KILLING INITIAL DATA ON TOTALLY UMBILICAL & COMPACT HYPERSURFACES

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& COMPACT HYPERSURFACES

Daniel Maerten

To cite this version:

Daniel Maerten. KILLING INITIAL DATA ON TOTALLY UMBILICAL & COMPACT

HYPER-SURFACES. 2009. �hal-00355882�

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DANIELMAERTEN

Abstra t. Inthisnote,wegiveageometri hara terizationofthe ompa tandtotally umbili alhypersurfa esthat arryanontriviallo allystati KillingInitialData(KID). More pre isely, su h ompa t hypersurfa es

(M

n

_{, g, cg)}

endowed with a Riemannian metri

g

and ase ondfundamentalform

cg

(where

c ∈ C

∞

(M )

apriori)have onstant mean urvatureandareisometri tooneofthefollowingmanifolds:

(i)

S

n

thestandardsphere,

(ii) anitequotientof awarpedprodu t

(S

1 _×

_Y,

_dt

2 _{+ h}

2 _(t)g

0 )

,where

(Y

n−1

_{, g}

0 )

is Einsteinwithpositives alar urvature.

Inparti ular,theyhaveharmoni urvatureandstri tlypositive onstants alar urva-ture.

1. Introdu tion

A good startingpoint for this arti leis thestudy ofthes alar urvatureappli ation

u : M −→ C

∞

(M ), g 7−→ Scal

g

,

where

M

denotes the one of the Riemannian metri s on

M

. If we denote by

U

g

the dierential of

u

at a metri

g ∈ M

, then for any

h ∈ Γ(S

2 _T

∗

_{M )}

we have (see [7℄ for details)

U

g

(h) = ∆(tr

g

(h)) + δ(δh) − hRic

g

, hi ,

where

∆

isthe positiveLapla ianwithrespe tto

g

,

h·, ·i

istheinnerprodu textendedto tensorsofanytype and

δ

isthe

g

divergen e operator dened as

δS(X

1 , · · · , X

p

) = −

n

X

i=1

∇

e

i

S(e

i

, X

1 , · · · , X

p

) ,

forany

(p + 1)

tensor

S (p ≥ 0)

andforanylo alorthonormal

g

basis

(e

i

)

n

i=1

. Theformal adjoint of

U

g

isgiven by

∀f ∈ C

∞

U

_g

∗

(f ) = Hess

g

f − f Ric

g

+(∆f )g .

Fromageometri pointofview,thefa tthat

Ker U

∗

g

6= {0}

isane essary onditionforthe s alar urvatureappli ation

u

nottobeasubmersionat

g

,andsoane essary onditionfor thelevelset

u

−1

_(Scal

g

_{) ⊂ M}

notto bea submanifold(inthesenseofinnitedimensional submanifolds, seeon eagain [7 ℄). Anatural issueisto study the ompa t and onne ted manifolds

(M

n

_{, g)}

thatadmit anon trivialfun tion solutionto theequation

Hess

g

f − f Ric

g

+(∆f )g = 0

(∗) ,

Date:January26, 2009.

Keywordsandphrases. S alar urvature,harmoni urvature,Einsteinmetri s,S hwarzs hilddeSitter metri s, onstraintsappli ation, Lafontaineequation.

2000 Mathemati sSubje tClassi ation. 53C24,53C25,53C80,83C05,83C15.

TheauthorbenetsfromtheANRgrantGéométriedesvariétésd'Einsteinnon ompa tesousingulières, n

◦

(3)

whi h has been extensively studied (notably by Lafontaine, that is why we will all

(∗)

the Lafontaine equation). As a matter of fa t, Fis her and Marsden [10℄ believed that thestandard sphere

S

n

wastheunique ompa t and onne ted Riemannian manifold (ex- ept the Ri i at ompa t manifolds) that have non trivial solutions to the Lafontaine equation[10 ℄. Their onje tureisfalse, sin eLafontaine listedin[12 ℄ allthe ompa t and onformallyatmanifoldsadmittingsolutionsto

(∗)

,inparti ularsomenonRi iat,non spheri al warped produ t metri s appear inthat list. However, without the onformally atnessassumption, we do not have to our disposal an exhaustive list of manifolds that havenontrivialsolutionsof

(∗)

. Thisquestionstillremainsopennowadays. Another inter-estinggeometri interpretationof

(∗)

thatisparti ularly relevant fromGeneralRelativity, dealswiththe lassi ation ofstati solutions ofthe Einsteinequations. Morepre isely,if

γ = −f

2 dt

2 + g

isastati Lorentzianwarpedprodu tmetri on

R

× M

,thenthemetri

γ

isEinstein (whi h is equivalent to be asolution of theEinstein va uum equations)ifand only if

f

is asolution of

(∗)

. Thus, determining thesolutions of

(∗)

is stri tly equivalent to lassifythestati solutions ofEinsteinva uumequations. Thisfa tis loselyrelatedto theLorentzian metri s obtained thanksto aKillingdevelopment (seebelowfor details). An important feature is that

(∗)

redu es (up to a normalization) to Obata's equation

Hess

g

f = −f g

whenthemetri

g

is assumedto be Einstein, andthereforeitis a general-ization of Obata's equation. It is well known that theexisten e of a non trivial fun tion solutionof Obata'sequation hara terizes thegeometry of thestandard sphere

S

n

[14 ℄. Whenwexametri

g

0 ∈ M

and restri t

u

tothe spa e

V

ofmetri sthathavethesame Riemannianvolume formthan

g

0

,namely

V

= {g ∈ M / dVol

g

= dVol

g

0 }

,thenif

u

isnot asubmersionat

g

then

Ker U

∗

g

0 6= {0}

with

U

∗

g

0

thetra elesspartof

U

∗

g

namely

∀f ∈ C

∞

U

_g

∗

(f )

₀

= Hess

g

f − f Ric

g

₀

+

∆f

n

g ,

where

Ric

g

0

denotes thetra eless partoftheRi i urvature. In[12 ℄,Lafontaine exhibited alarge family ofmetri s thatadmit non trivialsolutions to

Hess

g

_{f − f Ric}

g

0 +

∆f

n

g = 0

. One an also be interested inthe (positive) onstants that ould be the s alar urvature of a metri

g

among the metri s of a ertain xed volume, let us say

Vol(S

n

₎

. This is equivalent to restri t

u

to thespa e

C

= {g ∈ M /d Scal

g

= 0

and

Vol(M, g) = Vol(S

n

_)}

. In that ase, if

u

is not a submersion at

g

then there exists a non trivial solution

f

of

U

_g

∗

(f ) = Ric

g

₀

. Allthese results aresummarizedinthefollowing statement.

1.1.Theorem. Let

M

n

be a ompa t and onne ted manifold

(n ≥ 3)

.

(i) (Obata [14 ℄) If u is not a submersion at an Einstein metri

g ∈ M

then

(M

n

_{, g)}

is isometri to a standard sphere and

Ker U

∗

g

= Span{x

1 , x

2 , · · · , x

n+1

}

with

(x

i

)

n+1

i=1

the standard oordinates on

S

n

.

