# On rigidity of conformal submersions

Atreyee Bhattacharya and Sayoojya Prakash  Department of Mathematics, Indian
Institute of Science Education and Research Bhopal<EMAIL_ADDRESS>Department of Mathematics, Indian Institute of Science Education and Research
Bhopal<EMAIL_ADDRESS>
###### Abstract.

Conformal submersions are generalizations of Riemannian submersions where the
differential of the submersion map is a conformal isometry when restricted to
the horizontal distribution. Riemannian submersions have been instrumental in
the construction of new Riemannian metrics from existing ones. Therefore,
being generalizations of Riemannian submersions, conformal submersions are
potential tools for constructing new examples of (special) Riemannian
manifolds. A conformal submersion is said to be rigid if it reduces to a
Riemannian submersion up to homothety. In this paper, we study curvature
conditions that force conformal submersions to be rigid. Moreover, under
suitable circumstances, using these rigidity criteria for conformal
submersions, we conclude the rigidity of certain quasi-Einstein metrics.

###### Key words and phrases:

conformal submersions, quasi-Einstein, rigidity

###### 2020 Mathematics Subject Classification:

53C24, 53C15, 53C21, 53C25

##  1\. Introduction

Based on curvature properties, Riemannian manifolds are broadly classified
into following three categories: the space forms (manifolds with constant
sectional curvature), Einstein manifolds (manifolds with constant Ricci
curvature), and locally conformally flat manifolds (manifolds with vanishing
Weyl curvature). While the topology of a space form is completely well known
via its curvature properties, conditions under which a smooth manifold admits
an Einstein or a locally conformally flat metric, are yet to be fully
understood. Creating more examples of Einstein and locally conformally flat
manifolds is thus a necessary step to understand more about these spaces. To
this end, Riemannian submersions have been used as standard tools to construct
special classes of Einstein manifolds and classify them ([2, 21, 11, 22, 3]).
Conformal submersions are generalizations of Riemannian submersions where the
differential of the submersion map is a conformal isometry when restricted to
its horizontal distribution (see Definition 2.9 for details). The structural
equations (connecting the geometry of the total space, fibres and the base) of
a conformal submersion are immediate from the corresponding equations of
Riemannian submersions. Despite this, conformal submersions are far from being
well studied as compared to Riemannian submersions, apart from making sporadic
appearances in Riemannian geometry and theoretical physics ([20, 13, 14]). On
the other hand, as generalizations of Riemannian submersions, conformal
submersions are potential tools for the construction of more examples of
Riemannian manifolds. In view of this, one would like to understand geometric
conditions that force a conformal submersion to become a Riemannian submersion
up to homothety, and in that case, the conformal submersion under
consideration is called rigid. Recently, the rigidity of conformal submersions
was proved for a special class of contact metric manifolds ([1]) using
standard contact Riemannian geometry techniques. However, in general, not much
is known about the rigidity of conformal submersions that the authors are
aware of.

It is to be noted that in Riemannian geometry and geometric analysis,
analysing rigidity of standard geometric properties (such as curvature
properties, existence of special maps, etc.) of Riemannian manifolds has been
a key technique in identifying when a given manifold can admit a special
Riemannian metric. Although there are several interpretations of rigidity of a
geometric property, in this paper, we will restrict to the following. A
(geometric) property $\mathcal{P}$ on a Riemannian manifold $(M,g)$ is said to
be rigid if it has one of the following features:

  1. (1)

Certain constraints are imposed on (the underlying topology of) $M$ due to the
existence of $\mathcal{P}$. In other words, $(M,g)$ fails to possess
$\mathcal{P}$ unless it has the desired underlying structure.

  2. (2)

$\mathcal{P}$ generalises a property $\mathcal{P}^{\prime}$, but under some
conditions, the existence of $\mathcal{P}$ and $\mathcal{P}^{\prime}$ are
equivalent.

The above notions of rigidity will be referred to as Type (1) and Type (2),
respectively. While Type (1) is the strongest form of rigidity (e.g., constant
sectional curvature, Ricci curvature bounded below by a positive constant,
etc.) as it restricts the underlying topology of a manifold, rigidity of a
conformal submersion is of Type (2). In fact, in this paper, we discuss two
rigidity problems of Type (2), as elaborated below.

We first study curvature conditions that impose rigidity on conformal
submersions where the total space, fibres, and base are all closed orientable
Riemannian manifolds, and the total space has at least dimension four. Our
analysis mostly uses classical techniques from Riemannian geometry, such as
the weak maximal principle and the divergence theorem. Recall that in a
Riemannian submersion, the fundamental tensors $T$ and $A$ represent
obstructions to the fibres being totally geodesic and the integrability of the
horizontal distribution, respectively, and $N$ denotes the trace of $T$.
Analogously, there are fundamental tensors $T$ and $A$ for a conformal
submersion (see Section 2.2 for details). In particular, we prove the
following result.

######  Theorem 1.1.

Let $\pi\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with fibre
$F$, where the total space $(M,g)$ is the product of closed Einstein
manifolds. Let $\lambda_{i}$ denote the Einstein constant of the $i^{th}$
component of $(M,g)$, and $m_{i}$ and $k_{i}$ denote the respective dimensions
of horizontal and vertical distributions restricted to the same component.
Then $\pi$ is rigid if it satisfies one of the following three conditions.

  1. (1)

The horizontal distribution is integrable, and $N$ is parallel, together with
either of the following properties:

    1. (a)

The fibres are Einstein (with the same Einstein constant) and
$dim(B)\geq 2$,

    2. (b)

The base $(B,g_{B})$ is Ricci-flat and
$\sum_{i}\lambda_{i}m_{i}=-|T|^{2}-\frac{|N|^{2}}{\dim(F)}(\dim(M)+\dim(B)-4)$.

  2. (2)

The tensor $N$ vanishes together with either of the following conditions:

    1. (a)

The fibres are Einstein (with Einstein constant $\lambda_{F}$), $\dim(B)\geq
2$, and
$\sum_{i}\lambda_{i}k_{i}=\lambda_{F}dim(F)+|A|^{2}$,

    2. (b)

$(B,g_{B})$ is Ricci-flat and $\sum_{i}\lambda_{i}m_{i}+2|A|^{2}+|T|^{2}=0$,

  3. (3)

The fibres are totally geodesic, $(B,g_{B})$ is Ricci-flat with $\dim(B)\geq
2$, and
$\sum_{i}\lambda_{i}m_{i}+2|A|^{2}=0$.

Using Theorem 1.1, we make the following interesting observation.

######  Theorem 1.2.

Let $(M,g)$ be a closed Einstein manifold of dimension at least $4$, which is
the total space of a conformal submersion $\pi\colon(M,g)\rightarrow(B,g_{B})$
with integrable horizontal distribution and vanishing $T$. Then up to
homothety, $g$ is the unique Einstein metric in its conformal class whenever
the fibres are one dimensional or the base is Ricci-flat.

######  Remark 1.3.

Theorem 1.2 says in particular that a closed Ricci-flat Riemannian manifold
$(M,g)$ that is a Riemannian product, admits a unique Einstein metric in the
conformal class of $g$, up to homothety.

The next result deals with the rigidity of conformal submersions with locally
conformally flat total space.

######  Theorem 1.4.

Let $\pi\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with fibre
$(F,g_{F})$ such that $(M,g)$ is locally conformally flat and $dim(B)=dim(F)$.
If the scalar curvatures of the total space and fibres have particular signs
(not necessarily, the same sign) and $(F,g_{F},h,m,\lambda)$ is quasi-
Einstein, then $\pi$ is rigid if either of the following conditions hold.

  1. (1)

$T=0$, the scalar curvature of the base is non-positive, and $\lambda\leq 0$.

  2. (2)

$A=0$, parallel $T$, the scalar curvature of the base is non-negative, and
$\lambda\geq 0$.

######  Remark 1.5.

We also provide criteria for rigidity of conformal submersions similar to
Theorem 1.4 by assuming either the total space or the base to be quasi-
Einstein (see Theorems 4.5 and 4.8). In the following paragraph, quasi-
Einstein manifolds are briefly discussed; for more details on the same, we
refer to Section 2.3.

######  Remark 1.6.

The condition $dim(F)=dim(B)$ in Theorem 1.4 is necessary, and can be verified
using Example 4.10.

