^i2i2<0 and /°n<0 )
τ; the scalar curvature,
( , ) the inner product in fibres of a tensor bundle defined by g,
ζ , />; the global inner product for sections of a tensor bundle over My
M
S2\ the symmetric covariant 2-tensor bundle over M,
Hr(M)
 y the Hubert space of all /Γ-functions,
Hrg(M)\ the Hubert space of all /Γ-functions / such that \ /^=0,
J M
Hg(S2)j the Hubert space of all symmetric bilinear /ίr-forms h such that
<h,g>=0,
V; the covariant derivation,
δ; the formal adjoint of V with respect to < , >,
δ*; the formal adjoint of δ\Hr(S2),
Δ=δd the Laplacian operating on the space /Γ(M),
Δ=δV; the rough Laplacian operating on the space Hr(Tpq),
Hess= W; the Hessian on the space Hr(M),
3\ the ILH-manifold of all positive C°°-functions on M,
£Fr; the Hubert manifold of all positive /Γ-functions on M,
JK; the ILH-manifold of all C°°-metrics on M,
c_5$f the Hubert manifold of all Hr -metrics on M,
<3A,\\ the ILH-manifold of all C°°-metrics with volume 1,
c5/ί; the Hubert manifold of all ίίr-metrics with volume 1.
When we consider the metric space c5Ks, the covariant derivation, the cur-
vature tensor and the Ricci tensor with respect to an element g of JMS will be
denoted by V^, Rg or pg. By a deformation of g we mean a C°°-curve g(t): !^><3ά
such that £(())=£, where / is an open interval. The differential #'(()) is called
an infinitesimal deformation , or simply an i-deformation. If there is a 1 -parameter
family γ(£) of diffeomorphisms such that g(t) = γffig then the deformation
g(t) is said to be trival. If there is a 1-form ξ such that h=S*ξ, then the
i-deformation h is said to be trivaL On the other hand, an i-deformation h is
said to be essential if δλ=0.
Now, we give some fundamental propositions.
Lemma 1.1 [6,11.3]. Let E and F be vector bundles over M and f: E-*F
fiber preserving C°°-map.
is defined by φ(a)=fo<χ is C°°.
be a fi r  If s>~ , then the map φ: H\E)-*HS(F) which
DECOMPOSITION OF THE SPACE M 425
Proposition 1.2. Ifs>—, then the map D: JH. +1χH +1(Tζ)-*H'(Tζ+ύ
which is defined by D(g, ξ)=Vgξ is C°°.
Proof. Let g0 be a fixed C""-metric on M. We define the tensor field
T(g) by T(g)(X, Y) = (Vg)x Y — (Vgo)x Y for an ^ -metric g on M. Then we get
and (D(g,
By the definition of the //'-topology, we know that the map g-^(Vg())g is a
C°°-maρ from JU3+1 to //5( Γ°) . Hence Lemma 1 . 1 implies that the map : g -> T(#)
is a C°°-map from J#f+1 to HS(T\). Applying Lemma 1.1 to the above for-
mula, we see that the map : (T(g), ξ)-*D(g, ξ)—D(g0, ξ) is a C°°-map from
//S(Γ|)X//S+1(Γ^) to Hs(Tpq+1). But the map : ξ^D(g0, ξ) is a continuous
linear map from Hs+1(Tpq) to Hs(Tpq+1), hence the map: (T(g), ξ)->D(g, ξ) is C°°.
Thus we see that the map D is a composition of C°°-maps, and so is C°°.
Corollary 1.3. If ί>— , then the map : (g,f)->Vgf w a C°°-map from
to HS(M).
Proof. We apply Proposition 1.2 to the formula Δgf = —giiVidjf.
Corollary 1.4. If s>^- , then the maps : £->#, p, T are C~-mapsfrom ^Ms+2
to //S(ΓJ), HS(S2) and H*(M), respectively.
Proof. The smoothness of the map : g^R completes the proof. By
easy computation, we get the next formula :
+(T(g))'
ίm
(T(g}Yik-(T(g))ljm(T(g)γik.
Thus, applying Proposition 1.2, we see that the map : g—*R is C°°.
Lemma 1.5 [9,(19.5); 1,(2.11) (2.12)]. Let g(t) be a deformation of g.
If we set h=g'(0), then we have the following formulae;
-j-loV*) = Δtr h+8δh~(h, p) , (1.5.1)
at
A. \
 oPgω = ±{Zh+2Qh+2Lh-2S*8h-Hess tr h,} (1.5.2)
where 2(QK)ii=pikhkj+pj"hik and (Lh)tl=RltiJtl.