(ii) (Lafontaine[12℄)If uisnotasubmersion ata onformally at metri

g ∈ M

then

(M

n

_{, g)}

is isometri toone of the following manifolds: 1) A at ompa t manifoldand

Ker U

∗

g

= R

, 2) The standard sphere

S

n

and

Ker U

∗

g

= Span{x

1 , x

2 , · · · , x

n+1

}

. 3) A nite quotient of

(S

1 _{× S}

n−1

_{, dt}

2 _{+ g}

S

n−1

)

and

dim Ker U

∗

g

= 2

. 4) Anitequotientofawarpedprodu t

(S

1 _×S

n−1

_{, dt}

2 _+h

2 _(t)g

S

n−1

)

and

Ker U

∗

g

= Rh

′

. (iii) (Lafontaine [12℄) If

Y

n−1

is a ompa t and onne ted manifold that arries an Einstein metri of positive s alar urvature, then there exists an innite dimen-sionalset in

V

(S

1 _{× Y ) = V}

(4)

(iv) (Lafontaine[12℄)If

u

|C

isnotasubmersionata onformallyatmetri

g ∈ C

then

(M

n

_{, g)}

is isometri tothe standard sphere

S

n

.

(v) (BessièresLafontaineRozoy [4 ℄) If

n = 3

and

u

|C

is not a submersion at a metri

g ∈ C

then

(M

3 _{, g)}

isisometri to the standard sphere

S

3

.

A natural generalization of these questions is obtained by onsidering the natural ex-tension(due to theGeneral Relativity ontext) ofthe s alar urvature appli ation

u

: the onstraints appli ation

Φ

. More pre isely, if we onsider

(M

n

_{, g, k) ⊂ (N}

n+1

_{, γ)}

a Rie-mannian submanifold in a Lorentzian manifold, endowed with the indu ed metri

g

and these ond fundamental form

k

,the onstraintsappli ation isgivenby

Φ : (g, k) 7−→

Scal

g

+(tr

g

k)

2 − |k|

2 g

−2(δk + dtr

g

k)

=

Φ

1 (g, k)

Φ

2 (g, k)

∈ C

∞

(M ) × Γ(T

∗

M ),

where

(g, k) ∈ M × Γ(S

2 _T

∗

_{M )}

. Whenthe Lorentzian manifold

(N

n+1

_{, γ)}

isa solutionof theEinsteinequations (it isnot ne essarily the asefor our problem) that isto saywhen themetri

γ

satises

Ric

γ

−

1

2 Scal

γ

_{γ = T ,}

where

T

isthe stressenergytensor, thenthe Hamiltonian onstraint

Φ

1 (g, k)

orresponds to

T (∂

t

, ∂

t

)

and the moment onstraint

Φ

2 (g, k)

orresponds to (up to a multipli ative onstant)

T (∂

t

, ·)

|T M

(here

∂

t

denotes a unit normal to

M ֒→ N

). The other pie es of thetensor

T

areusually alled the Einstein evolutionequations and are hara terized by the o urren es of the

∂

t

derivatives of the metri

γ

(on the ontrary to the onstraints equations). The onstraintsequations arealsothetra edGaussand Codazziequations of theembedding

M ֒→ N

.

We denote by

L

(g,k)

(

= L

in short) the dierential of

Φ

at

(g, k)

, and by

L

∗

its formal adjoint. Analogouslyto thes alar urvatureappli ation

u

,

Φ

isnot asubmersionat some ouple

(g, k)

ifand only if

Ker L

∗

(g,k)

6= {0}

. The formal adjoint of

L

(g,k)

is given by the following rather ompli ated oupled dierential operator

(

L

∗

₁

(f, α) = E(f, α) − (tr

g

E(f, α)) g −

1 ₂

hL

∗

2 (f, α), ki + hα, Φ

2 (g, k)i + 2f Φ

1 (g, k)

g

L

∗

₂

(f, α) = −2(δ

∗

α + f k) + 2tr

g

(δ

∗

α + f k)g

,

where

(f, α) ∈ C

∞

_{(M ) × Γ(T}

∗

_{M )}

and

E(f, α) := Hess

g

f − f (Ric

g

+2(tr

g

k)k − 2k ◦ k) + L

α

k + (δα)k .

Here

L

stands for the Lie derivative,

k ◦ k

means

(k ◦ k)

ij

= k

ir

k

sj

g

rs

and nally

δ

∗

_α

is the symmetri part of the ovariant derivative

∇α

i.e.

δ

∗

_{α(X, Y ) =}

1

2 ∇

X

α(Y ) +

∇

Y

α(X)

. The ondition

Ker L

∗

(g,k)

6= {0}

is equivalent to the existen e of a non trivial ouple

(f, α) ∈ C

∞

_{(M ) × Γ(T}

∗

_{M )}

su h that

(

Hess

g

_{f + L}

α

k = f (Ric

g

+(tr

g

k)k − 2k ◦ k) −

_2(n−1)

1 hα, Φ

2 (g, k)i + 2f Φ

1 (g, k)

g

L

_α

_{g + 2f k = 0}

.

Inthis ontextwe anaddressthefollowingissue on erningtheexisten eofKillingInitial Data (KID, whi h are by denition [13℄ the non trivial elements of

Ker L

∗

(g,k)

or equiva-lently thenontrivial solutions ofthedierential systemabove).

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Question: Doesthe existen eof anon trivial KID(i.e. anelement in

Ker L

∗

(g,k)

) hara -terizethegeometry oftheRiemannian hypersurfa e

(M

n

_{, g, k)}

?

This problemisvery di ultingeneral, howeverBeig, Chru± ieland S hoen proved in [2℄ that KIDs are non generi for a large lass of sli es

(M

n

_{, g, k)}

. More expli itly, they proved that the family

n

(g, k) ∈ M × Γ(S

2 _T

∗

_{M )/ Ker L}

∗

(g,k)

= {0}

o

, was an open and dense setfor a

C

k,β

_{× C}

k,β

type topology. Unfortunately,they were not able to give the geometry of

(M

n

_{, g, k)}

under the ex eptional ondition that

Ker L

∗

(g,k)

is non zero. The aimof thepresent paperisto study aparti ular situation that generalizestheLafontaine equation, namelythe KID equations on a ompa t andtotally umbili al hypersurfa e. In otherwords, we onsider

(M

n

_{, g, k)}

su h that 1)

∃c ∈ C

∞

_{(M ), c 6≡ 0,}

_{k = cg}

, 2)

∃(f, α) 6≡ (0, 0) L

∗

(g,k)

(f, α) = 0

.

Itis learthatthestandard( onstantmean urvatureandtotallyumbili al)spheri alsli es indeSitterspa etimesatisfyallthese onditions. AnalogouslytotheFis herandMarsden onje ture,one ouldthinkthatthe onditions1)and2) hara terizethespheri alsli esin deSitter. This onje tureisfalseon eagainbe auseofaresultowedtoLafontaine. Tosee thatwe rst need to reformulate our problem. Invirtue ofof theumbili ity ondition1), themoment onstraintsbe omes

Φ

2 (g, cg) = −2(n − 1)dc

. TheHamiltonianonefor esthe s alar urvature of

g

to satisfy:

Φ

1 (g, cg) = Scal

g

_{+n(n − 1)c}

2

. The se ond KIDequation is exa tly

L

α

g + 2cf g = 0

, whi h means that

α

is a Killing onformal 1form (it is non isometri assoon as

f 6≡ 0

). Finally,therst KIDequation isequivalent to

Hess

g

f

= f (Ric

g

+nc

2 g − 2c

2 g) − L

α

(cg)

−

1 2(n − 1)