It turns out that under suitable circumstances, the rigidity of conformal
submersions, is naturally related to the rigidity of a special class of
Riemannian manifolds known as quasi-Einstein manifolds, a generalization of
Einstein manifolds. More precisely, a Riemannian manifold $(M,g)$ is said to
be quasi-Einstein if its Ricci curvature $Ric_{g}$ satisfies the identity:
$\ Ric_{g}-\lambda g=Hess(f)-\frac{1}{m}df\otimes df,$ for some $f\in
C^{\infty}(M)$, and constants $\lambda\in\mathbb{R}$ and $0<m\leq\infty$. A
quasi-Einstein manifold is called rigid if it reduces to being Einstein.
Quasi-Einstein manifolds and their rigidity are topics of interest both in
Riemannian geometry and theoretical physics (cf. [9, 15, 8, 19, 18, 17, 23, 5,
4]). In this paper, we obtain rigidity criteria for a closed quasi-Einstein
manifold by realizing it as the base or fibre of a suitable conformal
submersion and then using the aforementioned rigidity criteria for conformal
submersions. This seems to be a natural approach as Riemannian submersions
have been used to answer analogous questions in the context of Einstein
metrics (cf. [22, 3]). Since all closed quasi-Einstein manifolds of dimension
two are known to be rigid ([4, 9]), we consider quasi-Einstein manifolds of
dimensions at least $3$ and prove the following result.

######  Theorem 1.7.

A closed orientable quasi-Einstein manifold $(M,g,f,m,\lambda)$ is rigid if it
satisfies one of the following criteria.

  1. (1)

$m\geq 1$, $\lambda\geq 0$ and $(M,g)$ is the fibre (respectively, the base)
of a conformal submersion with integrable horizontal distribution, parallel
$T$, locally conformally flat total space having twice the dimension of $M$,
and where the base (respectively, the fibre) has non-negative scalar
curvature.

  2. (2)

$(M,g)$ is the fibre of a conformal submersion with integrable horizontal
distribution, vanishing $T$ and homogeneous Einstein total space, and where
the base is Einstein with a non-positive Einstein constant.

######  Remark 1.8.

We obtain rigidity results for a quasi-Einstein manifold similar to Theorem
1.7 by realizing it as the total space of a conformal submersion (see Theorem
5.2) and conclude the following result (see Example 5.3 for details).

######  Corollary 1.9.

Let $(M,g,h,m,\lambda)$ be a locally conformally flat quasi-Einstein manifold
of dimension $\geq 4,$ where $m\geq 1$ and $\lambda\geq 0$. If the universal
cover of $(M,g)$ is closed and the Riemannian product of two manifolds of
equal dimensions, then $(M,g)$ is a space form.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Riemannian submersions
      1. 2.1.1 Curvature formulae for Riemannian submersions.
    2. 2.2 Conformal submersions.
      1. 2.2.1 Structural equations and curvature formulas for conformal submersions.
    3. 2.3 Quasi-Einstein manifolds.
  3. 3 Conformal submersions with Einstein total space
    1. 3.1 When the horizontal distribution is integrable and $N$ is parallel.
    2. 3.2 When $N$ vanishes identically.
    3. 3.3 When the fibres are totally geodesic.
  4. 4 Conformal submersions with locally conformally flat total space
    1. 4.1 The total space is the product of locally conformally flat and Einstein manifolds.
    2. 4.2 When the total space is quasi-Einstein
    3. 4.3 When either the base or the fibre is quasi-Einstein.
  5. 5 An Application: Rigidity of quasi-Einstein manifolds

##  2\. Preliminaries

In this section, we recollect some basic concepts from Riemannian geometry
that are relevant to this paper (see [2, 6, 16] for details) and set up the
notations to be followed. Let $(M,g)$ denote a Riemannian manifold and
$\nabla$ the associated Levi-Civita connection. We will restrict to closed,
connected and orientable Riemannian manifolds.

######  Definition 2.1.

Given a Riemannian manifold $(M,g)$, its Riemann curvature $R$ is the $(0,4)$
tensor defined by

$R(X,Y,Z,W)=g(\nabla_{Y}\nabla_{X}Z-\nabla_{X}\nabla_{Y}Z+\nabla_{[X,Y]}Z,W),$

where $X,Y,Z$ and $W$ are arbitrary vector fields defined on an open subset of
$M$.

######  Remark 2.2.

The Riemann curvature tensor of $(M,g)$ satisfies the following orthogonal
decomposition (cf. [6])

(2.1) $R=\frac{s}{2n(n-1)}g\circ g+\frac{1}{n-2}(Ric-\frac{s}{n}g)\circ g+W$

where $Ric$, $W$ and $s$ denote the Ricci, Weyl and scalar curvatures of
$(M,g)$ respectively and $\circ$ denotes the Kulkarni-Nomizu product for
symmetric $(0,2)$ tensors of $(M,g)$ defined by,

$\displaystyle(h\circ k)(X,Y,Z,W)=$ $\displaystyle
h(X,Z)k(Y,W)+h(Y,W)k(X,Z)-h(X,W)k(Y,Z)$ $\displaystyle-h(Y,Z)k(X,W),\ \ \
\forall\,h,\,k\,\in\,S^{2}(M).$

######  Definition 2.3.

A Riemannian manifold $(M,g)$ is said to be

  1. (1)

a Space form if it has a constant sectional curvature.

  2. (2)

Einstein if its Ricci curvature is a scalar multiple of the Riemannian metric,
i.e., $Ric=\lambda g\ \ \text{ for some }\ \lambda\in\mathbb{R}.$

  3. (3)

locally conformally flat if given any point $p\in M$, there exists an open set
$\mathcal{U}$ containing $p$ and $f\in C^{\infty}(\mathcal{U})$ such that the
metric $e^{2f}g|_{\mathcal{U}}$ is flat.

######  Remark 2.4.

A Riemannian manifold $(M,g)$ of dimension at least $4$ is locally conformally
flat if and only if its Weyl tensor vanishes (cf. [6])

###  2.1. Riemannian submersions

Let $(M,g)$ and $(B,g_{B})$ be Riemannian manifolds and $\pi:M\rightarrow B$ a
smooth submersion. Given $p\in M$ and $\pi(p)=b$, let $F_{b}=\pi^{-1}(b)$
denote the fibre at $b$, $\mathcal{V}_{p}:=Ker(d\pi_{p})=T_{p}F_{b}$, and let
$\mathcal{H}_{p}$ be the orthogonal complement of $\mathcal{V}_{p}$ in
$T_{p}M$.
The distributions $\mathcal{V}:p\mapsto\mathcal{V}_{p},\ p\in M$, and
$\mathcal{H}:p\mapsto\mathcal{H}_{p},\ p\in M$ are called the vertical and
horizontal distributions of $\pi$ respectively.

######  Definition 2.5.

A (smooth) submersion $\pi\colon(M,g)\rightarrow(B,g_{B})$ as above is called
a Riemannian submersion if

$g|_{\mathcal{H}}=\pi^{*}g_{B}.$

######  Remark 2.6.

A vector field $\xi$ on $M$ is called vertical (respectively, horizontal) if
$\xi_{p}\in\mathcal{V}_{p}$( respectively, $\mathcal{H}_{p}$), $\forall\,p\in
M$.

######  Definition 2.7.

Given a Riemannian submersion $\pi\colon(M,g)\rightarrow(B,g_{B})$, the
fundamental tensors $T$ and $A$ are defined as follows,

(2.2) $\displaystyle
T_{E}F=\mathcal{H}\nabla_{\mathcal{V}E}\mathcal{V}F+\mathcal{V}\nabla_{\mathcal{V}E}\mathcal{H}F$
$\displaystyle
A_{E}F=\mathcal{H}\nabla_{\mathcal{H}E}\mathcal{V}F+\mathcal{V}\nabla_{\mathcal{H}E}\mathcal{H}F$

where $E,F\in\chi(M)$, $\nabla$ is the Levi-Civita connection of $(M,g)$ and
$\mathcal{H}E$ and $\mathcal{V}E$ respectively denote the horizontal and
vertical components of $E$.

######  Remark 2.8.

It follows that in a Riemannian submersion, the vanishing of tensors $T$ and
$A$ are respectively equivalent to the fibres being totally geodesic and the
horizontal distribution being integrable (cf. [2]).

