# Geometry of Positive Scalar curvature on Complete Manifold

Bo Zhu  School of Mathematics, University of Minnesota-Twin Cities, MN 55455,
USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the interplay of geometry and positive scalar
curvature on a complete, non-compact manifold with non-negative Ricci
curvature. In three-dimensional manifold, we prove a minimal volume growth, an
estimate of integral of scalar curvature and width. In higher dimensional
manifold, we obtain a volume growth with a stronger condition.

###### Key words and phrases:

Positive scalar curvature, Volume growth, Yau’s problem, Integral of scalar
curvature, Uryson width

###### 2020 Mathematics Subject Classification:

Primary 53C21

##  1\. Introduction

An important topic in geometric analysis is to understand the interplay
between curvature and geometry. One of the classical and widely popularized
results in this aspect is the Myers and Cheng’s maximal diameter theorem.

######  Theorem 1.1 (Maximal Diameter Theorem, [8, 25]).

Any complete Riemannian manifold $(M^{n},g)$ with the Ricci curvature
$Ric(g)\geq n-1$ has $\mathrm{Diam}(M)\leq\pi$ with equality if and only if
$M$ is a round sphere.

From the perspective of size geometry [11], i.e., diameter, volume, Uryson
width, Filling Radius, injectivity radius, etc. are called the size quantities
of a Riemannian manifold. Myers and Cheng’s maximal diameter theorem indicates
that the positivity of Ricci curvature completely controls its distance spread
in all directions, i.e., diameter. Here, we primarily focus on the size of
Riemannian manifold and metric structure of Riemannian manifold. A natural
problem is that how we could generalize theorem 1.1 to the scalar curvature in
some sense. It is clear that positive scalar curvature on a Riemannian
manifold can not determine its own distance spread fully. For instance,
$(\mathbb{S}^{2}\times\mathbb{R}^{n-2},g)$ has scalar curvature $2$ but no
control of the diameter if $g$ is the standard direct product of Riemannian
metric. Based on this basic example, we could never expect that the positivity
of scalar curvature on a Riemannian manifold can fully control its size. In
fact, the most promising expectation is that the positivity of scalar
curvature on a Riemannian manifold should have control on the size, which is
only related to $1$ or $2$ dimensional quantities. In this direction, Gromov
conjectures that

######  Conjecture 1.2 (Gromov, [14]).

Let $(M^{n},g)$ be a complete non-compact manifold with the scalar curvature
$Sc(g)\geq n(n-1)$. Then the macroscopic dimension of $M$ satisfies

$macrodim(M)\leq n-2.$

It remains open even for $marcodim(M)\leq n-1$. In the view of geometric
dimension theory, Conjecture 1.2 is equivalent to the statement that $M$ can
be approximated by $n-2$ dimensional polyhedrons within finite distance. For
the definition of macroscopic dimension, the reader can refer to the
references [11, 14], and we will not use the definition of macroscopic
dimension directly. However, in the spirit of it, Conjecture 1.2 provides the
insight for the following conjectures and the results of our paper.

Together with Theorem 1.1, it is proved that for the conjugate radius
$\mathrm{conj}(M)$ of a Riemannian manifold $M$,

######  Theorem 1.3.

[Maximal Conjugate Radius Theorem, [10]] Let $(M^{n},g)$ be a closed
Riemannian manifold with the scalar curvature $Sc(g)\geq n(n-1)$. Then
$\mathrm{conj}(M)\leq\pi,$ and equality holds if and only $M$ is isometric to
the round sphere $\mathbb{S}^{n}$.

Since the injectivity radius $\mathrm{inj}(M)\leq\mathrm{conj}(M)$, we obtain
that Theorem 1.3 implies that for any closed Riemannian manifold with
$Sc(g)\geq n(n-1)$, $\mathrm{Inj}(M)\leq\pi,$ and equality holds if and only
$M$ is isometric to the round sphere $\mathbb{S}^{n}$, which we would like to
name as maximal injectivity radius theorem. For the details of the proof of
Theorem 1.3, see [10, 19].

To my best knowledge, it remains open whether maximal conjugate or injectivity
radius still hold on any complete non-compact Riemannian manifold. In fact,
this question is deeply related to Conjecture 1.2. Here, we apply the
techniques used in the case of closed Riemannian manifold to the complete,
non-compact Riemannian manifold $(M^{n},g)$ and then obtain a local estimate
on the integral of scalar curvature.

######  Proposition 1.4.

Let $(M^{n},g)$ be a complete, non-compact Riemannian manifold. Then,

  * •

Suppose that $B(p,R)\subset M$ is a geodesic ball with center $p\in M$ and
radius $r>0$. If $Ric(g)\geq 0$ on $B(p,R)$, then we obtain,

(1) $\int_{B(p,R-c)}Sc\leq n(n-1)(\frac{\pi}{c})^{2}\mathrm{vol}B(p,R),\forall
c\leq\mathrm{conj}(M);$

  * •

If $Sc(g)\geq n(n-1)$ on $M$ and $Ric(g)\geq 0$ on $M$, then
$\mathrm{inj}(M)\leq\mathrm{conj}(M)\leq\pi$.

The conjugate radius estimate indicates that positive scalar curvature does
imply that a Riemannian manifold is curved or becomes thinner under the
assumption of non-negative Ricci curvature and strictly positive scalar
curvature. However, we still do not know whether Proposition 1.4 holds without
any assumptions on non-negative Ricci curvature. On the one hand, if one can
construct a complete, non-compact Riemannian manifold with positive scalar
curvature and its injectivity radius is infinity, then it will deduce a
negative answer to Conjecture 1.2; on the other hand, the local estimate (1)
indicates that the average of the integral of scalar curvature is bounded
above in terms of the lower bound of the conjugate radius. Also, it implies
that the volume growth and positive scalar curvature are intertwined locally.
However, the interplay is still a mystery on a global scale. Many years ago,
Yau proposed the following problem, which is involved with the volume growth
and positivity of scalar curvature on a complete, non-compact Riemannian
manifold with non-negative Ricci curvature.

######  Problem 1.5 (Yau [40]).

Let $(M^{n},g)$ be a complete, non-compact manifold with non-negative Ricci
curvature and $B(p,r)\subset M$ a geodesic ball with center $p\in M$ and
radius $r$. Do we have

(2) $\limsup_{r\rightarrow\infty}r^{2-n}\int_{B(p,r)}Sc<\infty?$

In fact, Yau proposed a more general version of this problem that is involved
with the $\sigma_{k},k=1,2,\cdots,n$ of Ricci tensor in [40]. Unfortunately,
Yang [38] constructs a counterexample on Kähler manifold to prove that the
general version of Yau’s Problem 1.5 does not hold for $k=1,2,\cdots,n-1$, Xu
[37] obtains an estimate involved with the integral of scalar curvature
towards the Problem 1.5 in the case of three-dimensional Riemannian manifold
by using the monotonicity formulas of Colding and Minicozzi [5]. However,
Problem 1.5 remains open. In fact, it has been shown [30] that the inequality
(2) holds if we impose a strong curvature condition non-negative sectional
curvature instead of non-negative Ricci curvature. Also, Naber [26] asks the
non-collapsing version of Yau’s Problem 1.5 that is a baby version, and
propose a local version of Yau’s Problem 1.5. Here, we propose a baby version
of Yau’s Problem 1.5 that is worthwhile of investigating as well.

######  Problem 1.6.

Suppose that $(M^{n},g)$ is a complete, non-compact Riemannian manifold with
$Ric(g)\geq 0$ and $Sc(g)\geq 1$. Do we have

$\limsup_{r\rightarrow\infty}\frac{\mathrm{vol}(B(p,r))}{r^{n-2}}<\infty?$

On the one hand, this problem can be regarded as a baby version of Yau’s
Problem 1.5: if Problem 1.5 holds, then Problem 1.6 holds; on the other hand,
from the perspective of size geometry, this problem can provide more valid
evidence for Gromov’s Conjecture 1.2, and it can be considered as a
quantitative version of 1.2 in the category of Riemannian manifolds with non-
negative Ricci curvature.

Here, this study primarily focuses on the case of three-dimensional, complete,
non-compact Riemannian manifolds. In fact, there are abundant of studies on
three-dimensional Riemannian manifolds, including the case of compact or non-
compact. For the closed three-dimensional Riemannian manifolds, the
topological classification of three-dimensional Riemannian manifolds is clear
according to Poincáre Conjecture. Moreover, the proof of Thurston’s
Geometrization conjecture [27, 28, 29] shows that a closed three-dimensional
Riemannian manifold admits a metric with positive scalar curvature if and only
if it is a connected sum of spherical 3-manifolds and some copies of
$S^{1}\times S^{2}$. For complete non-compact three-dimensional Riemannian
manifolds with non-negative Ricci curvature, Liu [20] proves that it is either
diffeomorphic to $\mathbb{R}^{3}$ or its universal cover splits. Therefore,
there are not many three-dimensional Riemannian manifolds with the properties
that they admit a complete Riemannian metric with non-negative Ricci curvature
and positive scalar curvature. Topologically, they are either $\mathbb{R}^{3}$
or $\mathbb{S}^{2}\times\mathbb{R}$. In a recent progress, Wang [35] shows
that any complete, non-compact contractible three-dimensional Riemannian
manifold with non-negative scalar curvature is homeomorphic to
$\mathbb{R}^{3}$. M-W [24] obtain a comparison theorem of positive Green
function and spectrum estimate on complete, non-compact, three-dimensional
Riemannian manifold with some extra topological assumptions.

However, our goal in the paper is to study the geometry of a complete, non-
compact, three-dimensional Riemannian manifold that positive scalar curvature
has influence on, rather than the topology of a complete, non-compact, three-
dimensional Riemannian manifold. According to the splitting theorem of
complete, non-compact Riemannian manifold with non-negative Ricci curvature
[7], we primarily focus on the geometry of positive scalar curvature on
$\mathbb{R}^{3}$.

First, we have the following observation on Yau’s Problem 1.5

######  Theorem 1.7.

Let $(M^{3},g)$ be a complete, non-compact three-dimensional Riemannian
manifold with a pole $p$ and $Ric(g)\geq 0$. Then

(3) $\limsup_{r\rightarrow\infty}\frac{1}{r}\int_{B(p,r)}Sc\leq 20\pi.$

This estimate confirms that Yau’s Problem 1.5 holds in the special case of
complete, non-compact three-dimensional Riemannian manifolds with a pole. But,
our assumption on the manifolds with pole is very artificial. For Yau’s
Problem 1.5, we will do more studies in our future works.

