#### Abstract

We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.

#### 1. Introduction

In comparison geometry of Ricci curvature, the classical Bishop-Gromov volume comparison has many applications, such as at least the linear volume growth of complete noncompact Riemannian manifolds with nonnegative Ricci curvature (see [1]), the upper bound of total Betti number (growth) of Riemannian manifolds (see [2–4]), and the finite topological type of complete noncompact Riemannian manifolds with nonnegative Ricci curvature or quadratic Ricci curvature decay (see [3, 5, 6]).

In [7], Lott and Shen establish a volume comparison estimate with quadratic Ricci curvature decay, and apply it to investigate the finite topological type of complete noncompact Riemannian manifolds with quadratic Ricci curvature decay, which generalizes a related result by Sha and Shen in [6].

In [8], we apply the volume comparison with asymptotically nonnegative Ricci curvature to investigate the corresponding topological results for manifolds with asymptotically nonnegative Ricci curvature.

In this paper, we will extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to general radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number and finite topological type of manifolds with non-asymptotically almost nonnegative Ricci curvature. (See Definitions 1.1 and 1.2 below for the notions of radially symmetric Ricci curvature lower bound, asymptotically almost nonnegative Ricci curvature, and non-asymptotically almost nonnegative Ricci curvature, resp.)

Note that quadratic Ricci curvature decay is non-asymptotically almost nonnegative Ricci curvature, so our result is a generalization of the corresponding result of Lott and Shen in [7] mentioned above. (See Theorem 1.7.)

*Definition 1.1. *Let be a complete -Riemannian manifold *, *, and *. * has a radially symmetric Ricci curvature lower bound, at the point if there exists a continuous function such that, for any tangent vector radial from the point *, *

One can refer to [9] for generalized space forms with radially symmetric curvature and the notion of tangent vector radial from a point.

*Definition 1.2. *Let a complete noncompact -Riemannian manifold *, **, * be a continuous positive function, and *. *(i) has almost nonnegative Ricci curvature if Furthermore, has asymptotically nonnegative Ricci curvature if and . has non-asymptotically almost nonnegative Ricci curvature if and .

The following is a volume comparison estimate for manifolds with general radially symmetric Ricci curvature lower bound, which is a generalization of that for manifolds with asymptotically nonnegative Ricci curvature and quadratic Ricci curvature decay by Zhu in [10] and Lott and Shen in [7], respectively.

Theorem 1.3. *Let be a complete -Riemannian manifold with a radially symmetric Ricci curvature lower bound at the point , and let , , , , be a measurable subset of the unit sphere in the tangent space :
**
Then
**
where is the unique solution of one of the following two equations:
**In particular, if is the unique solution of (1.4), then **
if is the unique solution of (1.5), then
** If there exist constants such that the unique solution of (1.4) or (1.5) satisfies , then one has a constant depending only on and such that *

*Remark 1.4. *The condition on in (1.5) constitutes an extra assumption imposed on the unique solution of (1.4). In Theorem 1.3, we do not require that the radially symmetric Ricci curvature lower bound corresponds to the generalized space forms with radially symmetric curvature lower bound. Our purpose is to establish a volume comparison estimate effectively.

Applying the generalized volume comparison estimate, we can now investigate the volume growth, total Betti number, and finite topological type of manifolds with non-asymptotically almost nonnegative Ricci curvature.

Theorem 1.5. *Let be a complete -Riemannian manifold with non-asymptotically almost nonnegative Ricci curvature at the point , and let be noncollapsing, that is, . If there exist constants such that, the unique solution of (1.4) or (1.5) satisfies , then one has a constant depending only on and such that for ,
*

Theorem 1.6. *Let be a complete -Riemannian manifold with non-asymptotically almost nonnegative Ricci curvature at the point , and has weakly bounded geometry, that is, sec, and .** If there exist constants such that the unique solution of (1.4) satisfies , then one has a constant depending only on , and such that, for ,** If there exist constants such that the unique solution of (1.5) satisfies , then one has a constant , depending only on , , , and such that, for ,*

Theorem 1.7. *Let be a complete -Riemannian manifold with non-asymptotically almost nonnegative Ricci curvature at the point , where , , and let be non-collapsing, that is, . If there exist constants such that the unique solution of (1.4) or (1.5) satisfies , then is of finite topological type with the additional assumption that
**
for some constant depending only on , , , and . *

#### 2. A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound

*Proof of Theorem 1.3. *Choose polar coordinate at . Define the function by the formula
Then
where is the distance from to the cut point in direction . It is well known (e.g., [11]) that satisfies the following:
Let be the unique solution of one of (1.4) and (1.5) (Note that, by the uniqueness of the solution of ordinary differential equation, the solution of (1.4) always exists.).