− 2(n − 1) hα, dci + 2f (Scal

g

_{+n(n − 1)c}

2 ₎

_g

Hess

g

f

= f (Ric

g

+nc

2 g − 2c

2 g) − cL

α

g − ∇

α

(cg) + hα, dci − f

Scal

g

n − 1

g − nf c

2 g

Hess

g

f

= f Ric

g

−f

Scal

g

n − 1

g ,

whi h is equivalent to

U

∗

g

(f ) = Hess

g

f − f Ric

g

+(∆f )g = 0

where we have used the relationobtained thanksto the tra edequation. Asa onsequen e, the onditions 1) and 2)are equivalent to thesystem

(Σ)

L

_α

_{g + 2cf g = 0}

U

∗

g

(f ) = 0

The onje ture à la Fis her and Marsden now reads as: the only metri

g

(ex ept the Ri i at) that has non trivial solutions to

(Σ)

is the standard metri on the sphere

S

n

. Thisis learly false sin e any ompa t manifold

(M

n

_{, g)}

having a non zeroKillingeld

α

provides anon trivial solutionof

(Σ)

oftheform

(0, α)

and whatever themean urvature

c

is. Consequently, from now on we dene as a non trivial KID a solution

(f, α)

su h that

f 6≡ 0

. Even underthis newdenitionofnon trivialKID,the onje tureà la Fis her andMarsden isfalse invirtueof thefollowing result(whi h isa orollaryof lassi ation resultsof Derdzinski about ompa t manifolds withharmoni urvature[8, 9℄).

(6)

1.2. Proposition (Lafontaine [12 ℄). Let

(M

n

_{, g)}

be a ompa t manifold

(n ≥ 3)

with harmoni urvature whi h arries a losed Killing onformal and non isometri 1-form, then

Ker U

∗

g

6= {0}

and

(M

n

_{, g)}

isisometri to one of the followingmanifolds: (i)

S

n

the standard sphere,

(ii) a nite quotient of a warped produ t

(S

1 _{× Y, dt}

2 _{+ h}

2 _(t)g

0 )

, where

(Y

n−1

_{, g}

0 )

is Einsteinwith positive s alar urvature.

Indeed,undertheassumptionsofProposition1.2,theharmoni urvatureandthese ond Bian hiidentity ompelthe s alar urvature

Scal

g

to bea (positive) onstant. Wedenote by

α

the losedKilling onformalandnonisometri 1formi.e.

dα = 0

and

L

α

g + ϕg = 0

(thisequationwasstudiedbyTashiro[15 ℄)fora ertainfun tion

ϕ 6≡ 0

. Itisprovedin[12 ℄ that

U

∗

g

(ϕ) = 0

,and therebythe ouple

(α, f :=

ϕ

2c

)

for any onstant

c ∈ R

∗

,is asolution of

(Σ)

as soon as

(M

n

_{, g)}

is isometri to one of the manifolds listed in Proposition 1.2 . Moreover, the sli e

(M

n

_{, g, cg)}

is onstant mean urvature and satises the va uum on-straintsequations withthepositive osmologi al onstant

Λ =

1

2 Scal

g

_{+n(n − 1)c}

2 _{> 0}

. Therstmainresultofthisarti leisto onsiderasystem

(Σ

1 )

slightly strongerthan

(Σ)

, sin ewemoreoverdemandthe Killing onformalformtobe losed,andgivethegeometry ofthemanifolds that arrynon trivialsolutions of thissystem.

1.3.Theorem. Let

(M

n

_{, g)}

(n ≥ 3)

with

C

3

metri , whi hhas a non trivial solution

(f, α)

of

(Σ

1 )

∇α + cfg = 0

_U

∗

g

(f ) = 0

,

for a ertain

c ∈ C

∞

_{(M ), c 6≡ 0}

. Then is a nonzero onstant and

(M

n

_{, g)}

is isometri toone of the following manifolds:

(i)

S

n

(S

1 _{× Y, dt}

2 _{+ h}

2 _(t)g

0 )

, where

(Y

n−1

_{, g}

0 )

From the KID point of view, the va uum sli e

(M

n

_{, g, cg)}

has positive onstant s alar urvature and onstant mean urvature given by

Scal

g

_{+n(n − 1)c}

2 _{= Φ}

1 (g, cg) = const.

. TheoriginaldenitionoftheKIDisowed toBeigandChru± iel[3℄andisthefollowing. ConsideraKillingeld

X

ontheambientLorentzian manifold

(N, γ)

andde omposeitin termsofalapsefun tion

f

andashift1form(orve tor)

α

. ThentheKillingequationof

X

restri tedto the Riemannian hypersurfa e

(M

n

_{, g, k)}

gives birthto a systemof equations satised bythe ouple

(f, α)

whi h is by denitiona Killing Initial Data. Now when the onstraints are va uum (possiblywith a osmologi al onstant) then being a KID in the originalsenseof [3℄isequivalentto belongto

Ker L

∗

(g,k)

. Theexisten eofa KIDallows us to make a Killingdevelopment of

(M

n

_{, g, k)}

i.e. to onstru ta Lorentzian manifold with topology

R

× M = ˜

N

whi h isendowed withthemetri

˜

γ = (|α|

2 − f

2 )dt

2 + 2dt ⊙ α + g ,

where thefun tions

f, α

i

, g

ij

aretriviallyextended alongthe produ t

R

× M

, and

dt ⊙ α

denotes the symmetri part of

dt ⊗ α

. By onstru tion, the metri

γ

˜

is stationary and the spatial sli es

{t = const.}

are isometri to our initial manifold

(M

n

_{, g, k)}

. The im-portant fa t is that the Lorentzian manifold

( ˜

N , ˜

γ)

is a solution of the va uum Einstein equations (possibly with a osmologi al onstant) that arries a Killing eld

∂

t

. In fa t,

(7)

∂

t

extends the ve tor eld

(f ν + α

i

_∂

i

) ∈ Γ(T ˜

N

|M

)

(where

ν

denotes a unitnormal to the sli e

{t = 0} ∼

= (M

n

_{, g, k)}

) along the wholeLorentzian manifold

( ˜

N , ˜

γ)

. As already said, thespa etime

( ˜

N , ˜

γ)

is stationaryby onstru tion, butwhen we assumethatthespatial part

α

is losed, it for es the Killing development to be lo ally stati , inthe sense of the lo alintegrability of the orthogonaldistribution of

∂

t

. In parti ular, by making aKilling development of the metri s that are lassied in the result above, we obtain a lass of stati solutions of the va uum Einstein equations with a positive osmologi al onstant: the S hwarzs hildde Sitter metri s. These metri s had been studied in [6 ℄ with

Λ < 0

(the S hwarzs hildanti de Sitter metri s), but all the omputations of Birmingham an be arriedoutwith

Λ > 0

. Asa on lusion,we an laimthattheS hwarzs hildde Sitter solutions are hara terized by the existen e of non trivial solutions of

(Σ

1 )

, and we will allthemetri slistedinthetheoremabovethespatialS hwarzs hildde Sittermetri s(

S

n

orrespondsto thespatialde Sittermetri s).

These ond mainresult on ernstherestri tionofthe onstraintsappli ationto theset

V

_{× Γ(S}

2 _T

∗

_{M )}

whi h gives rise to theproblem ofnding some

(f, α)

su h that

L

_α

_{g + 2cf g = 0}

Hess

g

f − f Ric

g

₀

+

∆f

_n

g = 0

⇐⇒

(

L

_α

_{g + 2cf g = 0}

U

∗

g

(f ) =

1 n

(n − 1)∆f − f Scal

g

_g

.

