The Fundamental Groups of m-Quasi-Einstein Manifolds

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


Introduction and Main Results
Ricci flow is introduced in 1982 and developed by Hamilton cf. 1 : g 0 g 0 .

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 identity Ric L X g cg 1.2

ISRN Geometry
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, L ōpez and Río have shown that if M, g is a complete manifold with Ric L X g ≥ 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 π 1 M is finite.Furthermore, Wylie 2 has shown that under these conditions if M is complete, then 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.
Then it has a finite fundamental group.

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. ∇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 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 f • p.Since p is a local isometry, then the same inequality holds, that is, Ric g ∇ f is bounded, it is followed from the above argument that M is compact.So π 1 M is finite.
Case 2. ∇f is unbounded.We will prove this case by following the proof of 2 .By Case  for any h ∈ π 1 M .Since the right-hand side is independent of h, this proves this case.

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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 an m-quasi-Einstein manifold if it satisfies the equation Ric Hessf − 1 m df ⊗ df cg 1.3 for some c ∈ R.Here m-Bakry-Emery Ricci tensor Ric m f 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 .
Now we will apply a similar argument like Case 1. Fix p ∈ M, and let h ∈ π 1 M identified as a deck transformation on M. Note that B p, 1 and B h p , 1 are isometric, and thus H p H h p .Also ∇ f p ∇ f h p .So we conclude that r 0 γ g ∇f, γ ≥ cd p, q − ∇f p − ∇f q , 2.4since g ∇f, γ ≤ ∇f γ .Hence, we have that for any p, q ∈ M