426 N. Koiso
2. A decomposition of the space <3M
We denote by Cr the space of all /Γ-metrics with constant scalar curvature
and with volume 1. Fix a C°°-metric g^ <3tt^ For an integer r> — +4
and £e<_5ϊ/ί, we define a C°°-map
In fact the map: (g,f)-*<rrg(f} is a C°°-map from <3tt\ X HgQ(M) to Hrg~\M] owing
to Corollary 1.3 and Corollary 1.4. First we show some lemmas.
Lemma 2.1. If we denote by Kr the subset of <Sά\ of all metrics g e <3tt\
such that σrg is an isomorphism, then K
r
 is open in <3tt\.
Proof. The map : g^σg is a C°°-map from <3tt\ to the space L(HgQ(M),
Hr-\M)) of all continuous linear maps from HgQ(M) to Hr~\M). On the other
hand the set of all isomorphisms is open in L(HgQ(M), Hg~4(M)), hence Kr
is open Jίίί.
Lemma 2.2. Let C be the subset of Jά of all metrics g with constant scalar
curvature rg such that τ^=0 or τg/(n — 1) is not an eigenvalue of Δ^. Then
Proof. Let^eC. Then g^CrΓ\ ^H9 and so it is sufficient to prove that
If/eKer σg then (n—l)(Δg)2f—τgAgf is a constant. By integration
we see
But here τg=Q or rg is not an eigenvalue of Δ^. Hence Δgf is a constant,
and so the assumption that f^HgQ(M) implies /=Q. Thus we see σg is
injective. On the other hand Im {(n— 1)(Δ^)2— τgΔg} =Hr~\M) implies σg
is surjective. Therefore Cc.Cr Π Kr Π c5K, and by the definition of £7 and Kr we
Lemma 2.3.(1) Cr Π Kr is an submanifold of
Proof. We define a C°°-map Δ^: M\-*Hr-\M) by
Then Cr=(Δτ)-1(0). By differentiation we get
(1) A.E. Fischer and J.E. Marsden [8, Theorem 3] show that the space R C becomes a
submanifold of JM.
DECOMPOSITION OF THE SPACE M 427
Let g e £r. Then we get
flί
If h is conformal, i.e., there is f^Hg(M) such that h=fgy by substituting to
the formula (1.5.1) we get
Thus we get Γ,(Δτ) (fg)=σg(f), and Γ,(Δτ) is surjective. This implies, by
implicit function theorem, CrΓ\Kr is a submanifold of c3/ί, and so of <3lf.
Lemma 2.4. Zfc/&ι* a C~-map %" : 3rx(C' 'f}Kr)-^^ίr by Xr(f,g)=fg.
then T(f> gfiC is an isomorphism.
Proof. Injectivity. We see
(T(ftg)Xr)(φ9h)=fh+φg.
Iffh+φg=Q, then φ£<ΞKer Γ^(Δτ), where φ=— φ//. Hence
therefore (n—
But here g^C, which implies $—0, and so λ=0, φ— 0.
Surjectivity. The equation ImT(ftg)%r=fTg(Cr) + Hr(M)g shows that
ImΓ(/ g))C is closed in Hr(S2). Hence, if T(/ }^%r is not surjective then there
exists a non-zero element h in Hr(S2) orthogornal tofTg(Cr) and Hr(M)g. We set
Then we get 71^(Cr)=Ker Γ^(Δ)=Ker Γ^(Δτ)=Ker ^ .^ On the other hand
Kg has surjective symbol. Hence [2, Corollary 6.9] implies that Hr(S2) has the
decomposition
Hr(S2) = Rg®Tg(Cr)®Im Kg* ,
where Kg* is the formal adjoint of Kg. fh is orthogonal to Tg(Cr) and Hr(M)gy
hence fh^ImKg*. If we setfh=Kg*(ty), then we see
Since f j i is orthogonal to Hr(M)g, we see
428 N. Koiso
By the assumption thatgeC, we see Δ^Λ/r=0 and so fh=Q, which contradicts
the assumption that AΦO.
Theorem 2.5,(2) The space C is an ILH-submanifold of JM and the map
% : SfxC-^i^M is a local ILH-dίffeomorphίsm into 3tt, where % is defined by
(For the notation ILH, see [5, pp. 168-169].)
REMARK 2.6. J.L. Kazdan and F.W. Warner [3, Theorem 1.1] show that
C is not empty.
REMARK 2.7. When w=2, this result is classical. That is, any metric g is
conformal to some metric with constant scalar curvature.