It ispointless to prove thatthe fun tion

c

is a onstant asin thesituation of the system

(Σ

1 )

,sin ethereexistsomeexamples(thestandardsphere

S

n

isoneofthem)ofmanifolds

(M

n

, g)

that have non trivial solutions

(f, α)

with a non onstant fun tions

c

. We will dis ussthis point later on. However, we an lassify thesolutions ofthis systemwhen we assumethat

c

isa nonzero onstant andthat theKilling onformalform

α

is losed. We surprisinglyobtain thesame family ofmanifolds than for thesystem

(Σ

1 )

.

1.4.Theorem. Let

(M

n

_{, g)}

(n ≥ 3)

with

C

3 (f, α)

of

(Σ

2 )

( ∇α + cfg = 0

U

g

∗

(f ) =

1 n

(n − 1)∆f − f Scal

g

,

fora ertain

c ∈ R

∗

. Then

(M

n

_{, g)}

is isometri toone of the following manifolds: (i)

S

n

(S

1 _{× Y, dt}

2 _{+ h}

2 _(t)g

0 )

, where

(Y

n−1

_{, g}

0 )

FromtheKIDpointofview,thesli e

(M

n

_{, g, cg)}

isaspatialS hwarzs hilddeSittermetri i.e. it is va uum and has positive onstant s alar urvature and onstant mean urvature given by

Scal

g

_{+n(n − 1)c}

2 _{= Φ}

1 (g, cg) = const.

.

The third main result on erns the restri tion of the onstraints appli ation to theset

C

_{× Γ(S}

2 _T

∗

_{M )}

whi h givesrise tothe problemofnding some

(f, α)

su h that

L

_α

_{g + 2cf g = 0}

U

_g

∗

(f ) = Ric

g

₀

,

whi hmustberelatedtotheresultindimension3ofBessièresLafontaineRozoyevo ated earlier. Here again, we study this system assuming the losed hara ter of the Killing onformalform

α

.

(8)

1.5.Theorem. Let

(M

n

_{, g)}

(n ≥ 2)

with

C

3

metri and onstant s alar urvature. Suppose there existsa nontrivial solution

(f, α)

of

(Σ

3 )

∇α + cfg = 0

_U

∗

g

(f ) = Ric

g

0 ,

c ∈ C

∞

(M ), c 6≡ 0

. Then isanonzero onstantand

(M

n

_{, g)}

isisometri tothestandard sphere

S

n

. Fromthe KID point of view, the sli e

(M

n

_{, g)}

isa spatial de Sitter metri . This arti le is organized as follows: in Se tion 2 we give some te hni al preliminary resultsthatwill be usedinSe tion 3inorderto prove themainTheorems. Se tion 3also ontains spe i resultsinsmalldimension

n = 2

or

3

. Finally,Se tion4isdevotedto the studyof anotherinteresting systemofequations.

2. Preliminary results In this arti le,

(M

n

_{, g)}

isa ompa t and onne ted

n

dimensional manifold withLevi Civita onne tion

∇

. A 1form

α

is said to be Killing onformal and non isometri ifit satisesthe equation

L

α

g = ψg

for a ertainfun tion

ψ 6≡ 0

. Su ha form

α

is not losed ingeneral, but ifitis, then

α

and

dψ

are loselyrelated.

2.1. Proposition. Let

(M

n

_{, g)}

(n ≥ 2)

. Suppose there exist a non trivial ouple

(ψ, α)

su h that

L

α

g = ψg

. If the Killing onformal

α

is losed then

α ∧ dψ = 0

.

Proof. The ovariant derivative

∇α ∈ Γ(⊗

2 _T

∗

_{M )}

is thesum of a symmetri part and a skewsymmetri partsin e

∇α =

1

2 dα + L

α

g

,where

dα(X, Y )

= ∇

X

α(Y ) − ∇

Y

α(X) = d

∇

α(X, Y )

L

_α

_{g(X, Y ) = ∇}

_X

_{α(Y ) + ∇}

_Y

_{α(X) = 2δ}

∗

_{α(X, Y )}

,

and

d

∇

_{S(X, Y, X}

1 , · · · , X

p

) := ∇

X

S(Y, X

1 , · · · , X

p

)−∇

Y

S(X, X

1 , · · · , X

p

)

forany

(p + 1)

tensor

S

. The ondition

d

2 _{α = 0}

implies

∇

Z

dα(X, Y ) = −d

∇

dα(X, Y, Z) .

Weapply the Ri i identityto

α

R(X, Y, Z, α) = ∇

2 _X,Y

α(Z) − ∇

2 _Y,X

α(Z)

= d

∇

(∇α)(X, Y, Z)

=

1

2 d

∇

_{(dα)(X, Y, Z) + d}

∇

_(δ

∗

_{α)(X, Y, Z)}

= −

1

2 ∇

Z

dα(X, Y ) + d

∇

_{(ψg)(X, Y, Z)}

= −

1

2 ∇

Z

dα(X, Y ) + dψ ∧ g(X, Y, Z) ,

where

(ω ∧ S)(X, Y, Z) := ω(X)S(Y, Z) − ω(Y )S(X, Z)

,for every

ω ∈ Γ(T

∗

_{M )}

and every

S ∈ Γ(S

2 T

∗

M )

. By plugging

Z = α

in this relation and assuming that

α

is losed, we immediatelyget

dψ ∧ α = 0

.

We an wonder if we an do without the losed assumption in the Lafontaine Propo-sition 1.2. The proposition above seems to show that we indeed annot. The following

(9)

2.2. Lemma. Let

(M

n

_{, g)}

(n ≥ 2)

that admits a losed Killing onformal and non isometri 1form

α

, i.e.

∇α = ψg

for a ertain

ψ ∈

C

∞

_{(M ), ψ 6≡ 0}

.

Thenthe following equations hold:

R(X, Y, Z, α) = dψ ∧ g(X, Y, Z)

(2.1)

dψ ∧ α = 0

(2.2)

Ric

g

(Y, α) = −(n − 1)dψ(Y )

(2.3)

∇

X

Ric

g

(Y, α) = −

(n − 1)U

g

∗

(ψ) + nψ Ric

g

−(n − 1)(∆ψ)g

(X, Y )

(2.4)

∇

α

Ric

g

(X, Y ) = −

(n − 2)U

_g

∗

(ψ) + nψ Ric

g

−(n − 1)(∆ψ)g

(X, Y )

(2.5)

d

∇

Ric

g

(α, X, Y ) = U

_g

∗

(ψ)(X, Y )

(2.6)

d

∇

Ric

g

(X, Y, α) = 0 .

(2.7)

Proof. Formula (2.1 -2.2) were proved in the previous proposition. Equation (2.3) is ob-tained by taking the tra e in (2.1 ). Equation (2.4 ) is the ovariant derivative of (2.3 ), namely

∇

X

Ric

g

(Y, α) = −(n − 1) Hess

g

ψ(X, Y ) − ψ Ric

g

(X, Y ) ,

where we have expressed

Hess

g

_ψ

in terms of

U

∗

g

(ψ)

. Equation (2.5 ) is a orollary of the rstvariationformulaofthe Ri i urvature( f. formula(d)ofTheorem1.174 in[5 ℄ with

L

_α

_{g = 2ψg}

) i.e.

∇

α

Ric

g

(X, Y ) = −(n − 2) Hess

g

ψ(X, Y ) − 2ψ Ric

g

(X, Y ) + (∆ψ)g(X, Y ) ,

where we have expressedagain

Hess

g

_ψ

interms of

U

∗

g

(ψ)

. Finally, Equation(2.6) is the resultofthe dieren e between (2.5 ) and (2.4 ),and Equation(2.7) istheskew symmetri

partof(2.4).

