TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 4, April 2007, Pages 1817–1828 S 0002-9947(06)04091-8 Article electronically published on November 3, 2006 SOME CHARACTERIZATIONS OF SPACE-FORMS STEFANO PIGOLA, MARCO RIGOLI, AND ALBERTO G. SETTI Dedicated to the memory of Franca Burrone Rigoli Abstract. Integral conditions on the traceless Ricci tensor are used to char- acterize Euclidean and hyperbolic spaces among complete, locally conformally flat manifolds of constant scalar curvature. The main tools are vanishing-type results for L p -solutions of a large class of differential inequalities. Further applications of the technique are also given. 1. Introduction A Riemannian manifold (M, 〈 , 〉) of dimension m is said to be locally conformally flat if a neighborhood of each point of M can be conformally immersed into the standard sphere S m 1 . When m ≥ 4, this is equivalent to the fact that the Weyl tensor identically vanishes. The category of locally conformally flat spaces contains the manifolds of constant sectional curvature, hence, in particular, the space-forms R m , H m -k 2 , S m k 2 . Note that, for m ≥ 3, according to the orthogonal decomposition of the Riemann tensor into its irreducible components, a conformally flat manifold has constant sectional curvature if and only if it is Einstein, i.e., the traceless part of its Ricci tensor is identically equal to zero. As a consequence, by the H. Hopf classification theorem, the space forms are (up to isometries) the only complete, simply connected, locally conformally flat, Einstein manifolds. In this paper we investigate other possible characterizations of space forms from the conformally flat viewpoint. In the late sixties M. Tani, [T], showed that the universal cover of a compact, orientable, m-dimensional, locally conformally flat Riemannian manifold (M, 〈 , 〉) with positive Ricci curvature and constant scalar curvature S is isometrically a sphere. This result has been generalized by S.I. Goldberg, [G], in the complete (non- necessarily orientable) case under the additional assumption that, for some ε> 0, (1.1) S 2 m − 1 −|Ricci| 2 ≥ ε> 0 on M (see also [H] and the further observations in [CW]). In fact, combining a classi- fication theorem by S. Zhu, [Z], with a celebrated global symmetry result due to Received by the editors January 29, 2005. 2000 Mathematics Subject Classification. Primary 53C21; Secondary 35J60, 35B05. Key words and phrases. Space forms, vanishing theorems, isolation phenomena. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 1817 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use