Moreover, we consider the baby version of Yau’s Problem 1.5, then we obtain

######  Theorem 1.8.

Let $(M^{3},g)$ be a complete, non-compact, three-dimensional Riemannian
manifold with $Ric(g)\geq 0$ and $Sc(g)\geq 6$. Then, for any $p\in M$, we
obtain

(4) $\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(p,R))}{R}<\infty,$

if $\text{vol}(B(q,1))\geq\epsilon>0$ for any $q\in M$.

Yau [39] proved that any complete, non-compact manifolds with non-negative
Ricci curvature have at least linear volume growth. Combining Yau’s result
with our Theorem 1.8, we obtain that any complete, non-compact three-
dimensional manifold with non-negative Ricci curvature and strictly positive
scalar curvature has linear volume growth or, namely, minimal volume growth.

As a byproduct of the proof of Theorem 1.8, we obtain the following volume
estimate in higher dimensional case,

######  Corollary 1.9.

Let $(M^{n},g)$ be a complete, non-compact Riemannian manifold with
$Ric(g)\geq 0$ and $Sc(g)\geq n(n-1)$. Then for any $p\in M$, we obtain

  * •

If $vol(B(q,1))\geq\epsilon>0$ for all $q\in M$, then

$\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(p,R))}{R^{n-1}}<\infty;$

  * •

if $\mathrm{inj}(M)\geq\epsilon>0$, then

$\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(p,R))}{R^{n-2}}<\infty.$

In fact, Anderson proved that [1] any complete, non-compact Riemannian
manifold $(M^{n},g)$ with positive Ricci curvature has $b_{1}(M)\leq n-3$ and
the rank of any free Abelian subgroup of $\pi_{1}(M)$ is at most $n-3$ for
which the estimate is optimal. From Corollary 1.9, we also expect that non-
negative Ricci curvature and strictly positive scalar curvature on a complete,
non-compact Riemannian manifold would imply $b_{1}(M)\leq n-3$ and it is
optimal as well. For relevant result, you may refer to [4]. In fact, it will
be interesting to study the dimension of harmonic functions with linear growth
on a complete, non-compact manifold with non-negative Ricci curvature and
strictly positive scalar curvature on higher dimensional Riemannian manifolds.

######  Remark 1.10.

In fact, Gromov [12] stated that, under the assumption that $K(g)\geq
0,Sc(g)\geq n(n-1)$ without any details, then

(5) $\sup_{p\in M}vol(B(p,r))\leq c_{n}r^{n-2}.$

For the proof of $(\ref{volumesection})$, you may refer to [30]. Furthermore,
Gromov [12] conjectures that the volume estimate 5 holds if we only
$Ric(g)\geq 0$. Here, our results can be regarded as a step to prove Gromov’s
conjecture under an extra condition volume non-collapse as $n=3$ or
injectivity radius non-collapse as $n\geq 4$.

Moreover, a complete, non-compact Riemannian manifold is said to be non-
parabolic if it admits a positive Green function, otherwise it is said to be
parabolic. By a result of Varopoulos [34], a complete, non-compact Riemannian
manifold with $Ric(g)\geq 0$ is non-parabolic if and only if

$\int_{1}^{\infty}\frac{r}{vol(B(p,r))}dr<\infty.$

Hence, the following conclusion is deduced

######  Corollary 1.11.

Let $(M^{3},g)$ be a complete, non-compact three-dimensional Riemannian
manifold with $Ric(g)\geq 0$, $Sc(g)\geq 6$ and
$\text{vol}(B(p,1))\geq\epsilon>0$ for all $p\in M$. Then $(M^{3},g)$ is
parabolic.

According to the result in [31], $(M^{3},g)$ admits no any nontrivial harmonic
functions with polynomial growth unless it splits. In fact, we would like to
believe that: on a complete, non-compact, $n$-dimensional Riemannian manifold
with non-negative Ricci curvature and strictly positive scalar curvature, then
the dimension of harmonic function with linear growth should be less or equal
to $n-2$ for any $n\geq 4$.

Finally, let’s study the width of the manifold. Sormani [32] proves for any
complete non-compact manifold with non-negative Ricci curvature and minimal
volume growth, Busemann function is proper. However, we can not expect that
the diameter of the level set of Busemann function has a uniformly bound there
and many examples are illustrated in [32]. Here, we prove an upper bound on
the width of the manifold.

######  Theorem 1.12.

Let $(M^{3},g)$ be a complete, non-compact three-dimensional Riemannian
manifold with $Ric(g)\geq 0$, $Sc(g)\geq 2$ and
$\text{vol}(B(p,1))\geq\epsilon>0$. Then, there exists a constant $c$ and
continuous function $f:M\rightarrow\mathbb{R}$ such that for any
$r\in\mathbb{R}$,

$\text{diam}(f^{-1}(r))\leq c.$

######  Remark 1.13.

Theorem 1.12 indicates that any 3 dimensional manifold $(M^{3},g)$ with non-
negative Ricci curvature and positive scalar curvature will wander around a
line. Hence, in the large scale, $M$ is a one dimensional line $\mathbb{R}$.
In fact, by the proof of Theorem 1.12, we know $f$ can be a Lipschitz function
such that $Lip(f)\leq 1$. Besides, it is possible that $f^{-1}(r)$ for some
$r$ may not be connected, and there is no reason to expect that $f^{-1}(r)$
for all $t\in\mathbb{R}$ are connected. Hence, for the case of discounted
level set, we also count the distance among different connected components of
the level set in the theorem.

$diam(f^{-1}(r))=\sup_{x,y}\\{d_{g}(x,y):f(x)=f(y)=r\\}.$

Here, $d_{g}$ is the distance induced by the Riemannian metric $g$ on $M$.
Finally, by the language of Uryson width, Theorem 1.12 implies that
$width_{1}(M)\leq c$ and $macrodim(M^{3})=1$.

The article is organized as follows. In section 2, we introduce some
preliminary materials for the proofs of main theorems and then prove
Proposition 1.4. In section 3, we prove Theorem 1.7, 1.8, Corollary 1.9 and
Theorem 1.12. Now, let’s briefly outline the proof of main results in this
work. For the proof of Theorem 1.7: first, by using the stability of the
geodesic ray, we deduce an upper bound on the integral of Ricci curvature in
the direction of the normal vector field over the geodesic ball centered at
the pole. Then, we make use of the geometrically relative Bochner formula and
Gauss Bonnet formula on a geodesic sphere to obtain an upper bound on the
integral of scalar curvature. In fact, the geometrically relative Bochner
formula plays a vital role in the proof to get rid of the second fundamental
form of the geodesic sphere and Theorem 1.7 will be much stronger than
Proposition 1.4. For the proof of Theorem 1.8: we first prove Lemma 2.12 which
relates volume growth to the Euclidean split of Ricci limiting space. Then, we
argue our theorem by contradiction. By combining Lemma 2.12 with Cheeger-
Colding theory and the result of M-T on the characterization of three-
dimensional Ricci limiting space, we obtain that, topologically, the Ricci
limiting space can only be: $\mathbb{R}^{3},\ \mathbb{R}^{2}\times S^{1}$,
then, we rule out both cases by using the torus band estimate. Hence, we prove
a minimal volume growth under strictly positive scalar curvature. Here, we
avoid defining a generalized scalar curvature on the Ricci limiting space,
which is a hard question for the author. Instead, we use the Lipschitz
structure of positive scalar curvature proved by G-WXY to rule out $2$ cases
of Ricci limiting space. Essentially, our argument works whenever the Ricci
limiting space is a topological manifold and hence we have a corresponding
result in the higher dimensional Riemannian manifolds with stronger condition
assumption. In fact, the method of using Lipschitz structure of strictly
positive scalar curvature is quite new to study the positivity of scalar
curvature on complete Riemannian manifolds with non-negative Ricci curvature.

Acknowledgement: The author would like to express his gratitude to his advisor
Prof. Jiaping Wang for his constant support. The author is also grateful to
Prof. Ruobing Zhang, Prof. Wenshuai Jiang for their help on metric geometry.
The author is indebted to Prof. Hanlong Fang and Pak-Yeung Chan, Zhichao Wang
for their discussions over the past years.

##  2\. Preliminaries and Notations

In this section, let’s make some preparations and prove some lemmas for the
proof of the main theorems in the paper. We will start by the stability of the
geodesics in a Riemannian manifold.

###  2.1. Variation of Geodesic

Suppose that $(M^{n},g)$ is a complete Riemannian manifold and
$\gamma:[a,b]\rightarrow M$ is a smooth curve in $M$ with
$\|\gamma^{\prime}(t)\|=1$ and $\gamma(a)=p,\gamma(b)=q,\ p,q\in M$. Then, we
consider a smooth variation of $\gamma(t)$:

$\gamma(t,s):[a,b]\times[-\epsilon,\epsilon]\rightarrow M.$

and $\gamma(t,0)=\gamma(t)$ and $\gamma(a,s)=p$ and $\gamma(b,s)=q$. We say
that $\gamma$ is a geodesic if $\gamma$ is a critical point of the length
functional

$L(s)=\int_{a}^{b}\|\frac{\partial\gamma(t,s)}{\partial t}\|dt.$

That is for any variation vector field $X(t)$ and
$X(t)=\frac{\partial\gamma(t,s)}{\partial s}|_{s=0}$ with $X(a)=X(b)=0$, we
have

(6) $0=L^{\prime}(0)=\int_{a}^{b}\langle\gamma^{\prime\prime}(t),X(t)\rangle
dt.$

It is equivalent to saying that $\gamma^{\prime\prime}(t)=0$ on $[a,b]$.

Moreover, we calculate the second variation of the arc length functional on
geodesic $\gamma$, then

(7) $L^{\prime\prime}(0)=\int_{a}^{b}\|\nabla X\|^{2}-\langle
R(X,\gamma^{\prime}(t))\gamma^{\prime}(t),X\rangle dt.$

Here, $\langle R(X,\gamma^{\prime}(t))\gamma^{\prime}(t),X\rangle$ is the
Riemannian curvature and $\nabla$ is the Levi-Civita connection on $(M,g)$. A
geodesic is said to be stable if the second variation is non-negative. i.e.,
$L^{\prime\prime}(0)\geq 0$. Since the calculations above are classical and
standard on any Riemannian geometry textbook, we omitted the details (See
[21]).