####  2.1.1. Curvature formulae for Riemannian submersions.

Let $p\in M$ and $\mathcal{U}$ a neighborhood of $p$ in $M$. Let $U,V$ be
vertical vector fields and $X,Y$ horizontal vector fields defined in
$\mathcal{U}$. Without loss of generality, one may assume that
$|U|=|V|=|X|=|Y|=1,|U\wedge V|=|X\wedge Y|=1$. Let $K_{p}$ and $\check{K}_{p}$
denote the sectional curvatures of $(M,g)$ and $(F_{\pi(p)},g_{F_{\pi(p)}})$
at $p$ respectively, and $\hat{K}_{\pi(p)}$ the sectional curvature of
$(B,g_{B})$ at $\pi(p)$. Then the following relations hold:

$\displaystyle K_{p}(U,V)$ $\displaystyle=$
$\displaystyle\check{K}_{p}(U,V)-g_{p}(T_{V}V,T_{U}U)+g_{p}(T_{U}V,T_{U}V)$
(2.3) $\displaystyle K_{p}(X,U)$ $\displaystyle=$ $\displaystyle
g_{p}((\nabla_{X}T)_{U}U,X)+|A_{X}U|_{p}^{2}-|T_{U}X|_{p}^{2}$ $\displaystyle
K_{p}(X,Y)$ $\displaystyle=$
$\displaystyle\hat{K}_{\pi(p)}(X,Y)-3|A_{X}Y|_{p}^{2}$

Let $r,\check{r},\hat{r}$ denote the Ricci curvatures of $(M,g)$, $(F,g_{F})$,
and $(B,g_{B})$, respectively. Let $p\in M$ and $\\{X_{i}\\}{i\in I}$,
$\\{U_{j}\\}_{j\in J}$ respectively denote local orthonormal frames of the
horizontal and vertical distributions at $p$. The Ricci curvatures
$r_{p},\check{r}_{p}$, and $\hat{r}_{\pi(p)}$ are related as follows:

$\displaystyle r_{p}(U,V)$ $\displaystyle=$
$\displaystyle\check{r}_{p}(U,V)-g_{p}(N,T_{U}V)+(\delta
T)(U,V)(p)+g_{p}(AU,AV)$ (2.4) $\displaystyle\ \ \ r_{p}(X,U)$
$\displaystyle=$ $\displaystyle
g_{p}((\check{\delta}T)(U),X)+g_{p}(\nabla_{U}N,X)-g_{p}((\hat{\delta}A)(X),U)$
$\displaystyle-2g_{p}(T_{U},A_{X})$ $\displaystyle r_{p}(X,Y)$
$\displaystyle=$
$\displaystyle\pi^{*}(\hat{r}_{\pi(p)})(X,Y)-2g_{p}(A_{X},A_{Y})-g_{p}(TX,TY)$
$\displaystyle+\frac{1}{2}(g_{p}(\nabla_{X}N,Y)+g_{p}(\nabla_{Y}N,X))$

where $N$ is the trace of $T$, i.e., $N=\sum_{j}T_{U_{j}}U_{j}$,
$(\delta T)(U,V)=\sum_{i}g((\nabla_{X_{i}}T)_{U}V,X_{i})$, and for any $(2,1)$
tensor $E$ on $M$
$\check{\delta}E=-\sum_{j}(\nabla_{U_{j}}E)_{U_{j}}$, and
$\hat{\delta}E=-\sum_{i}(\nabla_{X_{i}}E)_{X_{i}}$.
Finally, it follows that the scalar curvatures of $g$, $g_{F})$, and $g_{B}$
denoted respectively by $s,\check{s},\hat{s}$, are related as follows.

(2.5) $\displaystyle
s=\hat{s}\circ\pi+\check{s}-|A|^{2}-|T|^{2}-|N|^{2}-2\hat{\delta}N$

Here $\hat{\delta}N=-\sum_{j}(\delta T)(U_{j},U_{j})$. We refer to [2] for
further details.

###  2.2. Conformal submersions.

######  Definition 2.9.

Let $(M,g)$ and $(B,g_{B})$ be Riemannian manifolds. A smooth submersion
$\pi\colon M\rightarrow B$ is called a conformal submersion if
$g|_{\mathcal{H}}=e^{2f}\pi^{*}g_{B}$ for some $f\in C^{\infty}(M)$ where
$\mathcal{H}$ denotes the horizontal distribution of the submersion. In such a
case, we will denote the submersion map $\pi$ by $\pi_{f}$.

######  Remark 2.10.

  1. (1)

Given a conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ as in
Definition 2.9, consider the metric $g_{\pi}$ on $M$ given by
$g_{\pi}=e^{-2f}g.$ Then $\pi_{f}\colon(M,g_{\pi})\rightarrow(B,g_{B})$
defines a Riemannian submersion.

  2. (2)

Conformal submersions generalize Riemannian submersions, and when $f$ is
constant, a conformal submersion reduces to a Riemannian submersion up to
homothety.

This leads to the following definition.

######  Definition 2.11.

A conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is said to be
rigid if $f$ is a constant function.

####  2.2.1. Structural equations and curvature formulas for conformal
submersions.

Given a conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ as in
Definition 2.9, and the metric $g_{\pi}=e^{-2f}g$ on $M$, let
$\nabla,\tilde{\nabla}$ denote the Levi-Civita connections of $(M,g)$ and
$(M,g_{\pi})$ respectively. Let $T$ and $A$ denote the fundamental tensors of
the Riemannian submersion $\pi_{f}:(M,g_{\pi})\rightarrow(B,g_{B})$. Let $U,V$
denote vertical vector fields and $X,Y$ denote horizontal vector fields
defined in a neighborhood of an arbitrary point $p\in M$. Then, the following
structural equations hold:

(2.6) $\displaystyle\mathcal{H}\nabla_{U}V=$ $\displaystyle
T_{U}V-\mathcal{H}(\nabla f)g(U,V)$ $\displaystyle\mathcal{V}\nabla_{U}V=$
$\displaystyle\mathcal{V}\tilde{\nabla}_{U}V+U(f)V+V(f)U-\mathcal{V}(\nabla
f)g(U,V)$ $\displaystyle\mathcal{H}\nabla_{X}U=$ $\displaystyle A_{X}U+U(f)X$
$\displaystyle\mathcal{V}\nabla_{X}U=$
$\displaystyle\mathcal{V}\tilde{\nabla}_{X}U+X(f)U$
$\displaystyle\mathcal{H}\nabla_{U}X=$
$\displaystyle\mathcal{H}\tilde{\nabla}_{U}X+U(f)X$
$\displaystyle\mathcal{V}\nabla_{U}X=$ $\displaystyle T_{U}X+X(f)U$
$\displaystyle\mathcal{H}\nabla_{X}Y=$
$\displaystyle\mathcal{H}\tilde{\nabla}_{X}Y+X(f)Y+Y(f)X-\mathcal{H}(\nabla
f)g(X,Y)$ $\displaystyle\mathcal{V}\nabla_{X}Y=$ $\displaystyle
A_{X}Y-\mathcal{V}(\nabla f)g(X,Y).$

Let $p\in M$ and $\mathcal{U}$ a neighborhood of $p$ in $M$. Let
$U,V,W,W^{\prime}$ be vertical vector fields and $X,Y,Z,Z^{\prime}$ horizontal
vector fields defined in $\mathcal{U}$. Let $R$, $\check{R}$, and $\hat{R}$
denote the Riemannian curvature tensors of $(M,g)$,
$(F_{\pi_{f}(p)},g_{F_{\pi_{f}(p)}})$, and $(B,g_{B})$ respectively. Then the
curvature tensors are given below

(2.7) $\displaystyle R_{p}(U,V,W,W^{\prime})=$ $\displaystyle
g_{p}(\check{R}(U,V)W,W^{\prime})-g_{p}(T_{V}W^{\prime},T_{U}W)+g_{p}(T_{V}W,T_{U}W^{\prime})$
$\displaystyle+(T_{U}W)(f)(p)g_{p}(V,W^{\prime})-(T_{V}W)(f)(p)g_{p}(U,W^{\prime})$
$\displaystyle+g_{p}(U,W)T_{V}W^{\prime}(f)(p)-g_{p}(V,W)T_{U}W^{\prime}(f)(p)$
$\displaystyle+(g_{p}(U,W^{\prime})g_{p}(V,W)-g_{p}(U,W)g_{p}(V,W^{\prime}))|\mathcal{H}(\nabla
f)|_{p}^{2},$ (2.8) $\displaystyle R_{p}(X,Y,Z,Z^{\prime})=$ $\displaystyle
e^{2f(p)}\pi_{f}^{*}(\hat{R}_{\pi_{f}(p)})(X,Y,Z,Z^{\prime})-2g_{p}(A_{X}Y,A_{Z}Z^{\prime})$
$\displaystyle+g_{p}(A_{Y}Z,A_{X}Z^{\prime})-g_{p}(A_{X}Z,A_{Y}Z^{\prime})$
$\displaystyle-
Hessf(X,Z)(p)g_{p}(Y,Z^{\prime})-(X(f)Z(f))(p)g_{p}(Y,Z^{\prime})$
$\displaystyle+Hessf(Y,Z)(p)g_{p}(X,Z^{\prime})+(Y(f)Z(f))(p)g_{p}(X,Z^{\prime})$
$\displaystyle-
Hessf(Y,Z^{\prime})(p)g_{p}(X,Z)-(Y(f)Z^{\prime}(f))(p)g_{p}(X,Z)$
$\displaystyle+Hessf(X,Z^{\prime})(p)g_{p}(Y,Z)+(X(f)Z^{\prime}(f))(p)g_{p}(Y,Z)$
$\displaystyle+(g_{p}(X,Z)g_{p}(Y,Z^{\prime})-g_{p}(X,Z^{\prime})g_{p}(Y,Z))|\nabla
f|_{p}^{2}.$

