The Manifolds with Ricci Curvature Decay to Zero

The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.


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In 1971, J. Cheeger and D. Gromoll [1] proved the following classical: Cheeger-Gromoll Splitting Theorem: Let M be a complete Riemannian manifold with 0 RicM  .If there is a line in M, then the isometrically splitting M R N     : 0, is true.The proof of Cheeger-Gromoll Splitting Theorem is based on the sub-harmonicity of the Busemann functions, we will give some details in what follows.Let M be a noncompact complete Riemannian manifold and M    B be a ray of M. Busemann function  is defined as For a given point x in M and an arbitrary , let is embedding, so is a differential homeomorphic mapping.Thus for any . Let be the parallel vector field along .
By [2], one has The outline of the proof of Cheeger-Gromoll Theorem: then,  is subharmonic function, where S is a operator generalized from Laplace operator , one can refer to [2] for details.
, by 1), one is able B to prove that   and   are harmonic functions on

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The paper is supported by NSF of Fujian province and Pan Jinlong's SF of Jimei University, China.
grad f df  is a unite parallel it can be proved that vector field on M, where is the couple tangential vector field of    induced by the Riemannian metric.

4) By de Rham Partition Theorem, one has that
Cheeger-Gromoll Splitting Theorem and its proof are so excellent that there is few generalization can be found, the only result we known is in Cai's paper [3], in which a local splitting theorem was got.In order to narrate our main result, we quote the following.
Definition 1: Let M be a noncompact complete Riemannian manifold.Suppose that there are two continuous functions , such that for any normal shortest geodesic then we say M is with Ricci curvature decay to zero.
From the definition, according to (4), (5), it is clear of that A simple example of satisfying ( 4) is defined as we have: Theorem 1: Let M be with Ricci curvature decay to zero.If there is a line included in M, then isometrically is still true.By the way, when one discusses the relationship between a kind of curvature and topology of a Riemannian manifold, he generally assume that the sign of the kind of curvature is fixed, for examples, in [1] and [5], the authors assumed that the manifolds is with nonnegative Ricci curvature.If without this assumption, the corresponding problem seems more difficult, this is the reason we write out this short paper, though it is not easy to construct a manifold with Ricci curvature decay to zero for the time being.

The Proof of Theorem 1
Our argument follows closely that of Cheeger-Gromoll Splitting Theorem, but we should overcome some difficulties, especially in how to prove grad f is a unite parallel vector field of M.
Lemma 1: Let M be a complete noncompact Riemannian manifold with Ricci curvature decay to zero.Then the Busemann functions on M are subharmonic. Proof: be a ray in M. Just like above discussion, we can construct a smooth g supports By the assumption (4) and L'Hospital Rule, it is obvious that By ( 6) and ( 7), we have which means that B is subharmonic.a complete non-

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The proof of Theorem 1: Let M be compact Riemannian manifold with Ricci curvature decay to zero.Then the Busemann functions on M are subharmonic.If there is a line By maximum principle, B   and B   are harmonic fu he nctions on M. By canonical Weyl T orem (c.f.[2]), we know that they are all smooth in M. In simplicity, we set By (9), B has not critical point, this means M t is smooth hypersurface of M.
Supposed that X and Y are tangential vector fields on M, the Hessian of B satisfies In particular, by (9), This means the integral curves of grad B are the geod ve esics in M. It is clear that grad B is unit normal vector field of M t .By reviewing the definition of mean curvature of the horizontal hypersurface M t , we have N gradB  , by 1.5.8 of [4], Setting is a ray emanating from x, which is asymptotic to  .By [2,5], we know that X is an arbitrary tangential vector field on e M, , it is easily know that N gradB  is a unit parallel vector field of M. By de Rham Partition Theorem, we have M R N   .Thus we get Theorem 1.