Then, for any fixed point $x\in M$, we introduce the exponential map as
introduce

$exp:T_{x}M\rightarrow M,exp(v)=\gamma(1),v\in T_{x}M.$

######  Definition 2.1.

Let $(M^{n},g)$ be a complete Riemannian manifold. Then we define the
injectivity radius of $M$ as follows

$\mathrm{Inj}(M):=\inf_{x\in M}\sup_{r}\\{r:exp:B(x,r)\rightarrow exp(B(x,r))\
\text{is a diffeomorphism}\\}.$

and the conjugate radius of $M$ as follows

$\mathrm{conj}(M)=\inf_{x}\sup_{r}\\{r:exp:B(x,r)\rightarrow exp(B(x,r)),\
B(x,r)\ \text{contains no critical points of exp}.\\}$

Finally, $(M^{n},g)$ is said to be a manifold with a pole at $p$ if
$exp_{p}:T_{p}M\rightarrow M$ is a diffeomorphism.

By the definition of injectivity radius and conjugate radius of $(M,g)$, we
have $\mathrm{inj}(M)\leq\mathrm{conj}(M)$. Moreover, let $(M^{n},g)$ be a
complete Riemannian manifold and $\gamma_{x,v}(t)$ the unique geodesic with
initial conditions

$\left\\{\begin{array}[]{l}\gamma_{(x,v)}(0)=x;\\\
\gamma_{(x,v)}^{\prime}(0)=v,v\in T_{x}M.\end{array}\right.$

The initial problem is solvable uniquely by the classical ODE problem and
hence $\gamma$ always exists.

######  Definition 2.2.

For a given $t\in\mathbb{R}$, we define a diffeomorphism of the tangent bundle
$TM$

$\varphi_{t}:TM\rightarrow TM.$

as follows

$\varphi_{t}(x,v)=(\gamma_{(x,v)}(t),\gamma_{(x,v)}^{\prime}(t)).$

In fact, the family of diffeomorphism $\varphi_{t}$ is a flow. i.e., it
satisfies with $\varphi_{t+s}=\varphi_{t}\circ\varphi_{s}$ for any
$t,s\in\mathbb{R}$ since the uniqueness of the geodesic with respect to the
initial conditions. Besides, let $SM$ be the unit tangent bundle of $M$, that
is,

$SM=\\{(x,v):x\in M,v\in T_{x}M,\|v\|=1\\}.$

Since geodesics travel with constant speed, we have that $\varphi_{t}$ leaves
$SM$ invariant. Given $(x,v)\in SM$, we obtain that $\varphi_{t}(x,v)\in SM$
for all $t\in\mathbb{R}$. It is well known that any closed, compact Riemannian
manifold admits a complete geodesic flow. Finally, you may refer to the
textbook [3] for the details about the geodesic flow and the following
results.

######  Definition 2.3.

Let $(M^{n},g)$ be a complete Riemannian manifold. We define the Liouville
measure $L$ on $SM$. The measure $L$ is given locally by the product of the
Riemannian volume on $M$ and the Lebesgue measure on the unit sphere. That is,
for any subset $A=(U,A_{x})\subset SM$, where $U\subset M$ is a subset of $M$,
$A_{x}$ is a subset of the unit sphere of the tangent space at $x\in U$, $L$
is defined by

$L(A)=\int_{x\in U}\int_{A_{x}}d\mathbb{S}^{n-1}dvol(x).$

where $d\mathbb{S}^{n-1}$ is the usual Lebesgue measure on the unit sphere.

A well known result related to the Liouville theorem is,

######  Lemma 2.4.

Let $(M^{n},g)$ be a complete manifold and $\varphi_{t}$ the geodesic flow.
Then for any Borel set $B$ in $SM$ and $t\in\mathbb{R}$, we have,

$L(\varphi_{t}(B))=L(B).$

That is, geodesic flow preserves Liouville measure.

Finally, we need the following basic integral form of the scalar curvature
$Sc(g)$ in terms of the Ricci curvature $Ric(g)$.

######  Lemma 2.5.

Let $(M^{n},g)$ be a complete Riemannian manifold. Then for any $p\in M$, we
obtain,

(8)
$Sc_{p}=\frac{n}{\mathrm{vol}(\mathbb{S}^{n-1})}\int_{v\in\mathbb{S}^{n-1}}Ric_{p}(v)dv.$

###### Proof.

Let $\\{e_{i}\\}_{i=1}^{n}\in T_{p}M$ be an orthonormal coordinate such that
$Ric_{p}(e_{i})=\lambda_{i}e_{i}$. Then, for any $v\in\mathbb{S}^{n-1}$,

$v=x_{i}e_{i}.$

Hence

$\sum_{i=1}^{n}x_{i}^{2}=1,\ Sc_{p}=\sum_{i=1}^{n}\lambda_{i},$

and then

$\int_{v\in\mathbb{S}^{n-1}}Ric_{p}(v)dv=\int_{x\in\mathbb{S}^{n-1}}\sum_{i=1}^{n}x_{i}^{2}\lambda^{2}_{i}dx=\sum_{i=1}^{n}\lambda^{2}_{i}\int_{x\in\mathbb{S}^{n-1}}x_{i}^{2}dx=\frac{\mathrm{vol}(\mathbb{S}^{n-1})Sc_{p}}{n}.$

∎

###  2.2. Integral of Curvatures

######  Definition 2.6.

Let $(M^{n},g)$ be a complete, non-compact manifold, $\gamma(t)=\exp(tv),t\geq
0,v\in T_{p}M$ is called a ray if it is minimal on every interval

$d(\gamma(t),\gamma(s))=|s-t|,\ s,t>0,$

and the unit vector $v$ is called a direction of $\gamma(t)$. Assume that
$\gamma_{i}(t)=\exp(tv_{i}),v_{i}\in T_{p}M$ are rays, $\\{\gamma_{i}(t)\\}$
are independent and orthogonal if their directions $\\{v_{i}\\}$ are linearly
independent and mutually orthogonal at $p$. Moreover, we define the Busemann
function $B_{\gamma}(x)$ associated with any ray $\gamma(t)$

$B_{\gamma}(x)=\lim_{t\rightarrow\infty}(t-d(x,\gamma(t))).$

Here, $B_{\gamma}(x)$ is well-defined since $f(t)=t-d(x,\gamma(t))$ is
increasing in terms of $t$ and uniformly bounded from above.

######  Lemma 2.7.

Let $(M^{n},g)$ be a complete, non-compact manifold with a pole $p$ and
$Ric(g)\geq 0$. Then, for any ray $\gamma(t)$ with $\gamma(0)=p$,

(9)
$\frac{1}{r}\int_{0}^{r}t^{2}Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dt\leq
n-1.$

In particular, as $n=3$,

(10) $\frac{1}{r}\int_{B(p,r)}Ric(\nu,\nu)(x)d\mathcal{H}^{3}(x)\leq 8\pi.$

where $\nu$ is the outer unit normal vector field of the geodesic sphere.

###### Proof.

Assume that $\gamma(t)$ with $\gamma(0)=p$ is a ray and $V(t)$ is a smooth
vector field along $\gamma(t)$, we consider the variation of $\gamma(t)$:

$\gamma(t,s)=\exp_{\gamma(t)}(sV(t)),s\in\mathbb{R}.$

Since $\gamma(t)$ is a minimizing geodesic, then, on any interval $[0,b]$, we
obtain that the second variation of length functional is non-negative by 7.
i.e.,

$\int_{a}^{b}|\nabla V(t)|^{2}-K(\gamma^{\prime}(t),V(t))dt\geq 0.$

Hence,

$\int_{a}^{b}K(\gamma^{\prime}(t),V(t))dt\leq\int_{a}^{b}|\nabla V|^{2}dt.$

Then, for any $t\in[0,b]$, we assume that $\\{e_{i}(t)\\}_{i=1}^{n-1}$ is the
parallel vector field such that $\\{e_{i}(t),\gamma(t)\\}$ forms an
orthonormal base in $T_{\gamma(t)}M$. Now we fix any $x\in(0,b)$ and then take

$V_{i}(t)=\left\\{\begin{array}[]{ll}\frac{t}{x}e_{i}(t),&t\in[0,x];\\\
\frac{b-t}{b-x}e_{i}(t),&t\in[x,b].\end{array}\right.$

Then, we plug $V_{i}(t)$ into the inequality to obtain

$\frac{1}{x^{2}}\int_{0}^{x}t^{2}Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dt+\frac{1}{(b-x)^{2}}\int_{0}^{b-x}t^{2}Ric(\gamma^{\prime}(b-t),\gamma^{\prime}(b-t))dt\leq(n-1)(\frac{1}{x}+\frac{1}{b-x}).$

By taking $b\rightarrow\infty$ and the assumption that $Ric(g)\geq 0$, we
obtain

$\frac{1}{x^{2}}\int_{0}^{x}t^{2}Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dt\leq\frac{n-1}{x}.$

Hence, for any $r>0$

(11)
$\frac{1}{r}\int_{0}^{r}t^{2}Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dt\leq
n-1.$

Since $p\in M$ is a pole, we have for any $t>0$, the geodesic sphere $\partial
B(p,t)\subset M$ is a sphere and on which the unit normal vector field is
well-defined by $\nu$. As a result, we obtain $f(x)=Ric(\nu,\nu)(x)$ is a
well-defined, induced function on each point $x\in M$. Finally, for any $v\in
T_{p}M$, $\exp(tv)$ is a ray and hence we can obtain the estimate (11) on it.
Then, by the volume comparison theorem $Area(\partial B(p,t))\leq 4\pi t^{2}$
and coarea formula along the distance function, we obtain

$\frac{1}{r}\int_{B(p,r)}Ric(\nu,\nu)d\mathcal{H}^{3}\leq 4\pi(n-1).$

Here, $\mathcal{H}^{3}$ is the 3 dimensional Hausdorff measure on $M^{3}$. ∎

Then, let’s introduce the following type of geometrically relative Bochner
formula. Let $f$ be a smooth function defined on a complete Riemannian
manifold $(M^{n},g)$, we define the level set of $f$ as

$L_{t}^{f}=\\{x\in M:f(x)=t\\}.$

On each level set $L_{a}^{f}$, if it is a smooth $n-1$ dimension embedded
submanifold in $M$, we define the second fundamental form and mean curvature
of $L_{t}^{f}$ by $A$ and $H$ respectively with respect to the outer normal
vector field $u=\frac{\nabla f}{|\nabla f|}$. Hence,

$A=\nabla_{L^{f}_{t}}(u),\ H=\text{Div}(u).$

Here, $\nabla_{L^{f}_{t}}$ is the restriction of $\nabla$ on $L^{f}_{t}$.
Then, we introduce $G=H^{2}-|A|^{2}.$

Then, we have the following geometrically relative Bochner formula in [30].
Here, we give a different proof using Bochner formula of the vector field.