Similarly, one can obtain the expression for curvature terms, where there is a
mix of vertical and horizontal vector fields. We skip the details here.
Let $K_{p},\check{K}_{p},\hat{K}_{\pi_{f}(p)}$ respectively denote the
sectional curvatures of $(M,g)$ and $(F_{\pi_{f}(p)},g_{F_{\pi_{f}(p)}})$ at
$p$ and $(B,g_{B})$ at $\pi_{f}(p)$. The following relations hold.

(2.9) $\displaystyle K_{p}(U,V)=$
$\displaystyle\check{K}_{p}(U,V)-g_{p}(T_{V}V,T_{U}U)+g_{p}(T_{U}V,T_{U}V)$
$\displaystyle+T_{U}U(f)(p)+T_{V}V(f)(p)-|\mathcal{H}(\nabla f)|_{p}^{2}$
$\displaystyle K_{p}(X,U)=$ $\displaystyle
2X(f)(p)g_{p}(T_{U}U,X)+g_{p}((\nabla_{X}T)_{U}U,X)+|A_{X}U|_{p}^{2}$ (2.10)
$\displaystyle-|T_{U}X|_{p}^{2}+(\mathcal{H}\nabla_{X}X)(f)(p)-(X(f))^{2}(p)-X^{2}(f)(p)$
$\displaystyle+(\mathcal{V}\nabla_{U}U)(f)(p)-(U(f))^{2}(p)-U^{2}(f)(p)$
(2.11) $\displaystyle K_{p}(X,Y)=$ $\displaystyle
e^{-2f(p)}\hat{K}_{\pi_{f}(p)}(X,Y)-3|A_{X}Y|_{p}^{2}+|\nabla
f|_{p}^{2}-Hessf(X,X)(p)$
$\displaystyle-(X(f))^{2}(p)-Hessf(Y,Y)(p)-(Y(f))^{2}(p)$

Denoting the Ricci curvatures of $(M,g)$ and
$(F_{\pi_{f}(p)},g_{F_{\pi_{f}(p)}})$ at $p$ by $r_{p}$ and $\check{r}_{p}$
respectively, and the Ricci curvature of $(B,g_{B})$ at $\pi_{f}(p)$ by
$\hat{r}_{\pi_{f}(p)}$, we obtain the following relations

(2.12) $\displaystyle r_{p}(U,V)=$
$\displaystyle\check{r}_{p}(U,V)-g_{p}(N,T_{U}V)+(\delta
T)(U,V)(p)+g_{p}(AU,AV)+dim(F)T_{U}V(f)(p)$
$\displaystyle+g_{p}(U,V)N(f)(p)-dim(F)g_{p}(U,V)|\mathcal{H}(\nabla
f)|_{p}^{2}-g_{p}(U,V)\mathcal{H}(\Delta f)(p)$
$\displaystyle+g_{p}(U,V)dim(B)|\mathcal{V}(\nabla
f)|_{p}^{2}+dim(B)((\mathcal{V}\nabla_{U}V)(f)$ $\displaystyle-
UV(f)-U(f)V(f))(p);$ $\displaystyle r_{p}(X,U)=$ $\displaystyle
g_{p}((\check{\delta}T)(U),X)+g_{p}(\nabla_{U}N,X)-g_{p}((\hat{\delta}A)(X),U)-2g_{p}(T_{U},A_{X})$
(2.13) $\displaystyle-(dim(B)-3)A_{X}U(f)(p)-(dim(F)-1)T_{U}X(f)(p)$
$\displaystyle-(dim(M)-2)(Hessf(X,U)+X(f)U(f))(p);$ (2.14) $\displaystyle
r_{p}(X,Y)=$
$\displaystyle\pi_{f}^{*}(\hat{r}_{\pi_{f}(p)})(X,Y)-2g_{p}(A_{X},A_{Y})-g_{p}(TX,TY)+g_{p}(\nabla_{X}N,Y)$
$\displaystyle+\sum_{j}g_{p}(\nabla_{U_{j}}A_{X}Y,U_{j})-(dim(M)-2)(Hessf(X,Y)+X(f)Y(f))(p)$
$\displaystyle-dim(F)A_{X}Y(f)(p)+(dim(M)-2)g_{p}(X,Y)|\nabla f|_{p}^{2}$
$\displaystyle+X(f)(p)g_{p}(N,Y)+Y(f)(p)g_{p}(N,X)-g_{p}(X,Y)(N(f)+\Delta
f)(p).$

The scalar curvatures of $(M,g)$, $(F_{\pi_{f}(p)},g_{F_{\pi_{f}(p)}})$, and
$(B,g_{B})$ denoted by $s,\check{s},\hat{s}$ respectively, are related as
follows

$\displaystyle s$
$\displaystyle=e^{-2f}\hat{s}\circ\pi_{f}+\check{s}-|A|^{2}-|T|^{2}+div(N)-\hat{\delta}N$
(2.15) $\displaystyle+dimB(dimM+dimF-3)|\nabla f|^{2}+(2-2dimB+dimF)N(f)$
$\displaystyle-(2dimF-2)\mathcal{H}(\Delta f)-2dimB\Delta
f-(dimF-2+dim(F)^{2})|\mathcal{H}(\nabla f)|^{2}$

######  Remark 2.12.

In this paper, the following conventions will be followed.

  1. (1)

Given a vector field $X$ in a Riemannian manifold $(M,g)$, the divergence
$div(X)$ of $X$ is defined point-wise as
$div(X)=\sum_{i}g(\nabla_{e_{i}}X,e_{i})$, where $\\{e_{i}\\}$ is an
orthonormal basis at that point.

  2. (2)

Given $h\in C^{\infty}(M)$, $\Delta h:=div(\nabla h)$.

  3. (3)

Given a conformal submersion $\pi_{f}:(M,g)\rightarrow(B,g_{B})$ and $h\in
C^{\infty}(M)$, $\mathcal{H}(\Delta h):=\sum_{i}g(\nabla_{X_{i}}\nabla
h,X_{i})$ and $\mathcal{V}(\Delta h):=\sum_{j}g(\nabla_{U_{j}}\nabla h,U_{j})$
, where $\\{X_{i}\\}{i\in I}$, $\\{U_{j}\\}_{j\in J}$ respectively denote
local orthonormal frames of the horizontal and vertical distributions.

######  Remark 2.13.

In a conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$, the tensor
field $A$ corresponds to the obstruction of integrability of horizontal
distribution. However, the fibres of $\pi_{f}$ are totally geodesic if and
only if $\mathcal{H}\nabla_{U}V$ vanishes for all vertical vector fields
$U,V$. Unlike Riemannian submersions, the vanishing of $T$ does not imply the
totally geodesic nature of fibres in conformal submersions. In contrast, it
leads to another property of fibres, defined below.

######  Definition 2.14.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion. Then
fibres of $\pi_{f}$ are said to be totally umbilical (that is, all fibres are
totally umbilical submanifolds of $(M,g)$) if there exists a horizontal vector
field $X$ such that $\mathcal{H}\nabla_{U}V=g(U,V)X$ for all vertical vector
fields $U,V$ (cf. [24]).

######  Remark 2.15.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion. It
follows that if $T$ vanishes, then fibres of $\pi_{f}$ are totally umbilical.

###  2.3. Quasi-Einstein manifolds.

In this section, we briefly discuss a special class of Riemannian manifolds
known as quasi-Einstein manifolds, which are certain generalizations of
Einstein manifolds. We refer to [4, 5, 8, 23] for detailed discussions on
quasi-Einstein manifolds.