It is important to noti e thatwe neitherneed to x thevalue of

U

∗

g

(ψ)

nor tosuppose that the s alar urvature is a onstant to have the formulas of Lemma 2.2. The only assumptionistohavea losedKilling onformalandnonisometri 1formonthemanifold. Themanifold

(M

n

_{, g)}

issaidtohaveharmoni urvatureif

d

∇

_Ric

g

_{= 0}

. Inviewof(2.7),it seemsnaturaltoaskiftheexisten eofa losedKilling onformalandnonisometri 1form implies that

g

has harmoni urvature. It is not true ingeneral, but we an nonetheless dedu esome interesting information on

d

∇

_Ric

g

. 2.3. Lemma. Let

(M

n

_{, g)}

(n ≥ 2)

that admits a losed Killing onformal and non isometri 1form

α

, i.e.

∇α = ψg

for a ertain

ψ ∈

C

∞

_{(M ), ψ 6≡ 0}

.

Thenthe following equation holds: (2.8)

ψd

∇

_U

∗

g

(ψ) = dψ ∧ U

g

∗

(ψ) − ψ

2 d

∇

Ric

g

+

ψd(∆ψ) − (∆ψ)dψ

∧ g .

Proof. To proveEquation (2.8 ) we take the ovariant derivative of(2.1 )

∇

X

R(Y, Z, T, α) =

n

Hess

g

ψ(X, Y )g(Z, T ) − Hess

g

f (X, Z)g(Y, T )

o

(2.9)

(10)

Weworkon the expression

U

∗

g

(ψ)

,by omputing

d

∇

_U

∗

g

(ψ)

i.e.

d

∇

U

_g

∗

(ψ) = d

∇

Hess

g

ψ + d(∆ψ) ∧ g − ψd

∇

Ric

g

−dψ ∧ Ric

g

.

TheRi iidentityappliedto

dψ

gives

d

∇

_Hess

g

_{ψ(X, Y, Z) = R(X, Y, Z, ∇ψ)}

,andtherefore (2.10)

d

∇

_U

∗

g

(ψ) = R(·, ·, ·, ∇ψ) + d(∆ψ) ∧ g − ψd

∇

Ric

g

−dψ ∧ Ric

g

.

Besidesifweset

T = α

in(2.9 ) anduse

∇α = ψg

,wend(thankstothesymmetryofthe urvaturetensor)

∇

X

R(Y, Z, α, α) = X · R(Y, Z, α, α)

|

{z

}

=0

− R(∇

X

Y, Z, α, α)

|

{z

}

=0

− R(Y, ∇

X

Z, α, α)

|

{z

}

=0

−ψ R(Y, Z, X, α) + R(Y, Z, α, X)

|

{z

}

=0

.

Thanksto the relation

α ∧ dψ = 0

,we an useon e again(2.9 ) and theprevious formula toget

0 = ∇

X

R(Y, Z, ∇ψ, α) =

n

Hess

g

ψ(X, Y )g(Z, ∇ψ) − Hess

g

ψ(X, Z)g(Y, ∇ψ)

o

−ψR(Y, Z, ∇ψ, X)

= −dψ ∧ Hess

g

ψ(Y, Z, X) + ψR(Y, Z, X, ∇ψ) ,

thatwe write inshort

ψR(·, ·, ·, ∇ψ) = dψ ∧ Hess

g

_ψ

. Finally,by multiplying (2.10 )by

ψ

andusing our urvatureformulait omes out

ψd

∇

U

_g

∗

(ψ) = dψ ∧ Hess

g

ψ + ψd(∆ψ) ∧ g − ψdψ ∧ Ric

g

−ψ

2 d

∇

Ric

g

= dψ ∧ U

_g

∗

(ψ) − ψ

2 d

∇

Ric

g

+

ψd(∆ψ) − (∆ψ)dψ

∧ g .

2.4. Remark . Unfortunately, Formula (2.8) is more ompli ated than we ould have expe ted. Nevertheless, when

ψ

is an eigenfun tion for the Riemannian Lapla ian

∆

i.e.

∆ψ = λψ

for a onstant

λ ≥ 0

then

ψd(∆ψ) − (∆ψ)dψ = 0

, whi h learly simplies(2.8).

The following resultgivesadditionalinformation whenthes alar urvatureis supposed tobe a onstant.

2.5. Theorem. Let

(M

n

_{, g)}

(n ≥ 2)

with onstant s alar urvature that admits a losed Killing onformal and non isometri 1form

α

, i.e.

∇α = ψg

for a ertain

ψ ∈ C

∞

_{(M ), ψ 6≡ 0}

. Thenwe have: (i)

∆ψ =

Scal

g

n−1

ψ

, (ii)

Scal

g

_{> 0}

, (iii)

ψd

∇

_U

∗

g

(ψ) = dψ ∧ U

g

∗

(ψ) − ψ

2 d

∇

Ric

g

. Proof. (i)The tra eof Equation(2.5) leads to

d Scal

g

(α) = 2(n − 1)∆ψ − 2ψ Scal

g

= 0 ,

sin ethes alar urvatureis a onstant and (i)immediately follows. (ii)Thanks to (i) we know that

Scal

g

n−1

isan eigenvalue for the Riemannian Lapla ianon a ompa t and onne ted manifold and onsequently

Scal

g

_{≥ 0}

. In parti ular,

Scal

g

_{= 0}

(11)

andonlyiftheeigenfun tion

ψ

isa onstant. Butitisinfa timpossible. By ontradi tion, suppose

dψ = 0 =

1 nc

dδα

on

M

,so

dδα = 0

. Byintegrating byparts wend

0 =

Z

M

hdδα, αi =

Z

M

|δα|

2 ,

andthereby

δα = 0

on

M

whi h isimpossiblesin e

α

is nonisometri . We dedu ethat

ψ

annotbe a onstant and thus

Scal

g

_{> 0}

.

(iii)Thisis asimpli ation ofEquation (2.8)thanksto (i) and theprevious remark.

Wenishthisse tionbyre allinga lassi alresultofBourguignonwhi hisomnipresent throughout thisarti le.

2.6.Theorem(Bourguignon[7℄).Let

(M

n

_{, g)}

bea ompa tand onne tedmanifold

(n ≥ 2)

. Suppose that there existssome

ψ ∈ Ker U

∗

g

\ {0}

,then

Scal

g

isa nonnegative onstant and the level set

ψ

−1

₍₀₎

is either empty (

ψ

is a onstant) or omposed with embedded hyper-surfa esof M.

