^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

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.

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 [

In [

In [

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

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 [

Let

One can refer to [

Let

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 [

Let

In particular,

The condition

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.

Let

Let

Let

Choose polar coordinate

Then in the interval of

Note that in the interval of

Otherwise, suppose that

Thus consider the following lemma.

Let

We have

In particular, (1)

Let

When

When

(

Note that for

First let us recall Gromov's theorems [

Let

Let

By Theorem

Take

If there exist constants

If there exist constants

Similar to the above, there exists a constant

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

First of all, we recall some concepts (cf., e.g., [

A point

For every

Now we can introduce the following lemma.

Suppose that there is a

Another concept is the

The diameter growth function

(i) We first show that if a complete noncompact Riemannian manifold satisfies

As (i) in the proof of Theorem

Thus, there does not exist a critical point of

(ii) Given that

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