######  Definition 2.16.

A smooth metric measure space is a triple $(M,g,f)$, where $(M,g)$ is a
complete Riemannian manifold and $f\in C^{\infty}(M)$.
Given a smooth metric measure space $(M,g,f)$, its $m$-Bakry-Emery Ricci
tensor denoted as $Ric_{f}^{m}$ is defined by

$Ric_{f}^{m}=Ric+Hess(f)-\frac{1}{m}df\otimes df,$

where $Hess(f)(X,Y)=g(\nabla_{X}\nabla f,Y)$ and $0<m\leq\infty$.

######  Definition 2.17.

A smooth metric measure space $(M,g,f,m)$ is called a quasi-Einstein manifold
if $Ric_{f}^{m}=\lambda g$ for some $\lambda\in\mathbb{R}.$

######  Remark 2.18.

It is easy to see that a quasi-Einstein manifold $(M,g,f,m)$ becomes Einstein
if $f$ is constant. This motivates the following definition.

######  Definition 2.19.

A quasi-Einstein manifold $(M,g,f,m,\lambda)$ is said to be rigid if $f$ is a
constant function.

######  Remark 2.20.

From now on, we will restrict ourselves to closed-orientable Riemannian
manifolds and conformal submersions with total space of dimension at least
$4$.

##  3\. Conformal submersions with Einstein total space

In this section, we analyse the rigidity of conformal submersions where either
the total space or its universal cover is the Riemannian product of closed-
orientable Einstein manifolds. In both cases, the Ricci tensor turns out to be
parallel. This section is divided into three subsections. The first subsection
discusses the rigidity when the horizontal distribution is integrable and $N$
is parallel which is a more general assumption than that of $T$ being parallel
(cf. Lemma $4.6$ of [12]). In the second subsection, we discuss the rigidity
when $N$ vanishes (a weaker assumption than $T=0$, particularly when the
dimension of fibres is at least $2$) and the horizontal distribution is not
necessarily integrable. The last subsection deals with the rigidity when
fibres are totally geodesic. Throughout this section, we will have the
following two assumptions (even when it is not explicitly mentioned): (a). the
total space or its universal cover is the Riemannian product of closed
Einstein manifolds and (b). whenever fibres are assumed to be Einstein, all
the fibres have the same Einstein constant. For the sake of brevity, proofs of
all the results in this section are only showcased when the total space is
Einstein. However, the same idea also works when either the total space or its
universal cover is the Riemannian product of closed-orientable Einstein
manifolds.

###  3.1. When the horizontal distribution is integrable and $N$ is parallel.

The following theorem addresses the rigidity of conformal submersions with
Einstein fibres.

######  Theorem 3.1.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
integrable horizontal distribution and parallel $N$ and $dim(B)\geq 2$. Then
$\pi_{f}$ is rigid if the fibres are Einstein.

###### Proof.

Assume without loss of generality, that $(M,g)$ is Einstein with Einstein
constant $\lambda$, and $\lambda_{F}$ is the Einstein constant of the fibres.
First, we show that $\lambda=\lambda_{F}$. The following equation can be
derived by using Equation 2.12.

(3.1) $\displaystyle\lambda\,dim(F)=$
$\displaystyle\lambda_{F}\,dim(F)-|N|^{2}+(2dim(F)-dim(B))N(f)$
$\displaystyle+dim(F)(dim(B)-dim(F))|\mathcal{H}(\nabla
f)|^{2}-dim(F)\mathcal{H}(\Delta f)$
$\displaystyle+dim(B)(dim(F)-1)|\mathcal{V}(\nabla
f)|^{2}-dim(B)\mathcal{V}(\Delta f)$

Consequently, we observe that $\lambda\leq\lambda_{F}$ as $Hess(f)$ is non-
negative definite and $\nabla f$ vanishes at the point where $f$ attains its
global minimum. The equation below can then be derived from Equation 3.1 by
using the fact that $N$ is parallel.

(3.2) $\displaystyle\lambda\,dim(F)=$
$\displaystyle\lambda_{F}\,dim(F)-dim(F)div(\mathcal{H}(\nabla f))$
$\displaystyle-dim(B)div(\mathcal{V}(\nabla
f))+dim(B)(dim(B)-1)|\mathcal{V}(\nabla f)|^{2}$

Further, using the divergence theorem in Equation 3.2, it can be seen that
$\lambda\geq\lambda_{F}$. Thus, $\lambda=\lambda_{F}$. Therefore,
$\mathcal{V}(\nabla f)$ vanishes by employing the divergence theorem in
Equation 3.2. By substituting $\mathcal{V}(\nabla f)=0$ in Equation 3.2, and
applying the weak maximal principle, the function $f$ turns out to be
constant. This completes the proof. ∎

######  Remark 3.2.

In particular, Theorem 3.1 implies that in a conformal submersion with
integrable horizontal distribution and parallel $N$, if the total space and
fibres are both Einstein, then their Einstein constants must coincide.

The following rigidity result assumes the base of the given conformal
submersion to be Ricci-flat.

######  Theorem 3.3.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
integrable horizontal distribution and parallel $N$, where $(B,g_{B})$ is
Ricci-flat, then $\pi_{f}$ is rigid if and only if

$\sum_{i}\lambda_{i}m_{i}=-|T|^{2}-\frac{|N|^{2}}{dim(F)}(dim(M)+dim(B)-4),$

where $\lambda_{i}$ and $m_{i}$ are the Einstein constants and dimensions of
the horizontal distribution of the $i^{th}$ component of $(M,g)$ or its
universal cover, respectively.

###### Proof.

Assuming that $(M,g)$ is Einstein with Einstein constant $\lambda$, it follows
from Equation 2.14 that

(3.3) $\displaystyle\lambda\,dim(B)=$
$\displaystyle-|T|^{2}-\frac{|N|^{2}}{dim(F)}(dim(M)+dim(B)-4)-(dim(M)-2)$
$\displaystyle div(\mathcal{H}(\nabla
f))+(dim(M)-2)(dim(M)-1)|\mathcal{H}(\nabla f)|^{2}-dim(B)\Delta f.$

If $\lambda\,dim(B)=-|T|^{2}-\frac{|N|^{2}}{dim(F)}(dim(M)+dim(B)-4)$, then by
the divergence theorem and the weak maximal principle, we conclude that
$\pi_{f}$ is rigid. ∎

######  Remark 3.4.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
integrable horizontal distribution and parallel $N$. If both $(M,g)$ and
$(B,g_{B})$ are Einstein with Einstein constants $\lambda$ and $\lambda_{B}$,
respectively, then by Equation 2.14, $\lambda\leq c\lambda_{B}$ where $c>0$.
In particular, if $(B,g_{B})$ is Ricci-flat, then $\lambda$ is non-positive.
Additionally, if $(M,g)$ is a homogeneous Einstein manifold, then $(M,g)$ is
flat (See Theorem $7.4$ of [2]).

Since the Einstein constant of a compact homogeneous Einstein manifold, is
always non-negative, we have the following corollary.

######  Corollary 3.5.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
integrable horizontal distribution and parallel $N$, where either $(M,g)$ or
its universal cover is the Riemannian product of homogeneous Einstein
manifolds and $(B,g_{B})$ is Ricci flat, then $\pi_{f}$ is rigid if and only
if $T$ vanishes. In particular, this result holds true when the universal
cover of $(M,g)$ is a symmetric space.

###  3.2. When $N$ vanishes identically.

The following result is about the rigidity of conformal submersions with
Einstein fibres.

######  Theorem 3.6.

A conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ ($dim(B)\geq
2$) with Einstein fibres (with Einstein constant $\lambda_{F}$) and vanishing
$N$, is rigid if and only if
$\sum_{i}\lambda_{i}m_{i}=\lambda_{F}\,dim(F)+|A|^{2}$, where $\lambda_{i}$
and $m_{i}$ are the Einstein constants and dimensions of the vertical
distribution of the $i^{th}$ component of $(M,g)$ or its universal cover,
respectively.

###### Proof.

By using Equation 2.12,

(3.4) $\displaystyle(\lambda-\lambda_{F})dim(F)=$
$\displaystyle|A|^{2}-dim(F)div(\mathcal{H}(\nabla f))-dim(B)$ $\displaystyle
div(\mathcal{V}(\nabla f))+dim(B)(dim(B)-1)|\mathcal{V}(\nabla f)|^{2}$

We conclude the result by applying the divergence theorem and the weak maximal
principle in Equation 3.4. ∎

######  Remark 3.7.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
vanishing $N$. If both the total space and the fibres are Einstein, then the
Einstein constant of the total space is greater than the Einstein constant of
the fibres (follows from Equation 3.4).