Proof. Firstly, taking the tra e of

U

∗

g

(ψ) = 0

gives

∆ψ =

Scal

g

n−1

ψ

, so that it be omes

Hess

g

ψ − ψ Ric

g

+

Scal

_n−1

g

ψg = 0

. Se ondly, take

x ∈ M

and

t 7→ σ(t)

a geodesi urve su h that

σ(0) = x

. Consider thefun tion

F := ψ ◦ σ

whi hhasto satisfytheorder 2ordinary dierential equation

F

′′

(t) − F (t)

Ric

g

(σ

′

(t), σ

′

(t)) −

Scal

g

n − 1

g(σ

′

_{(t), σ}

′

_(t))

= 0 ,

withthe initial onditions

F (0) = ψ(x), F

′

_{(0) = d}

x

ψ(σ

′

(0))

. Now, ifone supposes that

x ∈ ψ

−1

(0)

is riti alfor

ψ

,then theinitial onditions vanishand

F

hasto be identi ally zero. Sin e we an over a dense part of

M

with geodesi s starting at

x

,

ψ

should also vanish on the whole

M

, whi h is not permitted. We on lude that if

ψ

is not onstant,

ψ

−1

₍₀₎

ontains no riti alpoint and sois omposed withembedded hypersurfa esof

M

. Besides

ψ

isanonzero onstant ifand onlyif

g

isRi i at,andinthat ase

ψ

−1

_{(0) = ∅}

. Asregards the s alar urvature, the tri kis to ompute thedivergen e of

U

∗

g

(ψ) = 0

0 = δU

_g

∗

(ψ) = δ Hess

g

ψ + Ric

g

(∇ψ) − ψδ Ric

g

−d(∆ψ) =

1

2 ψd Scal

g

_,

wherewehaveused

δ Ric

g

_{= −}

1

2 d Scal

g

( f. 3.135i)in[11 ℄),

δ Hess

g

_{f = d(∆f )−Ric}

g

_{(∇f )}

( f. theproof of 4.14in[11 ℄). In any ase

ψ

−1

_(R

∗

₎

isdense in

M

,thereby

d Scal

g

_{= 0}

on thewholemanifold. But

Scal

g

n−1

isan eigenvalueoftheRiemannianLapla ianona ompa t manifold,whi h implies

Scal

g

_{≥ 0}

.

3. Proof of the Theorems

3.1. The system

(Σ

1 )

. Werstproveageneral resultstatingthattheexisten eofanon trivial KID on a ompa t and totally umbili al hypersurfa e, has to be onstant mean urvatureinanydimension greaterthan 2.

3.1. Theorem. Let

(M

n

_{, g)}

(n ≥ 2)

whi h has a solution

(f, α), f 6≡ 0

, of

L

_α

_{g + 2cf g = 0}

U

∗

g

(f ) = 0

,

(12)

fora ertain

c ∈ C

∞

_{(M ), c 6≡ 0}

. Thenthe s alar urvature

Scal

g

isa positive onstant and

c

is a nonzero onstant.

Proof. ThankstoTheorem2.6 ,weknowthat

Scal

g

isanonnegative onstantandthatthe fun tion

f

isan eigenfun tion sin e

∆f =

Scal

g

n−1

f

. Therst variation Formula(2.5) of the Ri i urvatureappliessin e

L

α

g = −2cf g

:

∇

α

Ric

g

(X, Y ) = −(n − 2) Hess

g

(cf )(X, Y ) − 2cf Ric

g

(X, Y ) + (∆(cf ))g(X, Y ) ,

whose tra e is zero sin e

Scal

g

is onstant. We derive

∆(cf ) =

Scal

g

n−1

cf

. Suppose now that

Scal

g

_{= 0}

then

f c

should be onstant andsotheKilling onformalform

α

should be isometri ,whi hisin ontradi tion with

c 6≡ 0, f 6≡ 0

. Wehavethenprovedthat

Scal

g

_{> 0}

, andalsothatthe fun tion

cf

isan eigenfun tionwhi hliesinthesameeigenspa e than

f

. Now onsider anodaldomain of

f

that wedenote by

Ω

. Using Theorem2.6 , we on lude that

∂Ω

is aregularhypersurfa e of

M

,and

f

isasolution oftheusual Diri hlet problem on

Ω

sin e by onstru tion we have

∆f =

Scal

_n−1

g

f

on

Ω

f = 0

on

∂Ω

,

withpositive eigenvalue

Scal

g

n−1

> 0

. But, sin e

Ω

is a nodaldomain,

f

annot vanish and so

Scal

g

n−1

> 0

has to be the rst Diri hlet eigenvalue of thedomain

(Ω, g)

whi h is always simple. Now, the fun tion

(cf )

alsosatises

∆(cf ) =

Scal

_n−1

g

cf

on

Ω

cf = 0

on

∂Ω

,

and onsequently,thereexistsanon zero onstant

K ∈ R

∗

(otherwise

f

shouldbezeroon anopenset whi his ex ludedbyTheorem 2.6) su hthat

Kcf = f

. But

f

nevervanishes on

Ω

,sothat

c = K

−1

hasto be anon zero onstant on

Ω

. Thisargument istrueon any nodaldomain of

f

whose unionisa denseopenset in

M

,whi h means that

c

hasto bea nonzero onstant on the whole

M

.

As a onsequen e we an ompletely answer thequestion of hara terizing the totally umbili al KIDindimension 2.

3.2.Corollary. Let

(M

2 _{, g)}

be a ompa t and onne ted Riemannian surfa e whi h has a solution

(f, α), f 6≡ 0

, of

L

_α

_{g + 2cf g = 0}

U

_g

∗

(f ) = 0

,

for a ertain

c ∈ C

∞

_{(M ), c 6≡ 0}

. Then

c

is a non zero onstant and

(M

2 _{, g)}

is isometri tothe standard sphere

S

2

. In other words, any totally umbili al and ompa t hypersurfa e havinganontrivial KIDis isometri to aspatial (spheri al and onstantmean urvature) deSitter metri .

Proof. FromTheorem 3.1we knowthat

c

is anonzero onstant and

Scal

g

apositive on-stant. GaussBonnetformula laimsthat

M

2

ishomeomorphi to

S

2

andTheorem3.83in [11℄saysthat

M

2

isinfa tisometri toa standard 2dimensionalsphere (sin ethes alar urvatureand these tional urvature areequivalent for surfa es).

(13)

For manifolds of dimension greaterthan 3,we need to make an additional assumption (theKilling onformal1form

α

issupposedto be losed)inorder toobtaingeometri in-formation. WethenproveTheorem3.3that lassiesthe ompa tand onne tedmanifolds

(M

n

, g)

that arrynontrivial solutions of

(Σ

1 )

.

3.3.Theorem. Let

(M

n

_{, g)}

(n ≥ 3)

with

C

3

metri , whi hhas a solution

(f, α), f 6≡ 0

, of

(Σ

1 )

∇α + cfg = 0

_U

∗

g

(f ) = 0

,

for a ertain

c ∈ C

∞

_{(M ), c 6≡ 0}

. Then is a nonzero onstant and

(M

n

_{, g)}

is isometri toone of the following manifolds:

(i)

S

n

(S

1 _{× Y, dt}

2 _{+ h}

2 _(t)g

0 )

, where

(Y

n−1

_{, g}

0 )

(M

n

_{, g, cg)}

Scal

g

_{+n(n − 1)c}

2 _{= Φ}

1 (g, cg) = const.

. Proof. We anuse(iii)inTheorem2.5with

ψ = cf

and

c

anon zero onstant. Weobtain

0 = c

2 f d

∇

U

g

∗

(f ) = c

2 df ∧ U

g

∗

(f ) − c

2 f

2 d

∇

Ric

g

= −c

2 f

2 d

∇

Ric

g

,

sin e

U

∗

g

(f ) = 0

. It omes out

d

∇

_Ric

g

_{= 0}

on the dense open set

f

−1

_(R

∗

₎

, and by a ontinuationargument (

d

∇

_Ric

g

_{∈ C}

0

sin e themetri

g

isregular enoughi.e.

C

3

),we get

d

∇

Ric

g

= 0

on the whole

M

. We on lude by applying Proposition 1.2 whi h gives the announ ed lassi ation.

FromtheKIDpointofview,

(M

n

_{, g, k)}

istotallyumbili al withextrinsi urvature

k = cg

by onstru tion, and the onstant

c ∈ R

∗

is the mean urvature of the sli e. It is lear that it satisesthe va uum onstraint equations with the positive osmologi al onstant

Λ =

1 ₂

Scal

g

+n(n − 1)c

2 > 0

.