Proof. We fix a sufficiantly large integer r. By Lemma 2.2, Lemma
2.4 and the inverse function theorem there is an open neighbourhood Wr of
£F X C in 2"r X (Cr Π Kr) such that %r | Wr is a local diffeomorphism. We denote by
Cr the set of all metrics g e Cr (Ί -Kr such that there is an if '-function / such that
(f9g)ZΞWr. For an integer s^r we set Cs=Cr Π Q (C'Tl-K1). We easily see
that £5:D£S+1 and, by Lemma 2.1, that Cs is open in Cs Γ\KS. Moreover we see
5
— Cby Lemma 2.2, and thus we can define an ILH-structure on C as C—
Next we shall prove that the map X'lΞF'xC* : ΞFsxCs-^^ίs is a local
diffeomorphism. Lemma 1.1 implies the smoothness of this map. To prove
the smoothness of the inverse map, we choose an open covering {Wr
Λ
} of Wr
such that %r I Wr
Λ
 is a diffeomorphism. We apply the following lemma to
(xr\wryl.
Lemma 2.8 [4, Lemma 2.8]. Let E and F be vector bundles over M associ-
ated to the frame bundle of M. Ther*, there exists a cannonίcal linear map
77* : H\E)-*H\E) for a diffeomorphism ΎJ of M. Let A be an open set of
Hr(E) and φ : A-+Hr(F) be a C°°-map which commutes with any η*. If we set
As=AΠHs(E)for s^r, then φ(As)dHs(F) and the map φ\As : AS-+HS(F) is C°°.
If we set Im(%r| Wr
Λ
)=A and (%r| Wr
Λ
)"1=φ, then φ is a C°°-map from A
into Hr(M)xHr(S2) which commutes with the action of the diffeomorphism
group £D of M. Hence Lemma 2.8 implies that the map
(2) J.P. Bourguignon [7, VIII. 8. Proposition] shows that τ : <9tt-* £? is a submersion around
a metric g^JM such that τg is not non-negative constant.
DECOMPOSITION OF THE SPACE JM 429
(%Ί Wr
Λ
γ
ι\As : As-*Hs(M)xHs(S2)
is C°°. But here 3sxCs is a submanifold of Hs(M)xHs(S2), hence the map
()C\Wrγl\As :As-*3sχCs is C°°. Thus %s is a local diffeomorpnhism and
%=lim %5 is an ILH-diffeomorphism, which implies that C is an ILH-submani-
Corollary 2.9. Let g=fg, where f^^F and g^C. If g(t) is a deformation
of g with sufficiently small domain of ty then there exist a \-parameter family of
positive functions f(t) on My a \-parameter family of diffeomorphisms γ(£) of M and
a deformation g(t) in C such thatf(ϋ)=f, δ£'(0)=0 and g(t)=f(t)γ(t)*g(t).
Proof. By Theorem 2.5, g(t) is decomposed into f(t)g(t), where g(t) is
a deformation in C. Applying Slice theorem [4, Theorem 2.2] to g(t), we
get g(t)=y(t)*g(t), where g(t) is a deformation such that δ£'(0)=0. Also we
easily see that g(t) e C for each t .
The author wishes to express his thanks to Professor J.P. Bourguignon
for his kind informations concerning with Lemma 2.3 and Theorem 2.5.
OSAKA UNIVERSITY
References
[1] M. Berger: Quelques formules de variation pour une structure riemannienne, Ann.
Sci. Ecole Norm. Sup. 4e serie 3 (1970), 285-294.
[2] D.G. Ebin: The manifold of riemannian metrics, Global Analysis (Proc. Sympos.
Pure Math.) 15 (1968), 11-40.
[3] J.L. Kazdan and F.W. Warner: Scalar curvature and conformal deformation of
riemmanίan structures, J. Differential Geometry 10 (1975), 113-134.
[4] N. Koiso: Non-deformability of Einstein metrics, Osaka J. Math. 15 (1978), 419-
433.
[5] H. Omori: On the group of diffeomorphisms of a compact manifold, Global Analysis
(Proc. Sympos. Pure Math.) 15 (1968), 167-183.
[6] R.S. Palais: Foundations of non-linear functional analysis, Benjamin, New
York, 1968.
[7] J.P. Bourguignon: Une stratification de Γespace des structures riemanniennes,
Compositio Math. 30 (1975), 1-41.
[8] A,E. Fischer and J.E. Marsden: Manifolds of riemannian metrics ivith prescribed
scalar curvature, Bull. Amer. Math. Soc. 80 (1974)3 479-484.
[9] A. Lichnerowicz: Propagateurs et cummutateurs en relativite generate, Inst. Hautes
Etudes Sci. Publ. Math. 10 (1961), 293-344.



A decomposition of the space M  of Riemannian metrics on a manifold

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A decomposition of the space M of Riemannian metrics on a manifold

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