# The Fundamental Groups ofm-Quasi-Einstein Manifolds

Authors: Hee Kwon Lee
Journal: ISRN Geometry
 (2011)

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## Abstract

In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if them-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.


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## Body

## 1. Introduction and Main Results

Ricci flow is introduced in 1982 and developed by Hamilton (cf. [1]):(1.1)∂∂tg=-2Ric,g(0)=g0.
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 identity(1.2)Ric+LXg=cg
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+LXg≥cg 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 1≤m<∞ 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 anm-quasi-Einstein manifold if it satisfies the equation(1.3)Ric+Hessf-1mdf⊗df=cg
for some c∈R.
Herem-Bakry-Emery Ricci tensor Ricfm≐Ric+Hessf-(1/m)df⊗df 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)df⊗df≥cg. Then it has a finite fundamental group.


## 2. The Proof of Theorem1.2

The proof of Theorem1.2 is similar to the proofs of [2, 7].Proof.
We will prove it by dividing into two cases.Case 1.
∥∇f∥is 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
(2.1)∫0TRic(γ̇,γ̇)≥cT+1m∫0T(df(γ̇))2-∫0Tγ̇(g(∇f,γ̇))≥cT-g(∇f,γ̇)|0T.
Sinceg(∇f,γ̇)|0T is bounded we have that ∫0∞Ric(γ̇,γ̇)=∞. Hence, the claim is followed by the proof of [4, Theorem  1]. Let (M̃,g̃) be the Riemannian universal cover of (M,g), let p:(M̃,g̃)→(M,g) be a projection map, and let f̃ be a map f∘p. Since p is a local isometry, then the same inequality holds, that is, Ric(g̃)+Hesseg̃f̃-(1/m)df̃⊗df̃≥cg̃. Now, since ∥∇̃f̃∥ is bounded, it is followed from the above argument that M̃ 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 p∈M, define
(2.2)Hp≐max⁡{0,sup⁡{Ricy(v,v):y∈B(p,1),‖v‖=1}}.
Note that by [7, Lemma  2.2] we have
(2.3)∫0rRic(γ̇,γ̇)ds≤2(n-1)+Hp+Hq.
Assume that d(p,q)>1. On the other hand, from the inequality of Theorem 1.2, we have
(2.4)∫0rRic(γ̇,γ̇)ds≥cd(p,q)+1m∫0r(df(γ̇))2-∫0rγ̇(g(∇f,γ̇))≥cd(p,q)-‖∇f‖p-‖∇f‖q,
since g(∇f,γ̇)≤∥∇f∥∥γ̇∥. Hence, we have that for any p, q∈M(2.5)d(p,q)≤max⁡{1,1c(2(n-1)+Hp+Hq+‖∇f‖p+‖∇f‖q)}.
Now we will apply a similar argument like Case 1. Fix p̃∈M̃, and let h∈π1(M) identified as a deck transformation on M̃. Note that B(p̃,1) and B(h(p̃),1) are isometric, and thus Hp̃=Hh(p̃). Also ∥∇̃f̃∥p̃=∥∇̃f̃∥h(p̃). So we conclude that
(2.6)d(p̃,h(p̃))≤max⁡{1,2c(n-1+Hp̃+‖∇̃f̃‖p̃)}
for any h∈π1(M). Since the right-hand side is independent of h, this proves this case.



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*Source: 812541-2012-04-10.xml*