On the Structure of Riemannian Manifolds of Almost Nonnegative Ricci Curvature

We study the structure of manifolds with almost nonnegative Ricci curvature. We prove a compact Riemannian manifold with bounded curvature, diameter bounded from above, and Ricci curvature bounded from below by an almost nonnegative real number such that the first Betti number having codimension two is an infranilmanifold or a finite cover is a sphere bundle over a torus. Furthermore, if we assume the Ricci curvature is bounded and volume is bounded from below, then the manifold must be an infranilmanifold.


Introduction.
In this paper, we will consider a class of compact n-dimensional Riemannian manifolds (M, g) satisfying where K g , diam(M), and Ric(M) denote the sectional curvature, diameter, and Ricci curvature, respectively, of a Riemannian manifold (M, g), while D and Λ are positive real numbers and is usually a sufficiently small positive real number.
In [8], Gromov proved that there is an > 0 depending only on n and a given constant D > 0 such that if diam(M) ≤ D and Ric(M) ≥ − , then the first Betti number of M, b 1 (M), is bounded by n, that is, b 1 (M) ≤ n.Gallot [6] also gave an analytic proof for this.In [12], Yamaguchi has shown that if a Riemannian manifold (M, g) satisfies the conditions (1.1), then there is a smooth fibration where T b 1 (M) is the b 1 (M)-dimensional torus.This implies that if b 1 (M) = n, then M is diffeomorphic to the n-dimensional torus T n and if b 1 (M) = n − 1, then M is diffeomorphic to an infranilmanifold, that is, a finite covering space of M is a quotient of a simply connected nilpotent Lie group by a lattice.
In this paper, we study the structure of Riemannian manifolds satisfying the conditions (1.1) and whose first Betti number is n − 2 or n − 3.In case the first Betti number is n − 2, there are at least two known families of manifolds with metrics satisfying (1.1): infranilmanifolds and compact quotients of the product space M = S 2 ×R n−2 .We will see below that there are only such cases if the first Betti number b 1 (M) = n − 2. In case b 1 (M) = n − 3, since there is lack of examples, we only consider manifolds of dimension 4.
Throughout this paper, the dimension of manifolds is denoted by n unless otherwise stated.

Almost nonnegative Ricci curvature and the first Betti number.
In this section, we consider Riemannian manifolds (M, g) satisfying the conditions (1.1) with restriction on the first Betti number b 1 (M).First of all, we would like to mention a theorem due to Cheeger and Colding, which is crucially used in the proofs of our results.This theorem, conjectured originally by Gromov, says that the fundamental groups of a class of Riemannian manifolds with almost nonnegative Ricci curvature are almost nilpotent.
Theorem 2.1 [3].Given a positive integer n and D > 0, there exists Proposition 2.2.Given Λ > 0, D > 0, and a natural number n, there exists then M is a fiber bundle over T n−2 with the property that a finite cover of the fiber is diffeomorphic to T 2 or S 2 .
Proof.Choose >0 sufficiently small so that the properties in [12] and Theorem 2.1 hold.First note that, due to [12], M is a fiber bundle over T n−2 , that is, there is a fibration where T n−2 is the (n − 2)-dimensional torus.By the uniformization theorem, a finite cover F of F is diffeomorphic to S 2 , T 2 , or Σ, a surface of genus greater than or equal to 2. We will show that F cannot be diffeomorphic to Σ. Assume F is diffeomorphic to Σ.It follows from (2.3) that there is an exact sequence of homotopy groups (2.4) Since π 1 (M) is almost nilpotent by Theorem 2.1, the sequence (2.4) shows that π 1 (F ) is also almost nilpotent.However, since Σ is a surface of genus greater than or equal to 2, it is well known that π 1 (Σ) cannot be almost nilpotent.Hence the proof is complete.
Before going ahead, we state a basic algebraic lemma about a geometric group, which follows actually from [9].Lemma 2.3 [9,14].Let Γ be a finitely generated group of polynomial growth.Then it contains a torsion-free nilpotent subgroup of finite index.
A solvable group Γ is called polycyclic if there is a subnormal series where factors Γ i /Γ i+1 are all infinite-cyclic and e denotes the identity element in Γ .A solvable group is almost polycyclic if it contains a subgroup of finite index, which is polycyclic.The number of infinite cyclic factors is independent of the choice of finiteindex subgroup or subnormal series, and is called the Hirsch length of the group.Now we prove our main theorem as an application of Proposition 2.2 by using Lemma 2.3 and Theorem 2.1 Theorem 2.4.Given Λ > 0, D > 0, and a natural number n, there exists Proof.Choose > 0 sufficiently small so that Proposition 2.2 holds.Suppose (M, g) is a Riemannian n-manifold satisfying (2.2).M is a fiber bundle over T n−2 with a fiber being a quotient of S 2 or T 2 .It is enough to show that if the fiber is a quotient of T 2 , then M is an infranilmanifold.By Theorem 2.1 again, π 1 (M) is almost nilpotent.So, by Lemma 2.3, π 1 (M) has a torsion-free nilpotent subgroup of finite index Γ .From the above fibration, we have an exact sequence of homotopy groups where K is isomorphic to Z 2 ⊕ H and H is a finite group.Note that the universal covering M of M is diffeomorphic to R n and Γ has Hirsch length n.The nilpotent Malcev completion N of Γ can now be identified with M. So, M is a simply connected nilpotent Lie group with a lattice subgroup Γ .This means that M is an infranilmanifold.Remark 2.5.A converse of Theorem 2.4 holds, that is, any nilmanifold or any S 2bundle over T n−2 has Riemannian metrics which satisfy (2.2) for any .
Theorem 2.7.For given Λ > 0 and D > 0, there exists = (Λ,D) > 0 such that if (M 4 ,g) is a compact Riemannian 4-manifold satisfying then M is a fibration over S 1 whose fiber is homotopic to a spherical space form S 3 /Γ for some finite subgroup Γ acting on S 3 .
Proof.By [12], there exists an > 0 such that if (M n ,g) is a closed Riemannian manifold satisfying (2.7), then M is a fibration over S 1 , (2.8) On the other hand, by Theorem 2.1, π 1 (M) is almost nilpotent, and so it has a polynomial growth.It follows from Lemma 2.3 that π 1 (M) contains a torsion-free nilpotent subgroup Γ of finite index.Since b 1 (M) = 1, Γ is abelian, and so Γ Z.In fact, if Γ is not abelian, then it contains a subgroup which is isomorphic to the Heisenberg group (see Remark 2.8), and so the growth of Γ is at least 4 and b 1 (M) ≥ 2 (cf.[2, Section 7]).Thus, 1 (M) Z ⊕ H, where H is a finite group and π 1 (F ) is also finite group.Hence the universal cover F of F is a compact simply connected 3-manifold, and so F is a homotopy 3-sphere, that is, F is homotopic to S 3 /H for some finite group H acting on S 3 .Remark 2.8.In dimension n ≥ 5, replacing the condition on the first Betti number by b 1 (M) = n − 3, Theorem 2.7 does not hold anymore.For example, let N be the Heisenberg group and Γ its integer lattice.Then M := N/Γ is a compact orientable 3-dimensional nilmanifold.It is well known that b 1 (M) = 2 and M is an S 1 -bundle over T 2 .For a given > 0, since M is a nilmanifold, there is a metric g such that K g ≤ 24 2 , diam(M) ≤ 2. (2.10) Now consider the product (M × S 2 ) so that it satisfies the condition (2.7).It is easy to see that M × S 2 is a fibration over T 2 with fiber S 1 × S 2 .