######  Lemma 2.8.

Suppose that $(M^{n},g)$ is a complete Riemannian manifold and there exists no
critical point in $[a,b]$ for $f$. Then,

(12) $\int_{f^{-1}[a,b]}Ric(u,u)-G=\int_{L^{f}_{a}}H-\int_{L^{f}_{b}}H.$

###### Proof.

By Bochner formula of the vector field $X$ over $M$[22], we have

(13) $\frac{1}{2}\Delta|X|^{2}=|\nabla
X|^{2}+\text{Div}(L_{X}g)(X)-Ric(X,X)-\nabla_{X}\text{Div}(X)$

Now, we take $X=u=\frac{\nabla f}{|\nabla f|}$ in (13), then,

$Ric(u,u)=|\nabla u|^{2}+\text{Div}(L_{u}g)(u)-\nabla_{u}\text{Div}(u).$

For any $x\in L^{f}_{t}$, we obtain that $u(x)=\frac{\nabla f(x)}{|\nabla
f|(x)}$ is the unit normal vector field of $L_{t}^{f}$ at the point of $x\in
L^{f}_{t}$ and then we assume that $\\{e_{i},u\\}_{i=1}^{n-1}$ forms an
orthonormal base in $T_{x}M$, hence

$|A|^{2}=|\nabla u|^{2},\ H=\text{Div}(u).$

Moreover, by using integration by parts over $f^{-1}([a,b])$ for
$\nabla_{u}(\text{Div}(u))$, we obtain,

$\begin{array}[]{cl}&-\int_{f^{-1}([a,b])}\nabla_{u}(\text{Div}(u))\\\
=&-\int_{f^{-1}([a,b])}\nabla_{u}H\\\
=&-\int_{f^{-1}([a,b])}\text{Div}(Hu)-H\text{Div}(u)\\\
=&-\int_{f^{-1}([a,b])}\text{Div}(Hu)-H^{2}\\\
=&\int_{f^{-1}([a,b])}H^{2}+\int_{L^{f}_{a}}H-\int_{L^{f}_{b}}H.\\\
\end{array}$

For the term $\text{Div}(L_{u}g)(u)$,

$\begin{array}[]{cl}&\int_{f^{-1}([a,b])}\text{Div}(L_{u}g)(u)\\\
=&\int_{f^{-1}([a,b])}\nabla_{e_{i}}(L_{u}g)(e_{i},u)\\\
=&\int_{f^{-1}([a,b])}\nabla_{e_{i}}(L_{u}g(u,e_{i}))-(L_{u}g)(e_{i},\nabla_{e_{i}}u)\\\
=&\int_{f^{-1}([a,b])}\nabla_{e_{i}}((L_{u}g)u,e_{i}))-(L_{u}g)(e_{i},\nabla_{e_{i}}u)\\\
=&-2\int_{f^{-1}([a,b])}|A|^{2}.\\\ \end{array}$

Hence,

$\int_{f^{-1}([a,b])}Ric(u,u)=\int_{f^{-1}([a,b])}G-\int_{L^{f}_{b}}H+\int_{L^{f}_{a}}H$

∎

######  Remark 2.9.

From the perspective of function theory on a complete manifold: Sectional
curvature would impose the condition on the Hessian of functions or the second
fundamental form of level set. Ricci curvature would impose the condition on
the Laplacian of functions or the mean curvature of the level set. Lemma 2.8
seems trivial, but it is deep for the author, since the integral of the second
fundamental form can be expressed in terms of the mean curvature and the Ricci
curvature on the ambient Riemannian manifold.

###  2.3. Width and Positive Scalar Curvature

Let’s introduce the result related to the positive scalar curvature to our
article. The study of non-negative and positive scalar curvature is a very
important topic in geometry analysis, which is pioneered by the works [13, 33]
of Gromov-Lawson and Schoen-Yau. These works provide us two paths of the
understandings of the geometry and topology of scalar curvature bounded below:
spin techniques and minimal surface techniques. According to their works, it’s
known that $\mathbb{T}^{n}$ admits no complete Riemannian metric with non-
negative scalar curvature unless it is flat. In recent Gromov’s work [17], he
introduces the $\mu$ bubble, which is detailed by Zhu in his work [41] where
his result indicates that positive scalar curvature implies that $2$-systole
is bounded above in terms of the lower bound of the scalar curvature. After
that, many applications of $\mu$ bubble have been expanded to study the
existence of Riemannian metric with positive scalar curvature [9, 18]. Here,
we will use the following result, which relates the size to the positive
scalar curvature [16].

Suppose that $M^{n}=T^{n-1}\times I$ where $I$ is an interval $[a,b],a<b$.
Here, $M$ is called a torical band. We define

$\partial M=T^{n-1}\times\\{b\\}\bigcup
T^{n-1}\times\\{a\\}=:\partial_{+}M\bigcup\partial_{-}M.$

then,

(14)
$d(\partial_{+}M,\partial_{-}M)=\inf_{x\in\partial_{+}M,y\in\partial_{-}M}\\{d(x,y)\\}.$

Then, the following theorem holds

######  Theorem 2.10 (G-WXY [16, 17, 36]).

Let $(M,g)$ be a $n$-dimensional torical band with $Sc(g)\geq n(n-1)$. Then,

(15) $\ d(\partial_{+}M,\partial_{-}M)<\frac{2\pi}{n}.$

Theorem 2.10 plays a vital role in the proof of Theorem 1.8. Combing it with
the work [23] of McLeod-Topping, we avoid analyzing the singular Ricci
limiting space to achieve our goal.

######  Theorem 2.11 (Spherical Lipschitz Bound Theorem [16]).

Let $(M^{n},g)$ be a Riemannian manifold(possibly incomplete) with $Sc(g)\geq
n(n-1)$. Then, for all continuous maps $f$ from $M$ to the unit sphere
$\mathbb{S}^{n}$ (and also to the hemisphere to $\mathbb{S}^{n}_{+}$(1) of
non-zero degrees, we have,

$Lip(f)>\frac{c}{\pi\sqrt{n}}\mbox{ for the above }c>\frac{1}{3}.$

Finally, the following lemma is also needed for the proof of Theorem 1.8.

######  Lemma 2.12.

Let $(M^{n},g)$ be a complete, non-compact Riemannian manifold with
$Ric(g)\geq 0$. If there exists a sequence of $p_{i}\rightarrow\infty$ and
$R_{i}\rightarrow\infty$ such that

$\text{vol}(B(p_{i},R_{i}))=c(R_{i})R_{i}^{k},n-1\geq k\geq 1.$

with $c(R_{i})\rightarrow\infty$ as $i\rightarrow\infty$ and $vol(B(p,1))\geq
v>0$ for all $p\in M$, then there exists a sequence $q_{i}\in M$ such that
$(M,q_{i})$ pointedly Gromov Hausdorff converges to a length space
$(X\times\mathbb{R}^{l},p_{\infty})$ with

$l\geq k+1.$

Hence, it implies that there exists at least $k+1$ rays
$\\{\gamma_{l}^{(i)}\\}_{l=1}^{k+1}$ which are linearly independent and
orthogonal at $q_{i}$ in $M$ such that for all $l=1,2,\cdots,k,k+1$, the
length of $\gamma_{l}^{(i)}$, $L(\gamma_{l}^{(i)})\rightarrow\infty$ as
$i\rightarrow\infty$.

###### Proof.

Since we assume that $Ric(g)\geq 0$, by the precompactness theorem and
Cheeger-Colding theory [6], we obtain that, up to subsequence,
$(M,p_{i},vol_{i})$ converges to a metric measured length space
$(X,x_{\infty},\mu_{\infty})$ with a Borel measure $\mu_{\infty}$ on $X$.
Moreover, since it is assumed that $vol(B(p,1))\geq v>0$, for any geodesic
ball $B(p_{i},r)\subset M$ and $B(x_{\infty},r)\subset X$, we have,

$\lim_{i\rightarrow\infty}vol(B(p_{i},r))=\mu_{\infty}(B(x_{\infty},r)).$

Furthermore, let $\gamma_{i}:[0,\infty)\rightarrow M$ be a ray with
$\gamma_{i}(0)=p_{i}$, then we introduce that

$\sigma_{i}(t)=\gamma_{i}(t+R_{i}):[-R_{i},\infty)\rightarrow M.$

and we set $q_{i}=\sigma_{i}(0)$. By the assumption of the volume growth, we
have,

$vol(B(q_{i},2R_{i})\geq c(R_{i})R_{i}^{k}.$

Hence, we can replace $p_{i}$ by $q_{i}$ in the precompactness theorem. So we
have $\sigma_{i}$ will converge to a line $\sigma_{\infty}$ in $X$. By the
splitting theorem in the Ricci limiting space [6], we obtain

$X=X_{1}\times\mathbb{R}\ \text{and}\
\mu_{\infty}=\mu_{\infty}^{1}\times\mathbb{R}.$

If we let $q_{i}\rightarrow q_{\infty}$ and $R_{i}$ large, then

$\mu_{\infty}(B(q_{\infty},R_{i})\geq
c(R_{i})R_{i}^{k},c(R_{i})\rightarrow\infty,\ \text{as}\
R_{i}\rightarrow\infty.$

Here $c(r)$ may be different line by line. Hence, for metric ball
$B_{1}(q_{\infty}^{1},R_{i})\subset X_{1}$,

$\mu_{\infty}^{1}(B_{1}(q_{\infty}^{1},R_{i})\geq
c(R_{i})R_{i}^{k-1},\text{and}\ c(R_{i})\rightarrow\infty,\ \text{as}\
R_{i}\rightarrow\infty.$

