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 admitting a metric of constant scalar curvature and carrying a nontrivial conformal vector field . Among them, we cite Bochner and Yano (19532 BOCHNER S AND YANO K. 1953. Curvature and Betti Numbers. Ann Math Study. Princeton University Press, Princeton, New Jersey, US.), Goldberg and Kobayashi (19624 GOLDBERG SI AND KOBAYASHI S. 1962. The conformal transformation group of a compact Riemannian manifold. Amer J Math 84: 170-174.), Lichnerowicz (19556 LICHNEROWICZ A. 1955. Transformations infinitésimales conformes de certaines variétés riemannienne compacte. CR Acad Sci Paris 241: 726-729.), Nagano and Yano (19597 NAGANO T AND YANO K. 1959. Einstein spaces admitting a one-parameter group of conformal transformations. Ann Math 69: 451-461.), Obata (19628 OBATA M. 1962. Certain conditions for a Riemannian manifold to be isometric with a sphere. J Math Soc Japan 14: 333-340.), Obata and Yano (19659 OBATA M AND YANO K. 1965. Sur le groupe de transformations conformes d’une variété de Riemann dont le scalaire de courbure est constant. CR Acad Sci Paris 260: 2698-2700., 197010 OBATA M AND YANO K. 1970. Conformal changes of Riemannian metrics. J Differential Geom 4: 53-72.) and Tashiro (196511 TASHIRO Y. 1965. Complete Riemannian Manifolds and some vector fields. Amer Math Soc 117: 251-275.). 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 product , where is a compact Riemannian manifold of constant scalar curvature, while stands for the Euclidean circle. In his example the conformal vector field is , where is a unit vector field on and satisfies a certain ordinary differential equation, see Ejiri (19813 EJIRI N. 1981. A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J Math Soc Japan 33: 261-266.) 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 tensor (or of a vector field ) with respect to a vector field This is a method that uses the flow of to push values of back to and then differentiate. The result is called the Lie derivative of with respect to . A vector field on Riemannian manifold is called conformal if is a multiple of . 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 field with respect to is zero if and only if is invariant under the flow of . It turns out that the Lie derivative of a tensor has exactly the same interpretation. For more details see the book of Lee (20035 LEE JM. 2003. Introduction to smooth Manifolds. Springer-Verlag, New York, US.).
We highlight that Nagano and Yano (1959) have proved that the aforementioned conjecture is true when is an Einstein manifold, i.e., the Ricci tensor of metric satisfies . In this case, is constant for dimensions . Thus if is the conformal vector field with conformal factor that is, we deduce With this setting we define a conformal Ricci vector field on a Riemannian manifold as a vector field satisfying
for some smooth function 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 be a compact orientable Riemannian manifold with constant scalar curvature carrying a nontrivial conformal vector field which is also a conformal Ricci vector field. Then is isometric to a Euclidean sphere . Moreover, up to a rescaling, the conformal factor is given by
and is the gradient of the Hodge-de Rham function which in this case, up to a constant, is the function , where is a height function on a unitary sphere and is an appropriate constant.
We point out that Obata and Yano (1965) have obtained the same conclusion of the preceding theorem under the hypothesis for some smooth function defined in Moreover, when 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 of in our result is an essential hypothesis. In fact, let us consider the hyperbolic space as a hyperquadric of the Lorentz-Minkowski space and a nonzero fixed vector in . After a straightforward computation, it is easy to verify that the orthogonal projection of onto the tangent bundle provides a nontrivial conformal vector field on for an appropriate choice of . Consequently, since is Einstein, it follows that is also a nontrivial conformal Ricci vector field on .
PRELIMINARIES AND AUXILIARY RESULTS
To start with, we consider the Hilbert-Schmidt norm for tensors on a Riemannian manifold , i.e., the inner product It is important to notice that for an orthonormal basis , we can use the natural identification of -tensors with -tensors, , to write
We recall that the divergence of a -tensor on is the -tensor given by
where and
Let be a smooth vector field on , and let be its flow. For any , if is sufficiently close to zero, is a diffeomorphism from a neighborhood of to a neighborhood of , so pulls back tensors at to ones at .
Given a -tensor on , the Lie derivative of with respect to is the -tensor given by
Fortunately, there is a simple formula for computing the Lie derivative without explicitly finding the flow. For any -tensor on ,
where are any smooth vector fields on , and stands for the Lie bracket of and
In what follows we prove some lemmas and integral formulas which will be required later.
Lemma 1. For any symmetric -tensor on a Riemannian manifold and , holds
In particular, if is a conformal vector field with conformal factor then we have
Proof. Let be a geodesic orthonormal frame at . Then we have
Whence, we use that is symmetric to deduce
which completes the proof of the lemma.
Corollary 1. Under the assumptions of Lemma 1 we have provided that and
Another useful result is given by the following lemma.
Lemma 2. Let be a Riemannian manifold endowed with a symmetric -tensor Then it holds
In particular, if then
Proof. Let be a geodesic orthonormal frame at a point By the symmetry of , we have
that finishes the proof of the lemma.
Now we remind the reader that the traceless tensor of a symmetric -tensor on a Riemannian manifold is given by
With this setting we prove the next corollary.
Collary 2. Let be a Riemannian manifold endowed with a symmetric -tensor such that , with Then we have:
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If is a non null constant, then and
Proof. Since we have immediately the first item. Now, if is a non zero constant, then we have which implies finishing the proof of the corollary.