3.4. Remark . The fun tion h in the warped produ t metri s of (ii) in Theorem 3.3 has tosatisfy an order 2 dierential equation whi his given by the relation between

Scal

g

and

Scal

g

0

the s alar urvature of

(Y, g

0 )

. See [12℄ for further details.

In theKID ontext, Theorems 3.3givesrise to thefollowing natural issue. Question:Let

(M

n

_{, g)}

bea ompa tand onne tedmanifold

(n ≥ 3)

and

c ∈ C

∞

_{(M ), c 6≡ 0}

. Doestheexisten eofnon trivialsolutions (i.e.

f 6≡ 0

) of

L

_α

_{g + 2cf g = 0}

U

_g

∗

(f ) = 0

,

hara terize the spatial S hwarzs hildde Sitter metri s? In general, it is not lear sin e wea priorilose the important geometri urvature identity (2.1 ).

3.2. Thesystem

(Σ

2 )

. Asalreadysaidintheintrodu tion,we anfo usontherestri tion of the onstraints appli ation to

V

× Γ(S

2 _T

∗

_{M )}

where

V

= {g ∈ M / dVol

g

= dVol

g

0 }

. Inthis ontext, therelevant operator is

U

∗

g

0

sothatwe onsider theproblem of nding some

(f, α)

su h that

L

_α

_{g + 2cf g = 0}

(14)

where

c ∈ C

∞

_{(M ), c 6≡ 0}

. Thisproblem is learly equivalentto

(

L

_α

_{g + 2cf g = 0}

U

∗

g

(f ) =

n

1 (n − 1)∆f − f Scal

g

.

Themaindieren ewiththesystem

(Σ)

isthatthereisnohopeto provethatthefun tion

c

(remindthat

c

isthemean urvatureof thehypersurfa e

(M

n

_{, g, k = cg)}

) isa onstant. Indeed,let

(M

n

_{, g)}

be an Einstein manifoldwith nonnegative s alar urvature thathasa non isometri Killing onformal 1form

α

(su h manifolds do exist, the standard sphere

S

n

is one ofthem). Then any ouple

(f, α)

with

f

anynon zero onstant,hasto satisfy

L

_α

_{g = −2cf g}

and

U

∗

g

(f )

0 = 0 ,

where we have dened thefun tion

c

as

c :=

δα

f n

6≡ 0

sin e

α

is non isometri . Expe ting the non isometri Killing onformal 1form to be losed does not even guarantee that

c

shouldbea onstant. Youjusthaveto onsideronthestandardsphere

S

n

,thenontrivial ouples

(f, f dc)

where

f

isanynonzero onstantand

c ∈ Ker U

∗

g

= Span {x

1 , x

2 , · · · , x

n

}

, with

(x

i

)

n+1

i=1

the standard oordinateson

S

n

. Su h ouples satisfy

∇α = −cf g

and

U

∗

g

(f )

0 = 0 .

Nowifwegiveupthe losed hara terof

α

,thenwe an onsidersome nontrivial ouples

(f, f dc + β)

where

f

is any non zero onstant,

c ∈ Ker U

∗

g

= Span {x

1 , x

2 , · · · , x

n

}

and

β

anyKilling formon

S

n

. Thensu h ouples hasto verify

L

_α

_{g + 2cf g = 0}

and

U

∗

g

(f )

0 = 0 .

This spa e of solutions is parametrized by

R

∗

_{× R}

n+1

_{× so(n + 1)}

with the Lie algebra

so(n + 1) ∼

= R

n(n+1)

2

. It is the reason why we restri t our study to the ase where

c

is a non zero onstant. Then we an lassify thesolutions of this systemwhen we assume in additionthattheKilling onformalform

α

is losed,asitis laimedinthefollowingresult. 3.5.Theorem. Let

(M

n

_{, g)}

(n ≥ 3)

with

C

3 (f, α), f 6≡ 0

, of

(Σ

2 )

( ∇α + cfg = 0

U

_g

∗

(f ) =

1 _n

(n − 1)∆f − f Scal

g

,

fora ertain onstant

c 6= 0

. Then

(M

n

_{, g)}

isisometri to oneof the followingmanifolds: (i)

S

n

(S

1 _{× Y, dt}

2 _{+ h}

2 _(t)g

0 )

, where

(Y

n−1

_{, g}

0 )

(M

n

_{, g, cg)}

Scal

g

_{+n(n − 1)c}

2 _{= Φ}

1 (g, cg) = const.

. Proof. Our goal is to show that

U

∗

g

(f ) = 0

and then use Theorem 3.3. Equation (2.6 ) readsas

d

∇

Ric

g

(α, X, Y ) = −cU

_g

∗

(f )(X, Y ) = −

c

n

(n − 1)∆f − f Scal

g

g .

Let

x ∈ M

,thenthereareexa tly2 possibilities 1)

α

x

= 0

andthen

U

∗

(15)

2) Otherwise

α

x

6= 0

and then

−

c

n

(n − 1)∆f (x) − f (x) Scal

g

_(x)

_α

x

= d

∇

Ric

g

(α, α, ·)

x

= 0 ,

whi hentails

(n − 1)∆f (x) − f (x) Scal

g

_{(x) = 0}

and so

U

∗

g

(f ) = 0

at thepoint

x

. We an apply Theorem 3.3to on lude sin e

U

∗

g

(f ) = 0

onthewhole

M

.

3.6. Remark . In [12 ℄ Se tion E1, Lafontaine gives a family of non trivial solutions of theequation

Hess

g

_{f +}

∆f

n

g − f Ric

g

0 = 0

. These metri s are again warped produ t metri s

dt

2 _{+ h}

2 _(t)g

0

on

S

1 _{× Y}

, where

(Y, g

0 )

is Einstein withpositive s alar urvature, and

h

a periodi fun tion. Theorem3.5showsthat thenontrivial solutionsof

(Σ

2 )

are themetri s inthe lass exhibited by Lafontainein [12 ℄, that have positive onstant s alar urvature.

The ase of surfa es is easy to treat sin e the omputations of Lemmas 2.2 2.3 and Theorem2.5are valid indimension 2.

3.7.Corollary. Let

(M

2 _{, g)}

be a ompa t and onne ted Riemmaniansurfa e whi hhas a nontrivial solution

(f, α)

of

( ∇α + cfg = 0

U

g

∗

(f ) =

1 ₂

∆f − f Scal

g

,

fora ertain onstant

c 6= 0

. Then

(M

2 _{, g)}

isisometri to the standard sphere

S

2

.

Proof. The omputationsintheproof ofTheorem3.5arestillvalidindimension2sothat

U

∗

g

(f ) = 0

onthewhole

M

. We an on ludeusingthesurfa e versionofthe lassi ation resultonthe system

(Σ

1 )

.

In theKID ontext, Theorem3.5givesrise to thenatural issue Question:Let

(M

n

_{, g)}

be a ompa t and onne ted manifold. Doestheexisten eof non trivialsolutions (i.e.

f 6≡ 0

) of

(

L

_α

_{g + 2cf g = 0}

U

_g

∗

(f ) =

_n

1 (n − 1)∆f − f Scal

g

,

(with

c

a non zero onstant) hara terize the spatial S hwarzs hildde Sitter metri s for

n ≥ 3

(respe tively thespatialde Sittermetri s for

n = 2

)?

