Conformally Flat Manifolds with Nonnegative Ricci Curvature

We show that complete conformally flat manifolds of dimension n > 3 with nonnegative Ricci curvature enjoy nice rigidity properties: they are either flat, or locally isometric to a product of a sphere and a line, or are globally conformally equivalent to R n or a spherical spaceform Sn/Γ. This extends previous results due to Q.-M. Cheng and B.-L. Chen and X.-P. Zhu. In this note, we study complete conformally flat manifolds with nonnegative Ricci curvature. It is well known in dimension 2 that the sphere, the plane and their quotients are the only surfaces that can be endowed with a metric of nonnegative curvature. As Riemannian surfaces are always conformally flat, it seems natural to look at higher dimensional analogues of this fact. Schoen and Yau showed in [20] that conformal flatness together with nonnegative scalar curvature (or any variant of it involving the Yamabe constant) still allows much flexibility. On the contrary, if one concentrates on stronger curvature conditions such as Ricci curvature bounds, one might expect that they put some quite strong restrictions on the manifold. For instance, in case the manifold is closed, various characterizations of the spherical spaceforms, or S1×Sn−1, have been obtained, but with the help of some extra assumptions like constant positive, or constant nonnegative, scalar curvature, see for instance Q.-M. Cheng [7] for results and references. In the same vein, but in the non-compact case, B.-L. Chen and X.-P. Zhu proved in [5] that the only complete non-compact conformally flat manifolds with nonnegative Ricci curvature and fast curvature decay at infinity are the complete (non-compact) flat manifolds. One motivation for this result is that it stands as an analog on real manifolds of well-known rigidity and gap phenomena on Kähler manifolds with nonnegative holomorphic bisectional curvature [6, 13, 14, 15, 16, 17]. The goal of this paper is to point out a few facts on the geometry of complete conformally flat manifolds with nonnegative Ricci curvature with no other assumption. As we shall see, quite a bit can be said in this rather general setting on the topology of those manifolds, and also on their geometry, even without any strong assumption like compactness, constant scalar curvature, or fast curvature decay. More precisely, we prove: Both authors are supported in part by an aci program of the French Ministry of Research, and the edge Research Training Network hprn-ct-2000-00101 of the European Union. 1 2 GILLES CARRON AND MARC HERZLICH Theorem. Let (M, g) be a complete conformally flat manifold of dimension n > 3 with nonnegative Ricci curvature. Then, one of the following holds: 1. M is globally conformally equivalent to R with a conformal non-flat metric with nonnegative Ricci curvature; 2. M is globally conformally equivalent to a spaceform of positive curvature, endowed with a conformal metric with nonnegative Ricci curvature; 3. M is locally isometric to the cylinder R × S; 4. M is isometric to a complete flat manifold. Obviously, the last three cases do appear. In particular, since the round metric on spherical spaceforms has positive curvature, any small conformal deformation will keep nonnegative Ricci curvature as well, hence the second class is rather large. We shall show below that the first case also occurs, by exhibiting explicit examples of non-flat globally conformally flat metrics with nonnegative Ricci curvature on R. The most interesting part of our theorem seems to lie in the last two cases, where some strong rigidity is obtained. The philosophy of our result is then : a complete conformally flat manifold with nonnegative Ricci curvature is either (globally) topologically simple (diffeomorphic to a vector space or a quotient of a sphere), or it is (locally) metrically rigid. Hence, only very general assumptions on the conformally flat manifold are enough to yield strong constraints either on its topology, or on its geometry. B.-L. Chen and X.-P. Zhu’s paper [5] showed that, in case the metric satisfies fast curvature decay assumptions at infinity, the metric is flat. It is well known under very general circumstances that fast curvature decay at infinity implies strong constraints on the topology at infinity, but our work implies further that, in the conformally flat case, either the metric is already flat or the topology is precisely that of R. Their analysis is then relevant only in that last case, and their result amounts to say that curvature, if non identically zero, cannot decay too fast to zero at infinity. We will exhibit below examples of non-flat conformally flat metrics with nonnegative Ricci curvature on R that support the idea that the decay rate chosen by B.-L. Chen and X.-P. Zhu [5] is indeed the optimal one. It remains an interesting question to understand if further constraints can be deduced on non-flat conformally flat metrics with nonnegative Ricci curvature on R Acknowledgements. The authors thank J. Droniou and J. Lafontaine for useful discussions. 1. Non trivial conformally flat metrics on R with nonnegative Ricci curvature In this short section, we show that the Theorem above is indeed the best one can hope for. As a matter of fact, we exhibit examples of (rotationnally symmetric) metrics on R that are non flat, but complete, conformally flat and with nonnegative Ricci curvature. Hence this may exist, whereas existence of any analogous metric on quotients is forbidden by our result. A by-product is the optimality of the curvature decay rate imposed by NON-NEGATIVE RICCI CURVATURE 3 Chen and Zhu in [5] to get a flat metric: our example lies exactly on the threshold where the result in [5] does not hold anymore. Construction of the example. Let f be a real function. From Besse’s book [2, formula 1.159, page 59] we get the expression for the Ricci curvature of the metric eg0 (g0 the euclidean metric) is (1.1) Ric = −(n− 2)(Ddf − df ⊗ df) + (∆f − (n− 2)|df |)g0 which can be rewritten, in case f = f(r) is a radial function and h is the unit round metric on S, as Ric =− (n− 2) ( f dr − (f )dr + rf h ) + ( −f ′′ − (n− 1)rf ′ − (n− 2)(f ) ) (dr + rh) =− (n− 1) ( f ′′ + rf ′ ) dr − ( f ′′ + (2n− 3)rf ′ + (n− 2)(f ) ) rh. (1.2) The Ricci tensor is then nonnegative iff. (1.3) f ′′ + rf ′ 6 0 and (1.4) f ′′ + (2n− 3)rf ′ + (n− 2)(f ) 6 0. It will be convenient to work with f (r) = ra(r). With this choice, the conditions for Ricci to be nonnegative become: (1.5) a 6 0 and (1.6) a + (n− 2)r(2a+ a) 6 0. A function satisfying (1.5–1.6) is easily found: for instance, one can choose any smooth function a such that (1.7) a(r) = 0 ∀r ∈ [0, 1] , a(r) = − 2 ∀r > 2 so that a(r) 6 0 and −2 6 a(r) 6 0 everywhere. We then let