Here, $q_{\infty}^{1}$ is from
$q_{\infty}=(q_{\infty}^{1},x^{n}),x^{n}\in\mathbb{R}$. Hence, we obtain that
$(X_{1},\mu_{\infty}^{1})$ is a non-compact metric measured length space.
Then, there exists a ray in $X_{1}$, we can retake our base point in $M$ to
obtain a line associated with the ray as we did above by pulling back the ray
to $M$. Hence, we have

$(X_{1}=X_{2}\times\mathbb{R},\mu_{\infty}^{1}=\mu_{\infty}^{2}\times\mathbb{R}).$

Finally, we continue this process $k-1$ times to obtain the limiting space

$(X_{k-1}=X_{k}\times\mathbb{R},\ \mu^{k}=\mu_{\infty}^{k-1}\times\mathbb{R})$

and for any metric ball $B(q_{\infty}^{k},R_{i})\subset X^{k}$,

$vol(B(q_{\infty}^{k},R_{i})))\geq c(R_{i}),,c(R_{i})\rightarrow\infty,\
\text{as}\ R_{i}\rightarrow\infty.$

Here $q_{\infty}^{k-1}=(q_{\infty}^{k},x^{n-k+1}),x^{n-k+1}\in\mathbb{R}$.
Hence, $X_{k}$ is still non-compact, otherwise, its volume should be finite.
Hence, by the same argument above, we obtain that $X_{k}$ splits as
$X_{k+1}\times\mathbb{R}$. Hence, we finally find a sequence $q_{i}\in M$ such
that $(M,q_{i})$ pointedly Gromov Hausdorff converges to
$X\times\mathbb{R}^{k+1}$. We complete the proof of the lemma. ∎

###  2.4. Proof of Proposition 1.4

Before we are going to the proof, let’s first see the compact case: Setting
$l=\mathrm{conj}(M)$, we consider any geodesic ball $B(m,l),m\in M$ and any
$q\in\partial B(m,l)$, there exists arc length parameter
$\gamma:[0,l]\rightarrow M$ which is the shortest geodesic connecting $m,q$ .
Then, we consider the index form for any variational vector $X$ of $\gamma(t)$
with $X(0)=X(l)=0$ $0\leq I(X,X)=\int_{0}^{l}\|\nabla X\|^{2}-\langle
R(X,\gamma^{\prime}(t))\gamma^{\prime}(t),X\rangle ds.$ Hence,

(16) $\int_{0}^{l}\langle R(X,\gamma^{\prime}(t))\gamma^{\prime}(t),X\rangle
ds\leq\int_{0}^{l}\|\nabla X\|^{2}ds.$

If we pick $X=sin(\frac{\pi}{l}t)\nu$ with $\nu=\nu(t)$ a parallel unit vector
field along $\gamma(t)$, then

(17) $\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)\langle
R(\nu,\gamma^{\prime}(t))\gamma^{\prime}(t),\nu\rangle
dt\leq(\frac{\pi}{l})^{2}\int_{0}^{l}sin^{2}(\frac{\pi}{l})dt.$

By a direct calculation, we obtain
$(\frac{\pi}{l})^{2}\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)dt=\frac{\pi^{2}}{2l}$
and then

$\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)\langle
R(\nu,\gamma^{\prime}(t))\gamma^{\prime}(t),\nu\rangle
dt\leq\frac{\pi^{2}}{2l}.$

$\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dt\leq\frac{(n-1)\pi^{2}}{2l}.$

Integrating over the unit tangent bundle $SM$, we have,

(18)
$\int_{SM}\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dtdL\leq\int_{SM}\frac{(n-1)\pi^{2}}{2l}dL.$

For the integral on the left, we have,

$\displaystyle\ \ \
\int_{SM}\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dtdL$
$\displaystyle=\
\int_{0}^{l}\int_{SM}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dLdt$
$\displaystyle=\ \int_{0}^{l}\int_{m\in M}\int_{v\in
S_{m}M}sin^{2}(\frac{\pi}{l}t)Ric((\varphi_{t})_{\star}v,(\varphi_{t})_{\star}v)dvd\mathrm{vol}(m)dt$
$\displaystyle=\ \int_{0}^{l}sin^{2}({\frac{\pi}{l}}{t})dt\int_{m\in
M}\int_{S_{m}M}Ric(v)dvd\mathrm{vol}(m)$ $\displaystyle=\
\frac{\mathrm{vol}(\mathbb{S}^{n-1})}{n}\int_{0}^{l}sin^{2}({\frac{\pi}{l}}{t})dt\int_{m\in
M}Sc(m)d\mathrm{vol}(m)$ $\displaystyle\geq\
\frac{(n-1)l}{2}\mathrm{vol}(\mathbb{S}^{n-1})\mathrm{vol}(M)\ \ \text{since
$Sc\geq n(n-1)$}.$

By a direct calculation, we obtain that

$\int_{SM}\frac{(n-1)\pi^{2}}{2l}dL=\frac{(n-1)\pi^{2}}{2l}\mathrm{vol}(\mathbb{S}^{n-1})\mathrm{vol}(M).$

Hence,

$l\leq\pi.$

Finally, if $l=\pi$, then all above inequalities are equalities. Hence, $M$
has constant sectional curvature $K=1$ with
$\mathrm{Diam(M)}=\mathrm{Inj}(M)=\pi$. Hence, by Theorem 1.1, we obtain that
$M$ is isometric to the round sphere $\mathbb{S}^{n}$.

Now, let’s come back to the proof of Proposition 1.4: Rather than integrating
over the unit vector bundle $SM$, we consider the geodesic ball
$B=B(p,r)\subset M$ and $B_{-l}=B(p,r-l)$. Then we start from inequality (18),

$\int_{SB}\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dtdL\leq\int_{SB}\frac{(n-1)\pi^{2}}{2l}dL.$

$\displaystyle\ \ \
\int_{SB}\int_{0}^{l}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dtdL$
$\displaystyle=\
\int_{0}^{l}\int_{SB}sin^{2}(\frac{\pi}{l}t)Ric(\gamma^{\prime}(t),\gamma^{\prime}(t))dLdt$
$\displaystyle\geq\ \int_{0}^{l}sin^{2}({\frac{\pi}{l}}{t})dt\int_{m\in
M}\int_{S_{m}B_{-l}}Ric(v)dvd\mathrm{vol}(m),Ric(g)\geq 0\ \text{on}\ B(p,R)$
$\displaystyle=\
\frac{\mathrm{vol}(\mathbb{S}^{n-1})}{n}\int_{0}^{l}sin^{2}({\frac{\pi}{l}}{t})dt\int_{m\in
B_{-l}}Sc(m)d\mathrm{vol}(m).$

Hence,

$\int_{B(p,r-l)}Sc\leq n(n-1)\frac{\pi^{2}}{l^{2}}vol(B(p,r)).$

Moreover, if we assume that $Ric(g)\geq 0$ and $Sc(g)\geq n(n-1)$ on $M$, then
for any $r>l$,

$\frac{\mathrm{vol}{B(p,r-l)}}{\mathrm{vol}(B(p,r))}\leq\frac{\pi^{2}}{l^{2}}.$

By volume comparison theorem, we obtain,

$\lim_{r\rightarrow\infty}\frac{\mathrm{vol}{B(p,r-l)}}{\mathrm{vol}(B(p,r))}=1.$

Therefore $l\leq\pi$. We complete the proof of Proposition 1.4.

As a corollary, we have,

######  Corollary 2.13.

Let $(M^{n},g)$ be a complete, non-compact manifold with $Ric(g)\geq 0$ with
$inj(M)=c>0$. Then, for any $p\in M$,

$\frac{1}{\mathrm{vol}(B(p,c))}\int_{B(p,c)}Sc\leq\frac{2^{n}n(n-1)}{c^{2}}$

From the perspective of Cheeger-Colding theory and Anderson’s $C^{\alpha}$
convergence, it is too strong that we assume that the injectivity radius has a
uniformly lower bound. But, if you pay more attention to the generalized
scalar curvature on Ricci limiting space, we still do not have a systematic
way to introduce a useful scalar curvature on this singular space. Probably,
this inequality may help study the Ricci limiting space for non-collapsing
case in the future.

##  3\. Proof of Theorems

In this section, we will prove Theorem 1.7, 1.8 and 1.12. For the proof of
Theorem 1.7, we will use the geometrically relative Bochner formula along the
distance function and the stability of ray. For the proof of Theorem 1.8, we
will combine the Gromov-Hausdorff convergence with the estimate of torical
band to obtain the volume estimate. For the proof of Theorem 1.12, we analyze
the level set of Busemann function to obtain the existence of function
required.

###  3.1. Proof of Theorem 1.7

######  Theorem 3.1.

Let $(M^{3},g)$ be a complete, non-compact three-dimensional Riemannian
manifold with a pole $p$ and $Ric(g)\geq 0$. Then

(19) $\limsup_{r\rightarrow\infty}\frac{1}{r}\int_{B(p,r)}Sc\leq 20\pi.$

###### Proof.

Let’s first define $f(x):=d(p,x)$, on each level set $L_{t}^{f}$, we have the
following type of Gauss equation called S-Y trick on minimal surface [33]

$2\bar{K}=Sc-2Ric(\nu,\nu)+G.$

Here, $\bar{K}$ is the Gauss curvature of the level set $L_{t}^{f}$. Hence,

$Sc=2\bar{K}+2Ric(\nu,\nu)-G.$

Then integrating it over $B(a,b)=B(p,a,b)$, we obtain,

$\int_{B(a,b)}Scd\mathcal{H}^{3}=\int_{B(a,b)}(2\bar{K}+2Ric(\nu,\nu)-G)d\mathcal{H}^{3}.$

By Lemma 2.8, we obtain,

$\int_{B(a,b)}Gd\mathcal{H}^{3}=\int_{B(a,b)}Ric(\nu,\nu)d\mathcal{H}^{3}-\int_{L^{f}_{b}}H+\int_{L^{f}_{a}}H.$

Hence,

$\int_{B(a,b)}Sc\
d\mathcal{H}^{3}=2\int_{B(a,b)}\bar{K}d\mathcal{H}^{3}+\int_{B(a,b)}Ric(\nu,\nu)d\mathcal{H}^{3}+\int_{L^{f}_{b}}H-\int_{L^{f}_{a}}H.$

By the coarea formula and Gauss Bonnet theorem on each level set surface, we
have,

$\int_{B(a,b)}\bar{K}d\mathcal{H}^{3}=\int_{a}^{b}\int_{\partial
B(r)}\bar{K}d\mathcal{H}^{2}dr=\int_{a}^{b}2\pi\chi(\partial
B(r))dr=4\pi(b-a).$

Here, we used $\partial B(r)$ is a topological sphere for any $r\in[a,b]$.
Furthermore, by the volume comparison theorem and its proof, we have,

$\int_{L^{f}_{b}}H\leq 4\pi b.$

Combining above all estimate and Lemma 2.7, we get

$\int_{B(a,b)}Scd\mathcal{H}^{3}\leq 8\pi(b-a)+8\pi(b-a)+4\pi
b-\int_{L^{f}_{a}}H.$

By taking $b\rightarrow\infty$ and then $a\rightarrow 0$, we have,

$\limsup_{r\rightarrow\infty}\frac{1}{r}\int_{B(p,r)}Sc\ d\mathcal{H}^{3}\leq
20\pi.$

∎

###  3.2. Proof of Theorem 1.8

######  Theorem 3.2.

Let $(M^{3},g)$ be a complete, non-compact three-dimensional Riemannian
manifold with $Ric(g)\geq 0$ and $Sc(g)\geq 6$. Then, for any $p\in M$, we
obtain,

(20) $\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(p,R))}{R}<\infty,$

provided that $\text{vol}(B(q,1))\geq\epsilon>0$ for all $q$.

###### Proof.

Let’s show that it is sufficient to prove that there exists one $p\in M$ such
that

(21) $\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(p,R))}{R}<\infty.$

We assume that the estimate (21) holds for $p$, then for any $q\in M$, we have

$\frac{\text{vol}(B(q,r))}{r}\leq\frac{\text{vol}(B(p,r+2d(p,q)))}{r}=\frac{\text{vol}(B(p,r+2d(p,q)))}{r+2d(p,q)}\frac{r+2d(p,q)}{r}.$

Then, by taking the $\limsup$ on both sides, we have,

$\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(q,R))}{R}<\infty.$

Now, let’s prove inequality (21) by contradiction argument: Suppose that there
exists a sequence of $p_{i}\in M,R_{i}\in\mathbb{R}$ such that
$p_{i}\rightarrow\infty$ and $R_{i}\rightarrow\infty$ and

$\frac{\text{vol}(B(p_{i},R_{i}))}{R_{i}}\rightarrow\infty,i\rightarrow\infty.$

By the Lemma 2.12, there exists at least $2$ rays that are independent and
orthogonal at each $p_{i}$. By the main theorem in [23] and Cheeger-Colding
theory [6], there exists a subsequence $\\{q_{i}\\}$ of $\\{p_{i}\\}$ such
that

(22) $(M^{3},p_{i},g_{i})\rightarrow(M_{\infty},p_{\infty},d),$

in the sense of Gromov-Hausdorff convergence with the following properties
($**$):

  * •

$M_{\infty}$ is a smooth manifold. Notice that the topological regularity is
only known for $n=3$. Here, smooth manifold means that $M_{\infty}$ is a
smooth differential manifold topologically, and we do not know anything about
the deep metric structure of the Ricci limiting space. In fact, we mainly use
the topological structure in our paper;

  * •

For any $p_{\infty}$, there exists a sequence of smooth maps

$\varphi_{i}:B_{d}(p_{\infty},i)\rightarrow M_{i},$

such that $B_{d}(p_{\infty},i)$ is diffeomorphic onto
$\varphi_{i}(B_{d}(p_{\infty},p))$ and $\varphi_{i}(p_{\infty})=q_{i}$. Here
$B_{d}(p_{\infty},i)$ is the metric ball in $(M_{\infty},d)$ with radius $i$
and center $p_{\infty}$;

  * •

Under the above item, for any $R>0$,

$d_{g_{i}}(\varphi_{i}(x),\varphi_{i}(y))\rightarrow d(x,y),$

uniformly on $B_{d}(p_{\infty},R)$ as $i\rightarrow\infty$. Hence, the
convergence is at least $C^{0}$ convergence only in the sense of metric space.

######  Remark 3.3.

For the notation used in 22: actually, $g_{i}=g$, we write it as $g_{i}$ since
we want to match $g_{i}$ with the base point $p_{i}$. Moreover, we do not know
if the Riemannian metric $g_{i}$ will $C^{0}$-converge to a smooth Riemannian
metric.

Moreover, for the limiting space $M_{\infty}$, we have the following two cases
by Lemma 2.12: $M^{1}\times\mathbb{R}^{2}$, $\mathbb{R}^{3}$.

  * •

Case 1: If

$M_{\infty}\simeq M^{1}\times\mathbb{R}^{2}.$

isometrically, we will have $M^{1}$ is one dimensional, topologically, smooth
manifold. This implies that $M^{1}=\mathbb{S}^{1}$. Hence,
$M_{\infty}\simeq\mathbb{S}^{1}\times R^{2}$. Here, we merely obtain that the
manifold is smooth in the sense of topology. However, we did not know if the
metric $d_{\infty}$ on $M^{1}\times\mathbb{R}^{2}$ is induced by a smooth
Riemannian metric. Now we can overcome these difficulties as follows.

On the limiting space $\mathbb{S}^{1}\times\mathbb{R}^{2}$, we consider the
set

$T=B_{d}(p_{\infty},2R)-B_{d}(p_{\infty}.R).$

As $R$ is a large, fixed number, i.e., $R\geq 100$, $T$ is $T^{2}\times[R,2R]$
topologically and $d(\partial_{+}T,\partial_{-}T)=R$ under the metric $d$.
Since we know that, as $i\geq 10R$, $T\subset B_{d}(p_{\infty},i)$ and
$B_{d}(p_{\infty},i)$ is diffeomorphic onto
$\varphi_{i}(B_{d}(p_{\infty},i))$, we reach that $\varphi_{i}(T)$ is a
torical band in $M_{i}$ with a minor damage on the band distance. Moreover, we
can always perturb $\varphi_{i}(T)$ to $K$ such that $K$ becomes a smooth
manifold and its topology is kept fixed and

$d_{g_{i}}(\partial_{+}K,\partial_{-}K)\geq\frac{1}{2}R.$

Hence, we obtain a compact Riemannian manifold with boundary

$(K=\mathbb{T}^{2}\times I,g_{K}=g|_{K},Sc(g_{K})\geq
6,d(\partial_{+}K,\partial_{-}K)\geq 25).$

This contradicts with the Gromov’s torical band estimate Theorem 2.10.

  * •

Case 2: Otherwise, by Cheeger-Colding theory in [6],

$M_{\infty}\simeq\mathbb{R}^{3},$

isometrically. Since the limiting space is $\mathbb{R}^{3}$, we can always
pick a big torical band in $\mathbb{R}^{n}$ and then proceed the same argument
in case $1$ to reach a contradiction. Here, we will not repeat the argument
again since it is totally the same as case $1$. In fact, we may also use the
argument in the proof of volume non-collapse in the Corollary 1.9 below.

Together with all arguments above, we proved that for any $p\in M^{3}$,

$\limsup_{R\rightarrow\infty}\frac{\text{vol}(B(p,R))}{R}<\infty.$

This completes our proof. ∎

######  Remark 3.4.

In fact, by the proof, we see that the manifold with non-negative Ricci
curvature, positive scalar curvature and volume non-collapse is asymptotic to
$\mathbb{S}^{2}\times[R,\infty)$ at infinity or splits globally. In fact, this
can be also seen from the perspective of the minimal surface argument.

By the proof of Theorem 1.8, we have the following weaker version of volume
growth in higher dimension.

###  3.3. The Proof of Corollary 1.9

###### Proof.

  * •

Proof of volume non-collapsed case

As the proof of Theorem 1.8, we assume that the result does not hold. Hence,
there exists a sequence $p_{i}\in M$ such that $(M,p_{i})$ pointedly Gromov
Hausdorff converges to $\mathbb{R}^{n}$. However, we do not know if the
convergence is $C^{0}$ convergence or not in the sense of Riemannian metric
convergence. Hence, we can not directly use Gromov’s upper semi-continuity of
scalar curvature under Riemannian metric $C^{0}$ convergence. Instead, we make
use of an estimate in [16]

First, for all $\epsilon>0$, there exists a Lipschitz map $f$ from
$\mathbb{R}^{n}$ to the standard unit sphere $\mathbb{S}^{n}$ with $deg(f)\geq
1$ , $Lip(f)\leq\epsilon$ and $f$ is constant at infinity. Namely, $f$ is
constant on $B^{c}(R)\subset\mathbb{R}^{n}$. Then, since we have,
$(M_{i},r_{i})$ converges to $\mathbb{R}^{n}$ with respect to the Gromov
Hausdorff convergence. Hence, there exists a map
$\varphi_{i}:(M_{i},r_{i})\rightarrow(\mathbb{R}^{n},0)$ with Lipschitz
constant $Lip(\varphi_{i})\leq 2$ and $B(2R)\subset
Im(\varphi_{i}({B(r_{i},3R)})$ for large $i$. Here $B(r_{i},2R)$ is the
geodesic ball in $M_{i}$ centered at $r_{i}$. Finally, we construct a map
$F_{i}=f\circ\varphi_{i}:M_{i}\rightarrow\mathbb{S}^{n}$ with $Lip(F)\leq
2\epsilon$ with $Sc(M_{i})\geq n(n-1)$ and $deg(f_{i})\geq 1$. If we pick
$\epsilon$ small enough, then what we obtain contradicts with the Spherical
Lipschitz Bound Theorem 2.11 (cited from [16]). Hence,

$\limsup_{R\rightarrow\infty}\frac{vol(B(p,R))}{R^{n-1}}<\infty.$

  * •

Proof of injectivity radius non-collapsed case

As the proof of Theorem 1.8, we assume that the result does not hold. Hence,
there exists a sequence $p_{i}\in M$ such that $(M,p_{i})$
$C^{\alpha},\alpha\in(0,1)$ converges to a smooth manifold
$\mathbb{S}^{1}\times\mathbb{R}^{n-1}$ since we assume that the injectivity
radius has a uniformly positive lower bound [2]. Then, we take a torical band
$T^{n-1}\times[R,2R]$ in $\mathbb{S}^{1}\times\mathbb{R}^{n-1}$ for large $R$.
Since the convergence is $C^{\alpha}$, we have that the properties $(**)$ are
automatically satisfied by Anderson’s result in [2]. Hence, we will reach a
contradiction, since the following steps follow the same argument in the proof
of Theorem 1.8, we have

$\limsup_{R\rightarrow\infty}\frac{vol(B(p,R))}{R^{n-2}}<\infty.$

∎

###  3.4. Proof of Theorem 1.12

###### Proof.

Since we assume that $(M^{3},g)$ has nonnegative Ricci curvature, we have
$(M,g)$ has at most $2$ ends.

  * •

If $(M^{3},g)$ has $2$ ends, we have $(M,g)$ is split. i.e.

$M^{3}=\mathbb{S}^{2}\times\mathbb{R}.$

In this case, we take the function $f$ as the projection
$\mathbb{S}^{2}\times\mathbb{R}$ to $\mathbb{R}$. Since $Sc(g)\geq 2$, it is
trivial that for any $r\in\mathbb{R}$. $diam(f^{-1}(r))\leq 4\pi$ and
$f^{-1}(r)$ is a $2$ sphere and hence connected;

  * •

If $(M^{3},g)$ has $1$ end, we take any ray $\gamma(t)\in M$ and obtain the
associated Busemann function

$B_{\gamma}(x):M\rightarrow\mathbb{R}.$

Claim: $f(x)=B_{\gamma}(x)$ is a continuous function as required.

Assume that there exists a sequence $r_{i}\rightarrow\infty$ such that,
$diam(f^{-1}(r_{i}))\rightarrow\infty.$ By the proof of Theorem 1.8, there
exists a subsequence $p_{i}\in f^{-1}(r_{i})$ such that $(M,p_{i})$ pointedly
Gromov Hausdorff converges to a length space(smooth manifold)
$(M_{\infty}=X^{2}_{\infty}\times\mathbb{R},p_{\infty},d_{\infty})$ and
$X_{\infty}$ is a compact manifold. Then, we take a large metric ball
$B(p_{\infty},10R)\subset M_{\infty}$ such that for large $i$, $5R\geq
diam(f^{-1}(r_{i}))\geq R$ but the level set $f^{-1}(r_{i})$ is contained into
some neighborhood of $\gamma$: $N_{s}(\gamma)=\\{x\in M,d_{g}(x,\gamma)\leq
s\\}$. Here, $s$ only depends on $diam(X_{\infty}^{2})$.

Since we assume that $\text{diam}f^{-1}(r_{i})$ diverges to $\infty$ and we
know $\gamma(r_{i})\in f^{-1}(r_{i})$, then, for the large $i$ picked above,
we take a point $y_{i}\in f^{-1}(r_{i})$ such that
$d_{g}(y_{i},\gamma(r_{i}))$ is large. Moreover, we pick $q_{i}\in\gamma$ such
that $d(y_{i},\gamma)=d(q_{i},y_{i})$ that is small relative to the $r_{i}$.
Hence, for any large $t\geq r_{i}$, by the definition of Busemann function, we
have

$r_{i}=f(y_{i})=B_{\gamma}(y_{i})>t-d(y_{i},\gamma(t).$

If we initially pick $i$ large enough such that
$d_{GH}(B(p_{i},10R),B(p_{\infty},10R)$ is small, we obtain that for large $t$

$r_{i}\geq t-d(y_{i},\gamma(t))>r_{i}+1.$

Hence, we reach a contradiction. We conclude that there exists a constant $c$
such that $diam(f^{-1}(r))\leq c$.

∎

######  Remark 3.5.

Geometrically, the proof is very clear. If we keep
$X_{\infty}\times\mathbb{R}$ in mind, it would be natural to argue the level
set is uniformly bounded even the proof seems indirect.

######  Remark 3.6.

This article is a revised version of my original article, which was finalized
in May 2021 and submitted to a Journal in September 2021.

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# Deep transfer learning for detecting Covid-19, Pneumonia and Tuberculosis
using CXR images - A Review

Irad Mwendo  Department of Computer Science
Dedan Kimathi University of Technology
Nyeri, Kenya
<EMAIL_ADDRESS>Patrick Gikunda  Department of Computer Science
Dedan Kimathi University of Technology
Nyeri, Kenya
<EMAIL_ADDRESS>Anthony Maina  Department of Computer Science
Dedan Kimathi University of Technology
Nyeri, Kenya
<EMAIL_ADDRESS>
###### Index Terms:

Deep Learning, Transfer Learning, Covid-19, Pneumonia, Tuberculosis

Chest X-rays remains to be the most common imaging modality used to diagnose
lung diseases. However, they necessitate the interpretation of experts
(radiologists and pulmonologists), who are few. This review paper investigates
the use of deep transfer learning techniques to detect COVID-19, pneumonia,
and tuberculosis in chest X-ray (CXR) images. It provides an overview of
current state-of-the-art CXR image classification techniques and discusses the
challenges and opportunities in applying transfer learning to this domain. The
paper provides a thorough examination of recent research studies that used
deep transfer learning algorithms for COVID-19, pneumonia, and tuberculosis
detection, highlighting the advantages and disadvantages of these approaches.
Finally, the review paper discusses future research directions in the field of
deep transfer learning for CXR image classification, as well as the potential
for these techniques to aid in the diagnosis and treatment of lung diseases.

##  I Background Information

Due to constant exposure to air particles from the outside environment, human
lungs remain highly susceptible to airborne diseases and other injuries [1].
These diseases are commonly referred to as Respiratory Diseases (RD) and have
become the leading cause of death worldwide [2]. They include Covid-19 [3],
Pneumonia, and Tuberculosis [4] among others.

Covid-19, is a viral respiratory illness caused by the novel coronavirus SARS-
CoV-2, which was discovered in Wuhan, China, in December 2019. The virus
spread quickly around the world, resulting in a global pandemic that affected
millions of people and caused significant social and economic disruption[5].
As of March 8, 2023, there had been over 445 million confirmed cases of
COVID-19 worldwide, with over 6 million deaths [6].

Tuberculosis, or TB, is a bacterial infection that primarily affects the lungs
but can affect other parts of the body as well[7]. When an infected person
coughs or sneezes, the virus spreads through the air and can cause coughing,
fever, and weight loss. TB can be treated with antibiotics, even though the
drug resistance is a growing concern. An estimated 10 million people worldwide
became ill with tuberculosis in 2020, with 1.5 million dying as a result of
the disease[8] and due to Covid-19, more deaths are expected in 2022 and 2023.

Pneumonia is another type of respiratory disease that affects the lungs and is
caused by a virus, bacteria, or fungi[9]. The three types of Pneumonia affects
both children and adults even though, viral pneumonia is most common in
children below 5 years of age and adults with weak immunity [10]. On the other
hand, bacterial pneumonia remains the leading cause of mortalities in children
and a major cause of illnesses in adults.In United States (US), community
acquired pneumonia is estimated to be 4 - 5 million cases annually, with over
1 million hospitalizations and 50,000 deaths[11]. Fungal pneumonia is less
common, and is serious to people with weak immunity. In recent years, fungal
pneumonia incidences has increased due to decline in immunity as a result to
the growing infections of HIV/AIDS, and Cancer among other factors[12].

These lung conditions: Covid-19, Pneumonia, and Tuberculosis have serious
health consequences and can be fatal if untreated or mismanaged. However,
misdiagnosis[13], shortage of healthcare professionals[14], and the high cost
of treatment[15] continue to be significant challenges. To begin with, lung
diseases can be difficult to accurately diagnose, particularly in the early
stages, and are frequently misdiagnosed as other respiratory conditions,
causing disease progression by delaying proper treatment. Second, there is
frequently a shortage of healthcare professionals with specialized training in
the diagnosis and treatment of lung diseases, especially in low and middle-
income countries. Finally, the cost of treatment for lung diseases,
particularly Covid-19, Tuberculosis and Pneumonia, can be high and may not be
covered by health insurance, making it difficult for some patients to access
necessary care.

Healthcare technology has become increasingly important and widespread in the
medical industry in recent years, particularly in the management and treatment
of lung conditions. Chest X-rays, CT scans, and ultrasound are examples of
imaging technologies that have been widely used to diagnose and monitor lung
diseases. Chest X-rays and CT scans, for example, have been used to diagnose
and monitor opacity in COVID-19 patients, while ultrasound has been used to
diagnose and monitor pneumonia[16]. Telemedicine has also been widely used
during the COVID-19 pandemic for remote consultations, symptom monitoring, and
treatment. Patients suffering from lung diseases can now consult with
healthcare providers via video conferencing, reducing the risk of transmission
and ensuring timely treatment[17]. Finally, Artificial Intelligence (AI) and
Machine Learning (ML) have been used to analyze large datasets of patient
data, allowing for more precise diagnosis and treatment of lung diseases at a
reduced cost[18].

##  II Introduction

Chest X-ray (CXR) images are the most commonly used imaging for the diagnosis
of many lung conditions as mentioned in [3] and [4].CXR images are more
preferred to Computed Tomography (CT) scans as they expose the patients to
ionizing radiation[19]. These images requires experts for interpretation[20],
and as the number of patients suffering from these diseases rises, these
experts, who are few, are forced to overwork and in the process, result to
delayed diagnoses and timely reporting of results[21]. Also, due to
subjectivity nature of these diseases, experts are likely to produce varying
results for same CXR image interpretation[22].Treatment of lung diseases can
be expensive, particularly if one condition is overlooked and requires
additional healthcare costs down the line and to overcome such problems,
automation for multiple lung diseases detection can help streamline the
diagnostic process and enable healthcare providers make more informed
decisions about treatment without relying solely on experts. This can help to
reduce waiting times, improve patient outcomes, and reduce the burden on
radiologists while enabling faster and more efficient diagnoses. This can be
achieved with the help of Computer Aided Diagnosis (CAD)[23] which can be
improved through Computer Vision (CV) and Deep Learning (DL) approaches[24].

In the recent past, DL and traditional machine learning approaches have been
widely employed in the classification of medical images into either normal or
abnormal instances, allowing experts to specialize in the analysis of abnormal
images and classify them accordingly[21]. These classification are conducted
in two steps: the first step involves extracting image features with image
descriptors and second step involves feeding those features to a classifier
like Support Vector Machines (SVM), and K-Nearest Neighbors (KNN)[25].The
problem with traditional ML approaches is that the accuracy of classification
task depends on the manual feature extraction techniques [26].Another approach
employed for image classification tasks is DL, which allows automatic feature
extraction and classification in one step through use of convolution, pooling
and fully connected layers[26]. This has been evidenced in the works of [27]
to classify Pneumonia images and normal images, and [28] in detection of lung
cancer from normal instances among others.These DL approaches however are
computationally expensive[29] and suffers from over-fitting and low transfer
capability[26] and to achieve better results, they necessitates need for large
amounts of labelled datasets[30] which is sometimes, difficult to acquire.

To overcome the challenges with DL, Deep Transfer Learning (DTL), which can
also be used in medical image processing is used[31].DTL does not involve
training models from scratch, however, it uses pre-trained models similarly to
the ones trained on classical DL models like Convolution Neural Network(CNN)
to solve one problem, and then, use them to solve different but related
problems[32].These DTL approaches have been widely used in the medical
analysis of lung diseases which is evidenced in several works. For instance,
in the works of [33] to automatically classify and detect Covid-19, Pneumonia,
and Tuberculosis and similarly, in the works of [34] for the detection of
different types of Pneumonia from chest radiographs. Also, ensembled DTL
approaches were adopted by [35] to rapidly aid diagnosis of Covid-19 from
chest radiographic images which were also used by [36].From most of the works,
DTL possess several advantages over DL and traditional ML approaches which
are: (1) Requires small dataset for training [37], (2) High performance [38]
and, (3) less training time[39]. In solving domain problems, DTL adopts
strategies like freezing layers[40], fine tuning[41] and usage of fixed
feature extractor (freeze all layers except for final fully connected layer).
From the understanding of DTL, the research questions to be addressed are

  1. 1.

What is the role of DTL in the detection of lung diseases?

  2. 2.

What are the current state of the art DTL methods used?

  3. 3.

What are the advantages of using DTL for lung disease detection?

The rest of the paper is organized as follows: In Section III, we present the
methodological approach followed in filtering relevant state of the art works,
in Section IV we present relevant state of the art related works, and in
Section V we present the summary and Future works.

##  III Methodology

The study used a database with information from over 60 scientific articles
published between 2016 and 2023 to address the research questions in this
domain. These articles were obtained from a variety of sources, including
Google Scholar, IEEE Xplore, ScienceDirect, PubMed, and Springer. The articles
were chosen based on search phrases related to ”Covid-19”, ”Pneumonia”,
”Tuberculosis,” ”Lung diseases”, ”deep learning”, ”machine learning”,
”transfer learning”, ”deep transfer learning”, ”Chest x-rays”, and other
keyword combinations related to lung diseases’ detection. The search results
were reviewed to select those that were relevant to our work and to eliminate
those that were not. The analysis of the articles was done while taking into
consideration the lung disease detected in the study, data set used, state of
the art method used and the performance obtained based on different evaluation
metrics used.

##  IV Related Works

Many studies have been conducted over the past few years for the detection and
automatic classification of different lung conditions. Some of the studies
carried out are presented:
Panwar et al.[42] proposed a deep transfer learning model nCOVnet made from
the original VGG16 model to detect Covid-19 from Chest X-ray images.The only
data pre-processing technique used for images was resizing to 224x224x3, and
during the training of the model, data leakages were avoided by manually
splitting the train and test data set which was evaluated using AUC of ROC,
training accuracy and confusion matrix giving an overall accuracy score of
88%.The data set used was quite small comprising of 337 totals images, with
only 192 Covid-19 positive cases.

Three deep transfer learning approaches (ResNet, XCeption and DenseNet) were
studied in the detection of patients with Covid-19, Pneumonia, and TB using
Chest X-rays[43]. The authors used histogram equalization to minimize bias in
the data set, and at the same time, normalized the data. The study achieved
high performance accuracy of 98.2%, 94.21%, and 93.67%, obtained in training,
validation, and testing sets respectively with ResNet achieving the highest
overall performance. However, the data set used was little and imbalanced and
comprised of 1229 chest X-ray images with 42 COVID-Medium images, 40 COVID-
Mild images, 36 COVID-Severe images, 348 regular images, 263 tuberculosis
images, and 500 pneumonia images.

Oguz et al. [44] used 1345 CT scan images obtained from a research hospital
and applied them to different deep learning methods like ResNet-50,
ResNet-101, AlexNet, GoogleNet, and other classification methods like SVM,
Random Forest, Decision Trees, etc. to reduce diagnosis time for COVID-19.
Pre-processing activities involved reducing the dimensions of the CT images
through maximum pooling and applying the ReLU activation function to each
convolution layer. The experimental results have shown that ResNet-50 and SVM
performed better than other models with 96.3% accuracy and an F1-score of
95.87%. This work is however limited due to the size of data used and might
not generalize well to new data.

Kassania et al.[45] compared different deep transfer learning approaches
(MobileNet, DenseNet, Xception, ResNet, InceptionV3, InceptionResNetV2,
VGGNet, NASNet) in the automatic prediction of COVID-19 Pneumonia cases in
Chest x-rays and CT scans. The images used were resized and normalized. The
data set had 274 images with 137 images of positive Covid-19 cases and 137
images of health patients.Densenet-121 achieved high performance with a
classification accuracy of 99% followed by ResNet-50 with 98% classification
accuracy. Despite the high performance, the study focused on detecting two
conditions i.e. Covid-19 and Normal instances, and also, used little data set.

The more classes used the lower the accuracy of the model. This was evidenced
in the works of Hussain et al. [46] who proposed a deep CNN model ’CoroDet’
with 22 layers. The total data set used had a total of 7390 images where 2843
were for COVID-19 case,3108 for normal cases and 1439 for pneumonia (both
viral and bacterial) cases. Without pre-procesing the data, they applied their
deep CNN model to 2 (Covid-19 ,Pneumonia), 3(Covid-19, Pneumonia-Bacteria,
Normal), and 4(Covid-19, Pneumonia-Bacteria, Normal, Pneumonia-Viral) class
classifications.Their model achieved an accuracy of 99.1%, 94.2% and 91.2% for
2, 3 and 4 class classifications respectively. However, these results can be
improved by increasing the data, balancing the class instances and pre-
processing the data to minimize noise.

In the works of Haritha et al.[47], pre-trained models appeared to perform
better in the detection of Covid-19 cases. The researchers developed CheXNet
model from the original DenseNet121 to detect any anomalies in the chest x-ray
dataset which was re-scaled, and augmented through flipping, rotating, and
zooming to increase the size of the training set leading to achieving an
accuracy score of 99.9%. This was however limited due to use of small data
sets which comprised of 1824 Chest x-ray images (912 for Covid-19 cases and
912 for Non-Covid cases).

Sai et al.[48] proposed a deep transfer learning CNN method for detecting
viral Pneumonia, bacterial Pneumonia, and Covid-19. Using a pre-trained
ResNet-50V2 model, chest radiographs(5856 images) were trained on 100 epochs
resulting in accuracy and test score of 94%. The high accuracy was achieved
through an increase in the number of epochs and the addition of a dropout
layer after the dense layer.

In an attempt to classify whether a person has lung disease or not, Bharati et
al.[49] developed a hybrid deep-learning model using CNN, VGG16, data
augmentation, and spatial transformer networks. Using attributes of age, chest
x-ray images, a person’s gender, and view position, their hybrid model
obtained a validation accuracy of 73% despite the noisy and complex data used.
To improve detection accuracy, Rajagopal et al.[50] proposed the usage of an
Arithmetic Optimization algorithm (AOA) on deep CNN and further reduces the
noise of images by applying an “Anisotropic Diffusion Filter Based Unsharp
Masking and crispening scheme”. Lung Cancer, Tuberculosis, and Pneumonia are
also detectable through imaging, Khobragade et al.[51] proposed an automatic
detection of these diseases using an artificial feed-forward neural network.
To ensure better performance of the model, the histogram equalization
technique was used in the improvement of image intensity and contrast. This
resulted in an accuracy of 92%. An ensemble learning method of random forest,
SVM, and Logistic regression was utilized by Ravi et al.[52] in the lung
disease detection and achieved an accuracy of 98% while Bhosale et al.[53]
developed an ensemble deep transfer learning CNN model for the detection of
obstructive pulmonary diseases with an accuracy of 99.70%.

Artificial Neural Networks (ANN) aids in the detection of respiratory diseases
diagnosed with chest radiographs. Usha et al.[54] developed a model for
Tuberculosis detection. The contraction path technique was used with 4 blocks
each having convolution layers, activation functions, and max pooling. This
resulted in an accuracy of 81%. Non-Linear and Fisher’s discriminant
techniques were adopted by Seixas et al.[55] to detect pleural Tuberculosis
based on clinical grounds. Two Deep transfer learning CNN networks, AlexNet
and GoogleNet were used in the classification of chest radiographs to detect
pulmonary Tuberculosis according to Lakhani et al.[56] with radiologists
assisting in cases where discrepancies occurred. Ahsan et al.[57] applied deep
transfer learning model of VGG 16 to detect whether patients have Tuberculosis
or not. Sigmoid activation was used in the output layer while Adam optimizer
was used to update weighs resulting in an accuracy of 80% in the model. Liu et
al.[58] used shuffle sampling technique to deal with the unbalanced data set
in the classification of different TB manifestations, and achieved a
classification accuracy of 85.68%.

##  V Summary and Future works

The reviewed research works demonstrate the efficacy of employing deep
transfer learning approaches in the detection of lung diseases. Deep transfer
learning models mostly used include ResNet-50, VGG-16, InceptionV3, ResNet-18
and DenseNet-121 among others [48][49][59].

To improve the performance of models, the researchers have utilized techniques
like transfer learning, data augmentation and hyper-parameter tuning[59][60].
TL has been used as the base step to allow training of the pre-trained models
on new datasets [60]. Data augmentation techniques have been applied to
generate new training samples by applying transformations to the existing data
set [59][61].Hyperparameter tuning optimized the parameters that control the
behavior of deep learning models, such as the learning rate or the number of
layers [59][62]. Despite the advancements of research in this field, there are
still pre-existing gaps:

  1. 1.

The studies reviewed relatively use small data sets, for instance[42].

  2. 2.

Most of the studies reviewed focuse on detecting one lung disease [42][44].

  3. 3.

In the experimental analysis of different deep and transfer learning
approaches, data imbalance[58] issue remains a key problem and this could
result to biased results obtained in many researches.

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