Employing the divergence theorem together with Equations 3.4 and 2.12, and
Theorem 3.6, we obtain the following corollary.

######  Corollary 3.8.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
vanishing $N$, where $(M,g)$ and the fibres are Einstein, then it is
equivalent to have either $\lambda=\lambda_{F}$ or $A=0$, where $\lambda$ and
$\lambda_{F}$ are the Einstein constants of $(M,g)$ and the fibres,
respectively. Moreover, either of these conditions implies rigidity of
$\pi_{f}$ if $dim(B)\geq 2$.

Combining Remark 3.7 and Corollary 3.8, we conclude the following.

######  Corollary 3.9.

A conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ with vanishing
$N$, where $dim(B)\geq 2$, $(M,g)$ is Ricci-flat, and the fibres are
homogeneous Einstein manifolds, is rigid.

Equation 3.4 together with Bochner’s result (cf. Theorem $6.56$ of [2])
involving the first cohomology group $H^{1}(M;\mathbb{R})$ of a Riemannian
manifold, leads to the following interesting observation.

######  Corollary 3.10.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
vanishing $N$ where $(M,g)$ and the fibres are both Einstein. If fibres have a
non-negative Einstein constant, and the horizontal distribution is non-
integrable, then $H^{1}(M;\mathbb{R})=\\{0\\}$.

The following corollary is about the special case when the conformal
submersion has vanishing $T$ and Ricci flat fibres.

######  Corollary 3.11.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
vanishing $T$, where $(M,g)$ is Einstein and the fibres are Ricci flat. If
$(B,g_{B})$ is scalar flat, then the total space and the base are also Ricci-
flat. In that case, $\pi_{f}$ is rigid whenever $dim(B)\geq 2$.

###### Proof.

$(M,g)$ is Ricci-flat by Equation 2.2.1 and Remark 3.7. Now, the result
follows from Corollary 3.8 and Equation 2.14. ∎

The following theorem addresses the rigidity conditions when the base is
Ricci-flat.

######  Theorem 3.12.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
vanishing $N$, where $(B,g_{B})$ is Ricci-flat. Then $\pi_{f}$ is rigid if and
only if $\sum_{i}\lambda_{i}m_{i}+2|A|^{2}+|T|^{2}=0$, where $\lambda_{i}$ and
$m_{i}$ are the Einstein constants and dimensions of the horizontal
distribution of the $i^{th}$ component of $(M,g)$ or its universal cover,
respectively.

###### Proof.

Using Equation 2.14, we have

(3.5) $\displaystyle\lambda\,dim(B)=$
$\displaystyle-2|A|^{2}-|T|^{2}-(dim(M)-2)div(\mathcal{H}(\nabla f))$
$\displaystyle+(dim(M)-2)(dim(M)-1)|\mathcal{H}(\nabla f)|^{2}-dim(B)\Delta f$

Consequently, if $\lambda\,dim(B)+2|A|^{2}+|T|^{2}=0$, then $\pi_{f}$ is rigid
by the divergence theorem and the weak maximal principle. ∎

######  Remark 3.13.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
vanishing $N$. If both $(M,g)$ and $(B,g_{B})$ are Einstein with Einstein
constants $\lambda$ and $\lambda_{B}$, respectively, then $\lambda\leq
c\lambda_{B}$ for some $c>0$ (by Equation 2.14). In particular, if $(B,g_{B})$
is Ricci-flat, then $\lambda$ is non-positive. Additionally, if $(M,g)$ is
homogeneous, then it is flat.

The following rigidity result considers conformal submersion with a Ricci-flat
base.

######  Corollary 3.14.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
vanishing $N$, such that $(B,g_{B})$ is Ricci-flat and either $(M,g)$ or its
universal cover is the Riemannian product of compact homogeneous Einstein
manifolds. Then $\pi_{f}$ is rigid if and only if its horizontal distribution
is integrable and $T$ vanishes. In particular, this result is true when the
universal cover of $(M,g)$ is a symmetric space.

As a consequence of Theorem 3.12, together with Equations 2.12 and 2.14, we
obtain the following result.

######  Corollary 3.15.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
integrable horizontal distribution and vanishing $T$, where $(B,g_{B})$ is
Ricci-flat, then $\pi_{f}$ is rigid. In particular, if $(M,g)$ is Einstein,
then $(M,g)$ and the fibres are both Ricci-flat.

######  Remark 3.16.

It is well known that a Kähler manifold $(M,g)$ of dimension at least $4$ (see
[2] for the definition of Kähler manifolds) with harmonic Weyl tensor has
parallel Ricci tensor (Proposition $16.30$ of [2]). According to the de Rham
decomposition theorem (Theorem $56$ of [16]), the universal cover of $(M,g)$
splits into the Riemannian product of Einstein manifolds. Additionally, by
Corollary 3.15, a conformal submersion with integrable horizontal
distribution, vanishing $T$, Ricci-flat base, and total space $(M,g)$ is rigid
if $(M,g)$ has a closed universal cover.

######  Remark 3.17.

The rigidity of warped product Einstein metrics is a well-explored problem
([10, 7]). By our rigidity results (Theorem 3.1 and Corollary 3.15), we
retrieve the following result.
Let $(M,g)$ be Einstein and $f\in C^{\infty}(M)$. If $M\times_{e^{2f}}N$ is
Einstein, then $f$ is constant, and $N$ is Einstein.

Using Theorem 3.1 and Corollary 3.15, we conclude the following.

######  Theorem 3.18.

If $(M,g)$ is an Einstein manifold of dimension at least $4$, then up to
homothety, $g$ is the unique Einstein metric in its conformal class if $(M,g)$
is the total space of a conformal submersion with integrable horizontal
distribution and vanishing $T$, together with either one-dimensional fibres or
a Ricci-flat base.

######  Remark 3.19.

In particular, Theorem 3.18 implies that if $(M,g)$ is Ricci-flat, which is
also locally a Riemannian product, then up to homothety, $g$ is the only
Einstein metric in its conformal class.

###  3.3. When the fibres are totally geodesic.

The following theorem discusses rigidity conditions of a conformal submersion
with a Ricci-flat base in this case.

######  Theorem 3.20.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
totally geodesic fibres and $dim(B)\geq 2$, where $(B,g_{B})$ is Ricci-flat.
Then $\pi_{f}$ is rigid if and only if $\sum_{i}\lambda_{i}m_{i}+2|A|^{2}=0$,
where $\lambda_{i}$ and $m_{i}$ are the Einstein constants and dimensions of
the horizontal distribution of the $i^{th}$ component of $(M,g)$ or its
universal cover, respectively.

###### Proof.

WE use Equation 2.14 to see that

(3.6) $\displaystyle\lambda\,dim(B)=$
$\displaystyle-2|A|^{2}-(dim(B)-2)div(\mathcal{H}(\nabla f))-dim(B)\Delta f$
$\displaystyle+(dim(B)-2)(dim(B)-1)|\mathcal{H}(\nabla f)|^{2}$

Let $\lambda\,dim(B)+2|A|^{2}$ = 0. If $dim(B)=2$, then by the weak maximal
principle, $f$ is constant.
When $dim(B)>2$, using the divergence theorem in Equation 3.6, it follows that
$\mathcal{H}(\nabla f)=0$. Combining this with Equation 3.6 and applying the
weak maximal principle, $f$ becomes constant. ∎

######  Remark 3.21.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
totally geodesic fibres. If both $(M,g)$ and $(B,g_{B})$ are Einstein with
respective Einstein constants $\lambda$ and $\lambda_{B}$, then $\lambda\leq
c\lambda_{B}$ where $c>0$ (by Equation 2.14). In particular, if $(B,g_{B})$ is
Ricci-flat, then $\lambda$ is non-positive. Additionally, if $(M,g)$ is
homogeneous, then it is flat.

Using Equation 2.14 and Theorem 3.20, we conclude the following result.

######  Corollary 3.22.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
integrable horizontal distribution and totally geodesic fibres, where
$(B,g_{B})$ is Ricci-flat, then $\pi_{f}$ is rigid if $dim(B)\geq 2$. In
particular, if $(M,g)$ is Einstein, then $(M,g)$ is Ricci-flat.

######  Remark 3.23.

It should be noted that unlike Riemannian submersions, having totally geodesic
fibres and integrable horizontal distribution, does not necessarily mean that
the total space of a conformal submersion is locally a Riemannian product.

######  Remark 3.24.

If $(M,g)$ is a Kähler manifold of dimension at least $4$ with harmonic Weyl
tensor and a closed universal cover, then by Corollary 3.22, a conformal
submersion with integrable horizontal distribution, totally geodesic fibres,
Ricci-flat base, and $(M,g)$ as the total space, is rigid.

######  Remark 3.25.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
totally geodesic Einstein fibres. If in addition, $(M,g)$ is Einstein, then
its Einstein constant is greater than the Einstein constant of fibres (follows
from Equation 2.12).

Bochner’s result (cf. Theorem $6.56$ of [2]) and Equation 2.12 together lead
to the following remark.

######  Remark 3.26.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion with
totally geodesic Einstein fibres. If $(M,g)$ is Einstein, the Einstein
constant of fibres is non-negative and horizontal distribution is non-
integrable, then $H^{1}(M;\mathbb{R})=\\{0\\}$.

##  4\. Conformal submersions with locally conformally flat total space

In this section, we consider conformal submersions whose total spaces are
Riemannian products of locally conformally flat and Einstein manifolds (where
the Einstein components are possibly trivial), and study their rigidity. We
further assume that the base and the fibres have the same dimension. In such a
case, we provide suitable curvature conditions imposing rigidity.

###  4.1. The total space is the product of locally conformally flat and
Einstein manifolds.

In conformal submersions whose total spaces are Riemannian products of locally
conformally flat and Einstein manifolds (where the Einstein components are
possibly trivial), we make the following observations about the scalar
curvature of the components of the total space.

######  Proposition 4.1.

  1. (1)

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion such
that $(M,g)$ is locally conformally flat and $dim(B)=dim(F)$ where $F_{p}$
denotes the fibre at $p\in B$. Then the scalar curvature of $(M,g)$ satisfies
the following equation.

(4.1)
$\footnotesize\frac{ns}{2(n-1)}=2(|A|^{2}-|T|^{2}-\hat{\delta}N)-(n-4)N(f)-n\Delta
f+\frac{n(n-2)}{2}|\nabla f|^{2},$

where $n=dim(M)$.

  2. (2)

More generally, consider a conformal submersion
$\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ where $(M,g)$ is the Riemannian
product of locally conformally flat manifolds each of dimension at least $4$
such that the horizontal and the vertical distributions restricted to each
component of $(M,g)$ have the same dimension. Then,

(4.2)
$\footnotesize\sum_{i}\frac{n_{i}s_{i}}{2(n_{i}-1)}=2(|A|^{2}-|T|^{2}-\hat{\delta}N)-(n-4)N(f)-n\Delta
f+\frac{n(n-2)}{2}|\nabla f|^{2},$

where $n_{i}$ and $s_{i}$ respectively denote the dimension and the scalar
curvature of the $i^{th}$ component of $(M,g)$, and $n=dim(M)$.

  3. (3)

Let $\pi_{f}\colon(M\times N,g_{M}+g_{N})\rightarrow(B,g_{B})$ be a conformal
submersion where $(M,g_{M})$ is as in $(2)$. Let $(N,g_{N})$ be the Riemannian
product of Einstein manifolds where the tangent bundle of each component of
$(N,g_{N})$ is completely contained either in the vertical or the horizontal
distribution of $\pi_{f}$, and $dim(F)=dim(B)$ where $F_{p}$ denotes the fibre
at $p\in B$. Then,

(4.3) $\displaystyle\sum_{i}\frac{n_{i}s_{i}}{2(n_{i}-1)}=$ $\displaystyle
2(|A|^{2}-|T|^{2}-\hat{\delta}N)-(n-4)N(f)-(\frac{n+l}{2})(\Delta
f-\frac{n-2}{2}|\nabla f|^{2})$
$\displaystyle-\frac{e}{2}(div(\mathcal{H}\nabla
f)-\frac{n-2}{2}|\mathcal{H}\nabla f|^{2}),$

where $n_{i}$ and $s_{i}$ are the dimensions and the scalar curvatures of the
$i^{th}$ component of $(M,g)$, respectively and $l=dim(M)$, $e=dim(N)$ and
$l+e=n$.

###### Proof.

(1) Let $p\in M$. Let $\\{X_{i}\\}{i\in I}$, $\\{U_{j}\\}_{j\in J}$
respectively denote local orthonormal frames of the horizontal and vertical
distributions at $p$. Since the Weyl curvature $W=0$, using the orthogonal
decomposition of the Riemann curvature tensor given in Equation 2.1, it
follows that

(4.4) $\sum_{j\neq
k}R(U_{k},U_{j},U_{k},U_{j})=\frac{m-2}{n-2}r(U_{k},U_{k})+\sum_{j}\frac{r(U_{j},U_{j})}{n-2}-\frac{s(m-1)}{(n-1)(n-2)},$

where $m=n/2$.
Equations 2.7, 2.12 and 4.4 gives,

(4.5) $\displaystyle\sum_{j}\frac{r(U_{j},U_{j})}{n-2}=$
$\displaystyle\frac{m}{n-2}r(U_{k},U_{k})+\frac{s(m-1)}{(n-1)(n-2)}+g(T_{U_{k}},T_{U_{k}})$
$\displaystyle+(m-2)T_{U_{k}}U_{k}(f)+|\mathcal{H}(\nabla
f)|^{2}+\mathcal{H}(\Delta f)-\delta T(U_{k},U_{k})$
$\displaystyle-g(AU_{k},AU_{k})+m(Hessf(U_{k},U_{k})+U_{k}(f)^{2}-|\nabla
f|^{2}).$

Similarly, Equations 2.1, 2.8 and 2.14 imply

(4.6) $\displaystyle\sum_{i}\frac{r(X_{i},X_{i})}{n-2}=$
$\displaystyle\frac{m}{n-2}r(X_{l},X_{l})+\frac{s(m-1)}{(n-1)(n-2)}+g(TX_{l},TX_{l})-g(A_{X_{l}},A_{X_{l}})$
$\displaystyle-|\mathcal{H}(\nabla f)|^{2}-\mathcal{H}(\Delta
f)+m(Hessf(X_{l},X_{l})+X_{l}(f)^{2})$ $\displaystyle-(m-1)|\nabla
f|^{2}-g(\nabla_{X_{l}}N,X_{l})-2X_{l}(f)g(N,X_{l})+N(f)+\Delta f.$

Adding 4.5 and 4.6 gives,

(4.7) $\displaystyle\frac{s}{(n-1)(n-2)}=$
$\displaystyle\frac{mr(U_{k},U_{k})}{n-2}+\frac{mr(X_{l},X_{l})}{n-2}+g(TX_{l},TX_{l})+g(T_{U_{k}},T_{U_{k}})$
$\displaystyle+(m-2)T_{U_{k}}U_{k}(f)-g(A_{X_{l}},A_{X_{l}})-g(AU_{k},AU_{k})+m(Hessf(X_{l},X_{l})$
$\displaystyle+X_{l}(f)^{2})+m(Hessf(U_{k},U_{k})+U_{k}(f)^{2})-(n-1)|\nabla
f|^{2}$ $\displaystyle-\delta
T(U_{k},U_{k})-g(\nabla_{X_{l}}N,X_{l})-2X_{l}(f)g(N,X_{l})+N(f)+\Delta f.$

Taking summation over $\\{U_{k}\\}$’s and $\\{X_{l}\\}$’s,

(4.8) $\displaystyle\frac{ns}{2(n-1)}=$ $\displaystyle
2(|A|^{2}-|T|^{2}-\hat{\delta}N)-(n-4)N(f)-n\Delta f+\frac{n(n-2)}{2}|\nabla
f|^{2}.$

Proof of $(2)$ and $(3)$ are similar to $(1)$. ∎

Employing the divergence theorem and the weak maximal principle in Proposition
4.1 culminates in to the following result.

######  Theorem 4.2.

A conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ as described
in Proposition 4.1(2) is rigid if either of the following conditions holds.

  1. (1)

$T$ vanishes and $s\leq\sum_{i}\frac{s_{i}}{1-n_{i}}$ where $s_{i}$ and
$n_{i}$ are as mentioned in Proposition 4.1(2).

  2. (2)

$N$ is parallel, the horizontal distribution is integrable and
$s\geq\sum_{i}\frac{s_{i}}{1-n_{i}}$.