3.3. The system

(Σ

3 )

. The nextmainresultdealswiththerestri tionofthe onstraints appli ation to

C

× Γ(S

2 _T

∗

_{M )}

where

C

= {g ∈ M /d Scal

g

_{= 0}

and

Vol

g

= Vol(S

n

_)}

. In this ontext,

U

∗

g

hastobeequaltothetra elessRi i urvature,sothatweneedto onsider theproblem ofnding some

(f, α)

su h that

L

_α

_{g + 2cf g = 0}

U

g

∗

(f ) = Ric

g

0 .

We ould assume without loss of generality that

Scal

g

is a onstant, but it is in fa t not ne essary in virtue of the following general result (whi h is the analogous version of Theorem3.1for therestri tion to

C

× Γ(S

2 _T

∗

_{M )}

(16)

3.8.Theorem. Let

(M

n

_{, g)}

(n ≥ 2)

with

C

3

metri , whi hhas a solution

(f, α), f 6≡ 0

, of

L

_α

_{g + 2cf g = 0}

U

_g

∗

(f ) = Ric

g

₀

,

fora ertain

c ∈ C

∞

_{(M ), c 6≡ 0}

. Thenthe s alar urvature

Scal

g

isa positive onstant and

c

is a nonzero onstant.

Proof. We rst prove that

Scal

g

is a positive onstant. By omputing the divergen e of

U

∗

g

(f ) = Ric

g

0

we get

1

2 f d Scal

g

_{= δU}

∗

g

(f ) = δ Ric

g

0 =

n

1 −

1

2 d Scal

g

, whi h leads to the identity

f −

2−n

n

d Scal

g

= 0

. Let us dene

F

a losed subset as

F := f

−1

2−n

n

, and prove by ontradi tion that its interior

Int(F )

is empty. If

Int(F ) 6= ∅

then, on

Int(F )

we have

U

∗

g

(f ) = −

2−n

n

Ric

g

= Ric

g

₀

whose tra e gives

Scal

g

_{= 0}

. But on its omplement

M r Int(F )

, we have

d Scal

g

_{= 0}

and so

Scal

g

_{= 0}

on the whole

M

by a ontinuation argument. The rst variation Formula (2.5 ) of the Ri i urvature applies sin e

L

α

g = −2cf g

:

∇

α

Ric

g

(X, Y ) = −(n − 2) Hess

g

(cf )(X, Y ) − 2cf Ric

g

(X, Y ) + (∆(cf ))g(X, Y ) ,

whose tra e is zero sin e

Scal

g

_{≡ 0}

is a onstant. We derive

∆(cf ) =

Scal

g

n−1

cf = 0

, whi h impliesthatthe fun tion

cf

is a onstant. Thisis in ontradi tion withthefa tthat

α

is nonisometri and

f 6≡ 0, c 6≡ 0

. Therefore,

Int(F ) = ∅

andso

M r F

isanopenanddense subset of

M

where

d Scal

g

_{= 0}

. We nally obtain that

Scal

g

is a nonnegative onstant (sin e

∆(cf ) =

Scal

g

n−1

cf

) whi h annotbe zero(

α

isnon isometri ). The tra e of

U

∗

g

(f ) = Ric

g

0

gives

∆f =

Scal

g

n−1

f

so the fun tions

f

and

cf

belong to the sameeigenspa e ofpositiveeigenvalue

Scal

g

n−1

. Unfortunately,

f

−1

₍₀₎

thenodalsetof

f

has possiblyasingular (ordegenerate) partsin e theargument ofTheorem 2.6doesnot work anymore. We need to use a ru ial resulton nodal sets owed to Bär, namely Corollary 2 in [1 ℄. It states that the nodal set of

f

is the disjoint union

f

−1

_{(0) = N}

reg

∪ N

sing

with

N

reg

:=

x ∈ f

−1

(0)/d

x

f 6= 0

and

N

sing

:=

x ∈ f

−1

(0)/d

x

f = 0

that have the following properties:

(i)

N

reg

is omposed withsmooth embedded hypersurfa es (bytheimpli it fun tion theorem),

(ii)

N

sing

isa ountably

(n − 2)

re tiable setandthushasHausdordimension

n − 2

at most.

Wenoti ethat

N

reg

annotbeempty. Indeed(by ontradi tion)ifitwasemptythenthere shouldbeauniquenodaldomain

M r N

sing

(sin e

N

sing

hasHausdordimension

n − 2

at most,

M r N

sing

is onne ted). But

f

hasa vanishingintegral on

M

and hasa onstant signon the denseopen set

M r N

sing

,thereby

f ≡ 0

whi his not possible. Now onsider

Ω

any onne ted omponent ofthe open set

M r N

reg

. By onstru tion theboundary

∂Ω

issmoothand

f

isasolutionof theusualDiri hletproblemon

Ω

sin eby onstru tion we have

∆f =

Scal

_n−1

g

f

on

Ω

f = 0

on

∂Ω

,

withpositive eigenvalue

Scal

g

n−1

> 0

. But

f

hasa onstant sign on

Ω

(

f

an possiblyvanish but only on a set of Hausdor dimension less or equal to

n − 2

) and so

Scal

g

n−1

> 0

has to be the rst Diri hlet eigenvalue of the domain

(Ω, g)

whi h is always simple. Now, the

(17)

fun tion

cf

alsosatises

∆(cf ) =

Scal

_n−1

g

cf

on

Ω

cf = 0

on

∂Ω

,

and onsequently,thereexistsanonzero(sin eboth

f

and

cf

annotvanishonthewhole

Ω

) onstant

λ

su hthat

cf = λf

. Wehavethenprovedthat

c

hastobeanonzero onstant on(a denseopensubset of)

Ω

. Thisargument istrueon any onne ted omponent of the open set

M r N

reg

whose unionisa denseopen setin

M

,whi h meansthat

c

hasto bea nonzero onstant on thewhole

M

by a ontinuation argument.

Asa onsequen ewe an ompletelyanswerthequestionof hara terizingthemanifolds thathave non trivialsolutions of

L

_α

_{g + 2cf g = 0}

U

_g

∗

(f ) = Ric

g

₀

,

indimension 2and 3. 3.9. Corollary. Let

(M

n

_{, g)}

be a ompa t and onne ted Riemannian manifoldof dimen-sion

n = 2

or

3

, whi h has a solution

(f, α), f 6≡ 0

,of

L

_α

_{g + 2cf g = 0}

U

g

∗

(f ) = Ric

g

0 ,

for a ertain

c ∈ C

∞

_{(M ), c 6≡ 0}

. Then

c

is a non zero onstant and

(M

n

_{, g)}

is isometri tothestandard sphere

S

n

. Fromthe KIDpointof view,thesli e

(M

n

_{, g, cg)}

isa spatialde Sitter metri .

Proof. FromTheorem 3.8we knowthat

c

is anonzero onstant and

Scal

g

apositive on-stant.

For the ase of Riemannian surfa es i.e.

n = 2

, GaussBonnet formula laims that

M

2

is homeomorphi to

S

2

and Theorem 3.83 in [11 ℄ says that

M

2

is in fa t isometri to a standard2dimensional sphere (sin ethe s alar urvature andthese tional urvature are equivalent for surfa es). Thevolume normalization says thatitisexa tly

S

2

.

When the dimension is

n = 3

, then the theorem of BessièresLafontaineRozoy applies andso

M

3

is isometri to

S

3

.

For dimension greateror equal to 4, the problemis quite harder to solve. Here again, westudy this systemassuming the losed hara ter oftheKilling onformalform