Next, we apply the previous results to the Ricci tensor. First of all, given a conformal vector field on a Riemannian manifold such that we have the next well-known formulae, see e.g. Obata and Yano (1970).
and
where
We claim that if is compact with constant scalar curvature and is not constant, then equation (4) allows us to infer that is positive. Indeed, since is not constant belongs to the spectrum of the Laplacian of . Therefore, we deduce the next lemma.
Lemma 3. Let be a compact Riemannian manifold with constant scalar curvature such that and . Then we have and
Proof. Since and is a positive constant we get by Corollary 2 that and Therefore, applying Corollary 1 we have which completes the proof of the lemma.
Taking into account the second contracted Bianchi identity: and using the identity we obtain the following relation
Therefore, we can write
The next equation is well-known. For details of a more general case see for example Lemma in Barros and Gomes (20131 BARROS A AND GOMES JN. 2013. A Compact gradient generalized quasi-Einstein metric with constant scalar curvature. J Math Anal Appl 401: 702-705.).
Since , from (5) we have
Applying Lemma 1 to this identity we obtain
Comparing (6) and (7) we infer
In what follows we assume that is an orientable Riemannian manifold. If is not orientable, we take the orientable double covering of and induce, in the natural manner, the Riemannian metric on . Then and have the same local geometry.
As a direct consequence from (8) and Stokes’ Theorem we obtain the following lemma.
Lemma 4. Let be a compact orientable Riemannian manifold endowed with a conformal vector field of conformal factor , then
Before stating our next result we note that . Accordingly, the Bochner formula becomes
where in the last term we use equation (4). Then,
By integration,
In the notation of (2) we have . Comparing Lemma 4 with equation (9) we get the lemma.
Lemma 5. Let be a compact orientable Riemannian manifold endowed with a conformal vector field of conformal factor , then
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 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 field on such that and , for some smooth functions and on , where is non-constant. Consequently, from Lemma 3 we obtain and . Secondly, from item (1) of Lemma 5 and equation (4) we get
Taking into account that is non-constant the preceding identity allows us to achieve and Therefore, we can apply a classical result due to Obata (1962), for instance, to conclude that is isometric to a Euclidean sphere . Moreover, and (see equation (4)). Rescaling the metric we can assume that . Then we conclude that is the first eigenvalue of the unitary sphere Whence, there exists a fixed vector such that . Thus we have which gives . Setting we obtain
It is also true that . Besides, by Hodge-de Rham decomposition theorem we can write , for some vector field with and is the Hodge-de Rham function. So, which implies is constant. Note that this is sufficient to complete our proof.
A MORE GENERAL CASE
We notice that and give
where Therefore, we deduce
where
Let us suppose that and , where is a -tensor on . By using (3) and (4) we deduce
and
In particular, if is a non null constant and , we have
On the other hand, gives
and
Lemma 6. Let , be a compact orientable Riemannian manifold endowed with a conformal vector field whose conformal factor is If , then the following integral formula holds:
Proof. First we notice that from (12) we obtain
Therefore, using (13) in the first assertion of Lemma 5, we have
which completes the proof of the lemma.
Proceeding we use this lemma to obtain the following theorem.
Let , , be a compact orientable Riemannian manifold with constant scalar curvature . Suppose that there exists a nontrivial conformal vector field on such that and . If and , then is isometric to a Euclidean sphere.
Moreover, up to a rescaling, the conformal factor is given by and is the gradient of the Hodge-de Rham function which in this case, up to a constant, is the function , where is a height function on a unitary sphere and is an appropriate constant.
Proof. It follows from (10) that . Now we may use Lemma 6 to deduce that But, from (4) we have Therefore, we deduce , from which we may apply Obata’s Theorem, consult Obata (1962), to conclude that is isometric to a Euclidean sphere. Moreover, from (11) we have that is null tensor. So, the latter claim is proved following the same steps given in the proof of Theorem 1.
Notice that if , with , then the conditions of the previous theorem are verified. In particular, for , we have , 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.
REFERENCES
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1BARROS A AND GOMES JN. 2013. A Compact gradient generalized quasi-Einstein metric with constant scalar curvature. J Math Anal Appl 401: 702-705.
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2BOCHNER S AND YANO K. 1953. Curvature and Betti Numbers. Ann Math Study. Princeton University Press, Princeton, New Jersey, US.
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3EJIRI N. 1981. A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J Math Soc Japan 33: 261-266.
-
4GOLDBERG SI AND KOBAYASHI S. 1962. The conformal transformation group of a compact Riemannian manifold. Amer J Math 84: 170-174.
-
5LEE JM. 2003. Introduction to smooth Manifolds. Springer-Verlag, New York, US.
-
6LICHNEROWICZ A. 1955. Transformations infinitésimales conformes de certaines variétés riemannienne compacte. CR Acad Sci Paris 241: 726-729.
-
7NAGANO T AND YANO K. 1959. Einstein spaces admitting a one-parameter group of conformal transformations. Ann Math 69: 451-461.
-
8OBATA M. 1962. Certain conditions for a Riemannian manifold to be isometric with a sphere. J Math Soc Japan 14: 333-340.
-
9OBATA M AND YANO K. 1965. Sur le groupe de transformations conformes d’une variété de Riemann dont le scalaire de courbure est constant. CR Acad Sci Paris 260: 2698-2700.
-
10OBATA M AND YANO K. 1970. Conformal changes of Riemannian metrics. J Differential Geom 4: 53-72.
-
11TASHIRO Y. 1965. Complete Riemannian Manifolds and some vector fields. Amer Math Soc 117: 251-275.
Publication Dates
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Publication in this collection
Jul-Sep 2018
History
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Received
06 Sept 2017 -
Accepted
04 Apr 2018