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Volume 2011 (2011), Article ID 812541, 4 pages
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.
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. ): 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 is a Ricci soliton if , is a Riemannian manifold such that the identity holds for some constant and some complete vector field on . If , , or , then we call it shrinking, steady, or expanding. Moreover, if the vector field appearing in (1.2) is the gradient field of a potential function , one has and says is a gradient Ricci soliton. In 2008, Lōpez and Río have shown that if is a complete manifold with and some positive constant , then is compact if and only if is bounded. Moreover, under these assumptions if is compact, then is finite. Furthermore, Wylie  has shown that under these conditions if is complete, then is finite. Moreover, in 2008, Fang et al. (cf. ) 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  Case et al. have shown that an -quasi-Einstein with and has a positive scalar curvature. Let me introduce the definition of -quasi-Einstein.
Definition 1.1. The triple is an -quasi-Einstein manifold if it satisfies the equation
for some .
Here -Bakry-Emery Ricci tensor for is a natural extension of the Ricci tensor to smooth metric measure spaces (cf. [6, Section 1 ]). Note that if , then it reduces to a gradient Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of an -quasi-Einstein with .
Theorem 1.2. Let be a complete manifold with and . Then it has a finite fundamental group.
2. The Proof of Theorem 1.2
Proof. We will prove it by dividing into two cases.Case 1. is bounded. We claim that the bounded implies the compactness of . Let be a point in , and consider any geodesic emanating from and parametrized by arc length . Then we have
Since is bounded we have that . Hence, the claim is followed by the proof of [4, Theorem 1]. Let be the Riemannian universal cover of , let be a projection map, and let be a map . Since is a local isometry, then the same inequality holds, that is, . Now, since is bounded, it is followed from the above argument that is compact. So is finite.Case 2. is unbounded. We will prove this case by following the proof of . By Case 1, is noncompact. For any , define Note that by [7, Lemma 2.2] we have Assume that . On the other hand, from the inequality of Theorem 1.2, we have since . Hence, we have that for any , Now we will apply a similar argument like Case 1. Fix , and let identified as a deck transformation on . Note that and are isometric, and thus . Also . So we conclude that for any . Since the right-hand side is independent of , this proves this case.
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