1. Introduction and Main Results

Ricci flow is introduced in 1982 and developed by Hamilton (cf. [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 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 [2] has shown that under these conditions if is complete, then 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 -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

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. 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 [2]. 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.