Ricci curvature pinching.
If one replaces the lower bound on Ricci curvature by pinching and adds the lower volume bound, then one can prove that the second case in Theorem 2.4 does not happen.In [3], Cheeger and Colding extended the splitting theorem of sectional curvature version to that of Ricci curvature version.Namely, the splitting theorem does hold for the limit space of Gromov-Hausdorff convergent sequence each term of which satisfies a diameter upper bound and Ricci condition that Ric(M i ,g i ) ≥ − i → 0. Thus, using the abelian covering manifold which gives an extended version of splitting theorem and modifying the argument in [13] a little bit, one can easily prove the following lemma.

Lemma 3.1. Let M i be a sequence of compact Riemannian n-manifolds with Ric(M
and M i the universal cover of M i .Then, for any p i ∈ M i , ( M i , g i , p i ) subconverges to (R k × X 0 ,x 0 ,d) in the pointed Gromov-Hausdorff distance, where k ≥ b 1 , and X 0 is a compact length space.
We would like to remark that the dimension of the Euclidean factor is greater than or equal to the first Betti number.Theorem 3.2.Given Λ > 0, v > 0, D > 0, and a natural number n, there exists then M is an infranilmanifold.
Proof.Suppose the theorem does not hold.Then there are a sequence of positive real numbers i → 0 and a sequence of Riemannian n-manifolds (M i ,g i ) satisfying (3.1), but M i is not an infranilmanifold for all i.
With the volume condition, the standard Cheeger-Gromov compactness theorem [7,10] tells that there exists a subsequence of (M i ,g i ) converging to a smooth n-manifold with a C 1,α Riemannian metric (M, g) in the C 1,α topology with 0 < α < α.In particular, M i is diffeomorphic to M for all i sufficiently large.Furthermore, since | Ric(M i ,g i )| ≤ i → 0, the Ricci equation argument in harmonic coordinates [1] shows that the metric g is, in fact, C ∞ .Consequently, (M i ,g i ) subconverges to a smooth Ricci flat Riemannian manifold (M, g) in the C ∞ topology.This, together with the curvature condition, implies that the universal cover M i converges to the universal cover M (cf.[5,Theorem 2.7]).Now, applying Lemma 3.
where X 0 is a compact Riemannian manifold.Since g is Ricci-flat, X n−k 0 is also a Ricci flat manifold.Since n − k ≤ 2, X n−k 0 is a flat manifold, and so g is a flat metric on M. Therefore, M i admits a flat metric for i sufficiently large and so does M i .
On the other hand, since M i is not an infranilmanifold, Theorem 2.4 shows that M i is diffeomorphic to S 2 × R n−2 .So, S 2 × R n−2 admits a flat metric, but this is impossible because of the Cartan-Hadamard theorem.The proof is complete.Remark 3.3.In the collapsing case, the same result as Theorem 3.2 might also hold.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation