A new characterization of the Euclidean sphere

(Annals of the Brazilian Academy of Sciences)

Printed version ISSN 0001-3765 / Online version ISSN 1678-2690 http://dx.doi.org/10.1590/0001-3765201820170689

www.scielo.br/aabc | www.fb.com/aabcjournal

A new characterization of the Euclidean sphere

ABDÊNAGO A. BARROS1_{, CÍCERO P. AQUINO}2_{and JOSÉ N.V. GOMES}3 1_{Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici,}

Bloco 914, 60455-760 Fortaleza, CE, Brazil

2_{Departamento de Matemática, Universidade Federal do Piauí, Campus Universitário} Ministro Petrônio Portella, 64049-550 Teresina, PI, Brazil

3_{Departamento de Matemática, Universidade Federal do Amazonas, Av. General} Rodrigo Octávio, 6200, 69080-900 Manaus, AM, Brazil

Manuscript received on September 6, 2017; accepted for publication on April 4, 2018

ABSTRACT

In this paper, we obtain a new characterization of the Euclidean sphere as a compact Riemannian manifold with constant scalar curvature carrying a nontrivial conformal vector field which is also conformal Ricci vector field.

Key words:Conformal vector field, Scalar curvature, Euclidean sphere, Einstein manifold.

INTRODUCTION

In the middle of the last century many geometers tried to prove a conjecture concerning the Euclidean sphere as the unique compact orientable Riemannian manifold(Mn_{,g)admitting a metric of constant scalar} curvatureS and carrying a nontrivial conformal vector fieldX. Among them, we cite Bochner and Yano (1953), Goldberg and Kobayashi (1962), Lichnerowicz (1955), Nagano and Yano (1959), Obata (1962), Obata and Yano (1965, 1970) and Tashiro (1965). The attempts to prove this conjecture resulted into the rich literature which has currently been attracting a lot of attention in the mathematical community. We address the reader to the book of Yano (1970) for a summary of those results. Ejiri gave a counter example to this conjecture building metrics of constant scalar curvature on the warped productS1_×fFn−1, where

Fn−1 is a compact Riemannian manifold of constant scalar curvature, while S1 stands for the Euclidean circle. In his example the conformal vector field isX= f(d/dt), whered/dtis a unit vector field onS1and

f satisfies a certain ordinary differential equation, see Ejiri (1981) for details.

The primary concept involved in the study of this subject is of Lie derivatives. After all, what is the geometric meaning of the Lie derivative of a tensorT (or of a vector fieldY) with respect to a vector field

X?This is a method that uses the flow of_X to push values of_T back to _pand then differentiate. The result is called the Lie derivativeL_XTof_Twith respect to_X. A vector field_Xon Riemannian manifold_(Mn_,g)is called conformal ifL_Xgis a multiple of_g. There are important applications of Lie derivatives in the study of how geometric objects such as Riemannian metrics, volume forms, and symplectic forms behave under flows. For instance, it is well-known that the Lie derivative of a vector fieldY with respect toX is zero if and only ifY is invariant under the flow ofX. It turns out that the Lie derivative of a tensor has exactly the same interpretation. For more details see the book of Lee (2003).

We highlight that Nagano and Yano (1959) have proved that the aforementioned conjecture is true when (Mn_,g)is an Einstein manifold, i.e., the Ricci tensor of metric_gsatisfies_Ric=S

ng. In this case,Sis constant for dimensionsn≥3. Thus if_X is the conformal vector field with conformal factor_ρ,that is,L_Xg=2_ρg, we deduce L_XRic=2_ρRic.With this setting we define aconformal Ricci vector fieldon a Riemannian manifold(Mn_,g)as a vector field_X satisfying

L_XRic=2_βRic, (1)

for some smooth functionβ:_Mn_→R_.In particular, on Einstein manifolds this concept is equivalent to the classical conformal vector field. With this additional condition the aforementioned conjecture is true. More precisely, we have the following theorem.

Theorem 1. Let(Mn_{,g),n≥3,be a compact orientable Riemannian manifold with constant scalar}

curva-ture carrying a nontrivial conformal vector fieldXwhich is also a conformal Ricci vector field. ThenMnis

isometric to a Euclidean sphereSn_(r). Moreover, up to a rescaling, the conformal factor_ρis given by

ρ=τ−hv n

andX is the gradient of the Hodge-de Rham function which in this case, up to a constant, is the function

1

nhv, wherehv is a height function on a unitary sphereS

nand_τis an appropriate constant.

We point out that Obata and Yano (1965) have obtained the same conclusion of the preceding theorem under the hypothesisL_XRic=αg,for some smooth function_α defined in_Mn_.Moreover, when_Mn is an Einstein compact Riemannian manifold the result of Nagano and Yano (1959) is a consequence of Theorem 1. We also observe that the compactness ofMnin our result is an essential hypothesis. In fact, let us consider the hyperbolic spaceHn as a hyperquadric of the Lorentz-Minkowski space Ln+1 and_v a nonzero fixed vector inLn+1. After a straightforward computation, it is easy to verify that the orthogonal projection_v⊤of

vonto the tangent bundleTHnprovides a nontrivial conformal vector field on_Hnfor an appropriate choice ofv. Consequently, sinceHnis Einstein, it follows that_v⊤is also a nontrivial conformal Ricci vector field onHn.

PRELIMINARIES AND AUXILIARY RESULTS

To start with, we consider the Hilbert-Schmidt norm for tensors on a Riemannian manifold(Mn_,g), i.e., the inner producthT,Si=tr _{T S}∗_.It is important to notice that for an orthonormal basis_{e1, . . . ,e_n}, we can use the natural identification of(0_,2₎-tensors with₍1_,1₎-tensors,_{T(ei,ej) =g(Tei,ej)}, to write

hT,Si=

_∑

Ti jSi j=

∑

We recall that the divergence of a(1_,r)-tensor_T on_Mnis the₍0_,r)-tensor given by

(div_T)(v1, . . . ,vr)(p) =tr w7→(∇wT)(v1, . . . ,vr)(p)

,

wherep∈Mnand(v1, . . . ,vr)∈TpM×. . .×TpM.

LetX be a smooth vector field onMn, and letϕbe its flow. For anyp∈Mn, iftis sufficiently close to zero,ϕt is a diffeomorphism from a neighborhood ofpto a neighborhood ofϕt(p), soϕt∗pulls back tensors atϕt(p)to ones at p.

Given a(0_,r)-tensor_T on_Mn, the Lie derivative of_T with respect to_X is the_(0,r)-tensor_L

XT given by

(L_XT)p= d dt

t=0(ϕ

∗

tT)p=lim t→0 1

t

ϕ_t∗(T_ϕ(t,p))−Tp

.

Fortunately, there is a simple formula for computing the Lie derivative without explicitly finding the flow. For any(0_,r)-tensor_T on_Mn,

(L_XT)(Y1, . . . ,Y_r) = X(T(Y1, . . . ,Y_r))−T([X,Y1],Y2, . . . ,Y_r)−. . .

−T(Y1, . . . ,Y_r−1,[X,Y_r]),

whereY1, . . . ,Yrare any smooth vector fields onMn, and[X,Yi]stands for the Lie bracket ofX andYi. In what follows we prove some lemmas and integral formulas which will be required later.

Lemma 1. For any symmetric(0_,2₎-tensor_T on a Riemannian manifold_(Mn_,g)and_X∈X(M), holds

L_X|T|2₌2_hL_{XT,Ti −}2_hT2_,L_Xgi.

In particular, ifXis a conformal vector field with conformal factorρ,then we haveL_X|T|2₌2_hL_{XT,Ti −}

4_ρ|T|2_.

Proof.Let{e1, . . . ,en}be a geodesic orthonormal frame at p∈Mn. Then we have

L_X|T|2 ₌ L_X

_∑

Ti jTi j

=2

_∑

i,j

Ti jX(Ti j)

= 2

_∑

i,j

Ti j{(L_{XT)i j+T([X,ei],ej) +T(ei,[X,ej])}}

= 2

_∑

i,j

Ti j(L_{XT)i j−}2

_∑

i,j

Ti j{T(∇eiX,ej) +T(ei,∇ejX)}.

Whence, we use thatT is symmetric to deduce

L_X|T|2 ₌ 2_hT,L_{XTi −}2

_∑

i,j

Ti j{g(∇eiX,Tej) +g(Tei,∇ejX)}

= 2_hT,L_{XTi −}

∑

2_g(∇Te

jX,Tej)−

∑

i

2_g(Tei,∇Te iX)

= 2_hT,L_{XTi −}2

∑

i

(L_Xg)(Tei,Tei)

= 2_hT,L_{XTi −}2

_∑

i,j

T_{i j}2(L_{Xg)i j,}

Corollary 1. Under the assumptions of Lemma 1 we have L_X|T|2₌0_,provided that L_Xg=2_ρg and

L_{XT =}2_ρT.

