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ISRN Geometry
Volume 2011 (2011), Article ID 812541, 4 pages
http://dx.doi.org/10.5402/2011/812541
Research Article

The Fundamental Groups of m-Quasi-Einstein Manifolds

School of Mathematics, KIAS, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul 130–722, Republic of Korea

Received 21 November 2011; Accepted 8 December 2011

Academic Editors: A. Borowiec and A. M. Cegarra

Copyright © 2011 Hee Kwon Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of gradient shrinking Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of m-quasi-Einstein with a positive constant multiple.

1. Introduction and Main Results

Ricci flow is introduced in 1982 and developed by Hamilton (cf. [1]):𝜕𝜕𝑡𝑔=2Ric,𝑔(0)=𝑔0.(1.1) Recently, Perelman supplemented Hamilton’s result and solved the Poincaré Conjecture and the Geometrization Conjecture by using a Ricci flow theory. But in higher dimension greater than 4 classification using Ricci flow is still far-off. Most above all the classification of Ricci solitons, which are singularity models, is not completed. But there exist many properties of Ricci solitons. Here we say 𝑔 is a Ricci soliton if (𝑀,𝑔) is a Riemannian manifold such that the identityRic+𝐿𝑋𝑔=𝑐𝑔(1.2) holds for some constant 𝑐 and some complete vector field 𝑋 on 𝑀. If 𝑐>0, 𝑐=0, or 𝑐<0, then we call it shrinking, steady, or expanding. Moreover, if the vector field 𝑋 appearing in (1.2) is the gradient field of a potential function (1/2)𝑓, one has Ric+𝑓=𝑐𝑔 and says 𝑔 is a gradient Ricci soliton. In 2008, Lōpez and Río have shown that if (𝑀,𝑔) is a complete manifold with Ric+𝐿𝑋𝑔𝑐𝑔 and some positive constant 𝑐, then 𝑀 is compact if and only if 𝑋 is bounded. Moreover, under these assumptions if 𝑀 is compact, then 𝜋1(𝑀) is finite. Furthermore, Wylie [2] has shown that under these conditions if 𝑀 is complete, then 𝜋1(𝑀) is finite. Moreover, in 2008, Fang et al. (cf. [3]) have shown that a gradient shrinking Ricci soliton with a bounded scalar curvature has finite topological type. By [4, Proposition 1.5.6], Cao and Zhu have shown that compact steady or expanding Ricci solitons are Einstein manifolds. In addition by [4, Corollary 1.5.9 (ii)] note that compact shrinking Ricci solitons are gradient Ricci solitons. So we are interested in shrinking gradient Ricci solitons. In [6, page 354], Eminenti et al. have shown that compact shrinking Ricci solitons have positive scalar curvatures. In [6] Case et al. have shown that an 𝑚-quasi-Einstein with 1𝑚< and 𝑐>0 has a positive scalar curvature. Let me introduce the definition of 𝑚-quasi-Einstein.

Definition 1.1. The triple (𝑀,𝑔,𝑓) is an 𝑚-quasi-Einstein manifold if it satisfies the equation1Ric+Hess𝑓𝑚𝑑𝑓𝑑𝑓=𝑐𝑔(1.3) for some 𝑐𝑅.
Here 𝑚-Bakry-Emery Ricci tensor Ric𝑚𝑓Ric+Hess𝑓(1/𝑚)𝑑𝑓𝑑𝑓 for 0<𝑚 is a natural extension of the Ricci tensor to smooth metric measure spaces (cf. [6, Section  1 ]). Note that if 𝑚=, then it reduces to a gradient Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of an 𝑚-quasi-Einstein with 𝑐>0.

Theorem 1.2. Let (𝑀,𝑔,𝑓) be a complete manifold with 𝑐>0 and Ric+Hess𝑓(1/𝑚)𝑑𝑓𝑑𝑓𝑐𝑔. Then it has a finite fundamental group.

2. The Proof of Theorem 1.2

The proof of Theorem 1.2 is similar to the proofs of [2, 7].

Proof. We will prove it by dividing into two cases.Case 1. 𝑓is bounded. We claim that the bounded 𝑓 implies the compactness of 𝑀. Let 𝑞 be a point in 𝑀, and consider any geodesic 𝛾[0,)𝑀 emanating from 𝑞 and parametrized by arc length 𝑡. Then we have 𝑇01Ric(̇𝛾,̇𝛾)𝑐𝑇+𝑚𝑇0(𝑑𝑓(̇𝛾))2𝑇0̇𝛾(𝑔(𝑓,̇𝛾))𝑐𝑇𝑔(𝑓,̇𝛾)|𝑇0.(2.1)
Since 𝑔(𝑓,̇𝛾)|𝑇0 is bounded we have that 0Ric(̇𝛾,̇𝛾)=. Hence, the claim is followed by the proof of [4, Theorem  1]. Let (𝑀,̃𝑔) be the Riemannian universal cover of (𝑀,𝑔), let 𝑝(𝑀,̃𝑔)(𝑀,𝑔) be a projection map, and let 𝑓 be a map 𝑓𝑝. Since 𝑝 is a local isometry, then the same inequality holds, that is, Ric(̃𝑔)+Hessẽ𝑔𝑓(1/𝑚)𝑑𝑓𝑑𝑓𝑐̃𝑔. Now, since 𝑓 is bounded, it is followed from the above argument that 𝑀 is compact. So 𝜋1(𝑀) is finite.
Case 2. 𝑓 is unbounded. We will prove this case by following the proof of [2]. By Case 1, 𝑀 is noncompact. For any 𝑝𝑀, define 𝐻𝑝max0,supRic𝑦(𝑣,𝑣)𝑦𝐵(𝑝,1),𝑣=1.(2.2) Note that by [7, Lemma  2.2] we have 𝑟0Ric(̇𝛾,̇𝛾)𝑑𝑠2(𝑛1)+𝐻𝑝+𝐻𝑞.(2.3) Assume that 𝑑(𝑝,𝑞)>1. On the other hand, from the inequality of Theorem 1.2, we have 𝑟01Ric(̇𝛾,̇𝛾)𝑑𝑠𝑐𝑑(𝑝,𝑞)+𝑚𝑟0(𝑑𝑓(̇𝛾))2𝑟0̇𝛾(𝑔(𝑓,̇𝛾))𝑐𝑑(𝑝,𝑞)𝑓𝑝𝑓𝑞,(2.4) since 𝑔(𝑓,̇𝛾)𝑓̇𝛾. Hence, we have that for any 𝑝, 𝑞𝑀1𝑑(𝑝,𝑞)max1,𝑐2(𝑛1)+𝐻𝑝+𝐻𝑞+𝑓𝑝+𝑓𝑞.(2.5) Now we will apply a similar argument like Case 1. Fix 𝑀̃𝑝, and let 𝜋1(𝑀) identified as a deck transformation on 𝑀. Note that 𝐵(̃𝑝,1) and 𝐵((̃𝑝),1) are isometric, and thus 𝐻̃𝑝=𝐻(̃𝑝). Also 𝑓̃𝑝=𝑓(̃𝑝). So we conclude that 2𝑑(̃𝑝,(̃𝑝))max1,𝑐n1+𝐻̃𝑝+𝑓̃𝑝(2.6) for any 𝜋1(𝑀). Since the right-hand side is independent of , this proves this case.

References

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