Then in the interval of , , that is, . By the initial condition of and , . Thus, when ,
This shows that is nonincreasing in the interval of .

Note that in the interval of we must have . Thus it suffices to consider that is the unique solution of (1.4).

Otherwise, suppose that is the first point such that in , , and in . By , , is non-increasing in :
Let , then . This is a contradiction.

Thus consider the following lemma.

Lemma 2.1 (see [12]). * Let be positive functions on ; if is nonincreasing, then for all , , , , one has
*

We have where the last equality is due to Then by integration on , we have Similarly,

In particular, (1)

Let , then

When is the unique solution of (1.4), let ; by , then we have

When is the unique solution of (1.5), let , we have

() Choose ; for , an easy computation shows that

#### 3. Proof of Theorem 1.5

*Proof of Theorem 1.5. *Note that for there exists a point such that ; thus
And since does not collapse at infinity, that is, , for , we have
Thus, for , by Theorem 1.3(2), there is some constant such that . And note that for ,
Theorem 1.5 is obtained.

#### 4. Proof of Theorem 1.6

First let us recall Gromov's theorems [2]; one can refer to [13] for the details.

Theorem 4.1 (see [2]). * Let be an -dimensional complete Riemannian manifold with sectional curvature . Then there is a constant depending only on such that, for any and any bounded subset ,
**
where denotes the -neighborhood of in . *

Theorem 4.2 (see [2]). *Let be an -dimensional complete Riemannian manifold and let . For any fixed numbers and , let , , be a ball covering of with . Let , . Then
**
where is the smallest number such that, each ball intersects at most other balls .*

*Proof of Theorem 1.6. *By Theorem 4.1, there is a constant depending only on such that for all balls with radius in ,

Take , and let , , be a maximal set of disjoint balls with , and let , , , be the same as in Theorem 4.2. Then , , is a covering of . And let , be the same as in Theorem 4.2.

If there exist constants such that the unique solution of (1.4) satisfies , choosing , in Theorem 1.3(1), then, for ,
Then by the assumption that ,
Since each ball has radius , it follows from (4.3) and Theorem 4.2 that

If there exist constants such that the unique solution of (1.5) satisfies , choosing , in Theorem 1.3(1), then, for ,

Similar to the above, there exists a constant such that

#### 5. Proof of Theorem 1.7

We use critical point theory of the distance function to prove Theorem 1.7.

First of all, we recall some concepts (cf., e.g., [3, 7, 14]). Notice that the distance function is not a smooth function (on the cutlocus of ). Hence the critical points of are not defined in a usual sense. The notion of critical points of is introduced by Grove and Shiohama [15].

A point is called a *critical point* of if for any unit vector there is a minimizing geodesic from to such that .

For every , the open set contains only finitely many unbounded components, . Each has finitely many boundary components, . In particular, is a closed subset. Let us say that a connected component of is *good* if it is part of the boundary of an unbounded component of and there is a ray from passing through .

Now we can introduce the following lemma.

Lemma 5.1 (see (Lemma , [7]); cf., also [14]). * Suppose that there is a such that if then there is no critical point of on any good component of . Then has finite topological type.*

Another concept is the * diameter growth function *.

*Definition 5.2. *The diameter growth function is defined by
where the supremum is taken over all good components of and the diameter is measured using the metric on .

*Proof of Theorem 1.7. *(i) We first show that if a complete noncompact Riemannian manifold satisfies , where , , and the following diameter growth condition
where
then is of finite topological type.

As (i) in the proof of Theorem in [8], choose a good connected component of , for any , and a ray from passing through , choose such that , and suppose that is a critical point of , then
On the other hand, by the triangle inequality,
thus,
For large enough, by the assumption on the diameter growth, this is a contradiction.

Thus, there does not exist a critical point of on any good connected component. By Lemma 5.1, is of finite topological type.

(ii) Given that , choose a good connected component , of the boundary of an unbounded component of . For any , there is a continuous curve from to . Suppose that . Then there is a partition such that are disjoint and . Note that . Thus
Then, by Theorem 1.3(2), there is a constant such that if the volume growth satisfies
the diameter growth satisfies
Then by (i), Theorem 1.7 is obtained.

#### Acknowledgment

The authors would like to thank the referee for the comments and suggestions.