Another useful result is given by the following lemma.

Lemma 2. Let(Mn_,g)be a Riemannian manifold endowed with a symmetric_(0,2)-tensor_T.Then it holds

XhT,gi=hL_{XT,gi − hT,}L_Xgi.

In particular, ifL_{XT =}2_βT,then

2_βtr_{(T) =h∇(}tr_(T)),Xi+hT,L_Xgi.

Proof.Let{e1, . . . ,e_n}be a geodesic orthonormal frame at a pointp∈Mn.By the symmetry ofT, we have

hL_XT,gi=tr₍L_{XT) =}

∑

X(Tii) +2_T(∇e iX,ei)

= XhT,gi+2_hT,∇Xi

= XhT,gi+hT,L_Xgi,

that finishes the proof of the lemma.

Now we remind the reader that the traceless tensor of a symmetric(0_,2₎-tensor_T on a Riemannian manifold Mn,gis given by

˚

T =T−tr(T)

n g. (2)

With this setting we prove the next corollary.

Corollary 2. Let(Mn_{,g)be a Riemannian manifold endowed with a symmetric (0,2)-tensorT such that} L_{XT =}2_βT, withL_Xg=2_ρg.Then we have:

1. 2 _β−ρ

tr_{(T) =h∇(}tr_(T)),Xi.

2. Iftr_(T)is a non null constant, then_{β =ρ} andL_XT˚ ₌2_ρT˚_.

Proof.SinceL_Xg=2_ρgwe have immediately the first item. Now, iftr_(T)is a non zero constant, then we haveβ =ρ,which impliesL_XT˚ ₌L_XT−tr(T)

n LXg=2ρT˚,finishing the proof of the corollary.

Next, we apply the previous results to the Ricci tensor. First of all, given a conformal vector fieldXon a Riemannian manifoldMnsuch thatL_Xg=2_ρg,we have the next well-known formulae, see e.g. Obata and Yano (1970).

L_{XRic=−(n−}2_)∇2_{ρ−(∆ρ)g,} (3)

L_XS=−2_(n−1_)∆ρ−2_Sρ, (4) and

L_XG=−(n−2₎

∇2ρ−1 n(∆ρ)g

, (5)

whereG=Ric−S

ng=Ric˚ .

We claim that ifMnis compact with constant scalar curvatureSandρ is not constant, then equation

(4) allows us to infer thatSis positive. Indeed, sinceρ is not constant S

Lemma 3. Let (Mn_{,g) be a compact Riemannian manifold with constant scalar curvature such that} L_XRic=2_βRicandL_Xg=2_ρg. Then we have_β=ρandL_X|G|2₌0_.

Proof.SinceL_XRic=2_βRicand_S=tr_(Ric)is a positive constant we get by Corollary 2 that_{β =ρ} and

L_XG=2_ρG.Therefore, applying Corollary 1 we have L_X|G|2

=0_, which completes the proof of the lemma.

Taking into account the second contracted Bianchi identity: div_{(Ric) =} 1

2dS and using the identity div₍S

ng) =

1

ndSwe obtain the following relation

div_{(G) =}n−2 2_n dS. Therefore, we can write

ρdiv_{(G)(∇ρ) =}n−2

4_n h∇S,∇ρ 2

i.

The next equation is well-known. For details of a more general case see for example Lemma1 in Barros and Gomes (2013).

div_{(ρG(∇ρ)) =ρ}div_{(G)(∇ρ) +ρh∇}2_{ρ,Gi+G(∇ρ,∇ρ).} (6)

SincehG,gi=0, from (5) we have

hL_{XG,Gi=−(n−}2_)h∇2_ρ,Gi.

Applying Lemma 1 to this identity we obtain

h∇2ρ,Gi=− 1 n−2

1

2LX|G| 2

+2_ρ|G|2_. (7)

Comparing (6) and (7) we infer

div_{(ρG(∇ρ)) =} n−2

4_n h∇S,∇ρ 2

i − 1 n−2

ρ

2LX|G| 2

+2_ρ2_|G|2

+G(∇ρ,∇ρ). (8)

In what follows we assume that(M,g)is an orientable Riemannian manifold. IfMis not orientable, we take the orientable double covering_M˜ _{ofMand induce, in the natural manner, the Riemannian metricg˜on}

˜

M. Then(M,g)and(_M˜_{,g)˜ have the same local geometry.}

As a direct consequence from (8) and Stokes’ Theorem we obtain the following lemma.

Lemma 4. Let (Mn_,g) be a compact orientable Riemannian manifold endowed with a conformal vector

fieldX of conformal factorρ, then

Z

M

G(∇ρ,∇ρ)dM= 1 n−2

Z

M

ρ

2LX|G| 2

+2_ρ2_|G|2_dM+n−2 4_n

Z

M

ρ2∆SdM.

Before stating our next result we note that|∇2

ρ−∆ρ_n g|2

=|∇2

ρ|2

−1_n(∆ρ)2. Accordingly, the Bochner formula becomes

1

2∆|∇ρ| 2

= G(∇ρ,∇ρ) +S n|∇ρ|

2

+|∇2ρ−∆ρ n g|

2

+1 n(∆ρ)

2

−h∇ρ,∇ Sρ

n−1+

L_XS

2_(n−1₎

where in the last term we use equation (4). Then,

1

2∆|∇ρ| 2

= G(∇ρ,∇ρ) +|∇2ρ−∆ρ n g|

2

−S |∇ρ|

2

+ρ∆ρ

n(n−1₎

− 1

2_n(n−1₎

(L_{XS)∆ρ+nh∇S,∇ρ}2_{i+nh∇ρ,∇(}L_XS)i.

By integration,

Z

M

G(∇ρ,∇ρ) +|∇2ρ−∆ρ n g|

2

+ 1

2_n(ρ 2

∆S+ (L_XS)∆ρ

dM=0_. (9)

In the notation of (2) we have|∇˚2_ρ|2

=|∇2

ρ−∆ρ_n g|2. Comparing Lemma 4 with equation (9) we get the lemma.

Lemma 5. Let(Mn_{,g) be a compact orientable Riemannian manifold endowed with a conformal vector}

fieldXof conformal factorρ, then

1. R M ρ n−2 1 2LX|G|

2

+2_ρ|G|2

+|∇˚2_ρ|2

+S

2div(ρ∇ρ) +

L_XS

2n ∆ρ

dM=0_.

2. R M

ρ

n−2hLXG,Gi+| ˚

∇2_ρ|2

+S

2div(ρ∇ρ) +

L_XS

2_n ∆ρ

dM=0_.

Proof.First assertion is a direct combination of Lemma 4 and equation (9), while the second one follows from Lemma 1 and the first assertion. Indeed, from this latter lemma we have 1

2LX|G| 2

+2_ρ|G|2_=hL_XG,Gi, which completes our proof.

We are in the right position to prove our main result.

PROOF OF THEOREM 1

Firstly, we observe that we are supposing that there exists a vector fieldXonMnsuch thatL_Xg=2_ρgand

L_XRic=2_βRic, for some smooth functions_ρ and_β on_Mn, where_ρ is non-constant. Consequently, from Lemma 3 we obtainβ =ρ andL_X|G|2₌0. Secondly, from item (1) of Lemma 5 and equation (4) we get

Z

M

2

n−2ρ 2

|G|2+|∇2ρ+ Sρ n(n−1₎g|

2

dM=0_.

Taking into account that ρ is non-constant the preceding identity allows us to achieve G = 0 and

∇2

ρ =−_n(n−S1₎ρg.Therefore, we can apply a classical result due to Obata (1962), for instance, to con-clude thatMnis isometric to a Euclidean sphereSn_(r). Moreover,_∇2_ρ=∆ρ

n gand∆(∆ρ) + S

n−1∆ρ=0(see equation (4)). Rescaling the metric we can assume thatS=n(n−1₎. Then we conclude that_∆ρ is the first eigenvalue of the unitary sphereSn_.Whence, there exists a fixed vector_v∈Snsuch that_∆ρ=hv=−1

n∆hv. Thus we have∆(ρ+1_nhv) =0_,which gives_ρ=τ−1

nhv. Settingu=−ρwe obtain

L_∇ug=2_∇2_u=−2_∇2_ρ=2_ρg=L_Xg.

It is also true that L_∇uRic=2_ρRic. Besides, by Hodge-de Rham decomposition theorem we can write

A MORE GENERAL CASE We notice thatL_XG=−(n−2_)∇˚2_ρand_L

XG=LXRic−1_nLX(Sg)give

L_{XRic =} 1

nLX(Sg)−(n−2)

˚

∇2_ρ

= 2_ρ S

ng

+1

n(LXS)g−(n−2)

˚

∇2_ρ

= 2_{ρ Ric−G}

+1

n(LXS)g−(n−2)

˚

∇2_ρ

= 2_ρRic+1

n(LXS)g+T, whereT =−2_ρG−(n−2_)∇˚2_ρ.Therefore, we deduce

L_XRic=2_ρRic+1

n(LXS)g+T, wheretr_{(T) =}0_.

Let us suppose thatL_XRic=2_βRic+T andL_Xg=2_ρg, where_T is a₍0_,2₎-tensor on_Mn. By using (3) and (4) we deduce

tr_{(T) =−}2_(n−1_)∆ρ−2_βS and

L_XS=tr_{(T) +}2_(β−ρ)S. In particular, ifSis a non null constant andρ6=0, we have

tr_{(T) =}0 if and only if _{β =ρ.} (10)

On the other hand,L_XG=L_XRic−1

nLX(Sg)gives

L_XG=2_βG+T−tr(T)

n g (11)

and

L_X|G|2₌4_(β−ρ)|G|2₊2_hT,Gi. (12) Lemma 6. Let(Mn_,g),n≥3, be a compact orientable Riemannian manifold endowed with a conformal

vector fieldX whose conformal factor is ρ.If L_XRic=2_βRic+T, then the following integral formula holds:

Z

M

1

n−2 2β ρ|G| 2

+ρhT,Gi

+|∇2ρ−∆ρ n g|

2

+S

2div(ρ∇ρ) +

L_XS

2_n ∆ρ

dM=0_.

Proof.First we notice that from (12) we obtain

ρ

2LX|G| 2

+2_ρ2_|G|2₌2_{β ρ|G|}2_+ρhT,Gi. (13)

Therefore, using (13) in the first assertion of Lemma 5, we have Z

M

1

n−2 2β ρ|G| 2

+ρhT,Gi

+|∇2ρ−∆ρ n g|

2

+S

2div(ρ∇ρ) +

L_XS

2_n ∆ρ

dM=0

which completes the proof of the lemma.

Theorem 2. Let(Mn_{,g),n≥3, be a compact orientable Riemannian manifold with constant scalar}

cur-vatureS. Suppose that there exists a nontrivial conformal vector fieldX onMnsuch thatL_Xg=2_ρgand

L_XRic=2_βRic+T. Iftr_{(T) =}0andR M

2_ρ2

|G|2

+ρhG,TidM≥0, then_Mnis isometric to a Euclidean

sphere.

Moreover, up to a rescaling, the conformal factorρ is given byρ =τ−hv

n andX is the gradient of

the Hodge-de Rham function which in this case, up to a constant, is the function 1

nhv, wherehvis a height

function on a unitary sphereSnand_τis an appropriate constant.

Proof.It follows from (10) thatβ=ρ. Now we may use Lemma 6 to deduce that|∇2

ρ−∆ρ_n g|2

=0_.But, from (4) we have∆ρ=− S

n−1ρ.Therefore, we deduce∇ 2

ρ=− S

n(n−1)ρg, from which we may apply Obata’s Theorem, consult Obata (1962), to conclude thatMnis isometric to a Euclidean sphere. Moreover, from (11) we have thatT is null tensor. So, the latter claim is proved following the same steps given in the proof of Theorem 1.

Remark 1. Notice that if T =λ ρG, with (2_{+λ) ≥}0, then the conditions of the previous theorem are verified. In particular, for λ =−2, we have L_XRic=2_ρS

ng, which allows us to obtain the result due to

Obata and Yano (1965).

ACKNOWLEDGMENTS

The authors would like to thank the referees for their careful reading and useful comments which improved the paper. José N.V. Gomes would like to thank the Department of Mathematics at Lehigh University, where part of this work was carried out. He is grateful to Huai-Dong Cao and Mary Ann for their warm hospitality and constant encouragement. The authors are partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), do Ministério da Ciência, Tecnologia e Inovação (MCTI) do Brasil.

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