Additionally, if $(M,g)$ is locally conformally flat, then $(M,g)$ is scalar
flat and locally a Riemannian product.

We further observe the following scalar curvature properties for the total
space of a conformal submersion given in Theorem 4.2.

######  Proposition 4.3.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion such
that $(M,g)$ is as defined in Theorem 4.2. Moreover, if the scalar curvature
of each component of $(M,g)$ have the same sign, then:

  1. (1)

The scalar curvature of $(M,g)$ is non-negative if $T$ vanishes.

  2. (2)

The scalar curvature of $(M,g)$ is non-positive if $N$ is parallel and the
horizontal distribution is integrable.

  3. (3)

$(M,g)$ is scalar flat, and $\pi_{f}$ is rigid if the horizontal distribution
is integrable and $T$ vanishes.

###### Proof.

The following is a proof for locally conformally flat total space. The general
case follows analogously.
$(1)$ If $T$ vanishes,

(4.9) $\displaystyle\frac{ns}{2(n-1)}=$ $\displaystyle 2|A|^{2}-n\Delta
f+\frac{n(n-2)}{2}|\nabla f|^{2}.$

The scalar curvature of $(M,g)$ is non-negative by the divergence theorem.
$(2)$ If $N$ is parallel and the horizontal distribution is integrable,

(4.10) $\displaystyle\frac{ns}{2(n-1)}=$
$\displaystyle-2|T|^{2}-\frac{2(n-4)}{n}|N|^{2}-n\Delta
f+\frac{n(n-2)}{2}|\nabla f|^{2}.$

Multiply Equation 4.10 by $e^{-\frac{n-2}{2}f}$,

(4.11) $\displaystyle\frac{ns}{2(n-1)}e^{-\frac{n-2}{2}f}=$
$\displaystyle-2e^{-\frac{n-2}{2}f}|T|^{2}-\frac{2(n-4)}{n}e^{-\frac{n-2}{2}f}|N|^{2}-ndiv(e^{-\frac{n-2}{2}f}\nabla
f).$

The scalar curvature of $(M,g)$ is non-positive by the divergence theorem.
$(3)$ If the horizontal distribution is integrable and $T=0$,

(4.12) $\displaystyle\frac{ns}{2(n-1)}e^{-\frac{n-2}{2}f}=$ $\displaystyle-
ndiv(e^{-\frac{n-2}{2}f}\nabla f).$

$(M,g)$ is scalar flat by the divergence theorem, and $\pi_{f}$ is rigid by
the weak maximal principle. ∎

More generally, if the total space of a conformal submersion is as in
Proposition 4.1(3), we obtain the following rigidity result similar to Theorem
4.2.

######  Theorem 4.4.

A conformal submersion $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ as described
in Proposition 4.1(3) is rigid if either of the following conditions holds.

  1. (1)

$T$ vanishes and $s_{M}\leq\sum_{i}\frac{s_{i}}{1-n_{i}}$.

  2. (2)

$N$ is parallel, the horizontal distribution is integrable and
$s_{M}\geq\sum_{i}\frac{s_{i}}{1-n_{i}}$.

where $s_{i}$ is the scalar curvature of the $i^{th}$ component of $(M,g_{M})$
and $s_{M}$ is the scalar curvature of $(M,g_{M})$.

In the next two subsections, we prove more rigidity results by putting further
conditions on the total space or the base of a conformal submersion as
described in this part.

###  4.2. When the total space is quasi-Einstein

######  Theorem 4.5.

Let $\pi_{f}:(M,g)\rightarrow(B,g_{B})$ be a conformal submersion where
$(M,g)$ is as described in Theorem 4.2 with the scalar curvatures of each
component of $(M,g)$ having the same sign. If $(M,g,h,m,\lambda)$ is also
quasi-Einstein, then it is Ricci-flat and $\pi_{f}$ is rigid if one of the
following conditions holds.

  1. (1)

$\lambda$ is non-negative, $N$ is parallel, and the horizontal distribution is
integrable.

  2. (2)

$\lambda$ is non-positive, and $T=0$.

  3. (3)

The horizontal distribution is integrable, and $T=0$.

Furthermore, if $(M,g)$ is locally conformally flat, then $(M,g)$ is locally a
Riemannian product.

###### Proof.

The following is proof for locally conformally flat total space. The general
case follows analogously.
We have,

(4.13) $Ric+Hessh-\frac{1}{m}dh\otimes dh=\lambda g.$

By taking trace,

(4.14) $\lambda n-s=\Delta h-\frac{1}{m}|\nabla h|^{2}$ (4.15)
$e^{-h/m}(\lambda n-s)=div(e^{-h/m}\nabla h).$

Using the divergence theorem and Proposition 4.3, $\lambda=s=0$. Applying the
weak maximal principle in Equation 4.14, $h$ turns out to be constant, and
hence $(M,g)$ is Ricci-flat. By Theorem 4.2, we can conclude that $(M,g)$ is
locally a Riemannian product, and $\pi_{f}$ is rigid. ∎

###  4.3. When either the base or the fibre is quasi-Einstein.

The following lemma can be derived by using the same method employed to prove
Proposition 4.1 and hence the proof is skipped..

######  Lemma 4.6.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion such
that $(M,g)$ is locally conformally flat and $dim(B)=dim(F)$ where $F_{p}$
denotes the fibre at $p\in B$. Then,

(4.16) $\displaystyle e^{-2f}\hat{s}\circ\pi_{f}+\check{s}=$
$\displaystyle\frac{5n-4}{n}|A|^{2}-\frac{3n-4}{n}(|T|^{2}+\check{\delta}N)-div(N)$
$\displaystyle-\frac{n^{2}-12n+16}{2n}N(f)-(n-2)div(\mathcal{V}(\nabla f))$
$\displaystyle+\frac{(n-2)(3n-4)}{4}|\mathcal{V}(\nabla f)|^{2}.$

We further observe some properties of the scalar curvatures of the base and
fibres of a conformal submersion as given in Lemma 4.6.

######  Proposition 4.7.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion as in
Lemma 4.6. If the scalar curvatures of the fibres (respectively, the base)
have a particular sign and

  1. (1)

$T=0$, and the scalar curvature of the base (respectively, the fibres) is non-
positive, then the scalar curvatures of the fibres (respectively, the base)
are non-negative.

  2. (2)

$A=0$, parallel $T$, and the scalar curvature of the base (respectively, the
fibres) is non-negative, then the scalar curvatures of the
fibres(respectively, the base) are non-positive.

###### Proof.

$(1)$. By substituting $T=0$ in Equation 4.16,

(4.17) $\displaystyle e^{-2f}\hat{s}\circ\pi_{f}+\check{s}=$
$\displaystyle\frac{5n-4}{n}|A|^{2}-(n-2)div(\mathcal{V}(\nabla
f))+\frac{(n-2)(3n-4)}{4}|\mathcal{V}(\nabla f)|^{2}.$

Applying the divergence theorem in Equation 4.17 gives the result.
$(2)$. By substituting $A=0$ and $\nabla T=0$ ($\nabla T=0$ implies
$|T|^{2}=N(f)$) and multiplying $e^{-\frac{3n-4}{4}f}$ in Equation 4.16,

(4.18) $\displaystyle
e^{-\frac{3n-4}{4}f}\hat{s}\circ\pi_{f}+e^{-\frac{3n-4}{4}f}\check{s}=$
$\displaystyle-\frac{(n-2)(n-4)}{2n}|T|^{2}e^{-\frac{3n-4}{4}f}-(n-2)(div(e^{-\frac{3n-4}{4}f}\mathcal{V}(\nabla
f)).$

The result follows by applying the divergence theorem in Equation 4.18. ∎

Employing the divergence theorem and the weak maximal principle in Lemma 4.6,
Proposition 4.1 and then using Proposition 4.7 we conclude the following
result.

######  Theorem 4.8.

Let $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ be a conformal submersion as in
Lemma 4.6. If the scalar curvatures of the fibres (respectively, the base)
have a particular sign and $(F,g_{F},h,m,\lambda)$ (respectively,
$(B,g_{B},h,m,\lambda)$) is quasi-Einstein, then $(F,g_{F},h)$ (respectively,
$(B,g_{B},h)$) is Ricci-flat, $(B,g_{B})$ (respectively, $(F,g_{F})$) is
scalar flat, and also $f$ is constant along fibres whenever either of the
following conditions holds.

  1. (1)

$T=0$, the scalar curvature of the base (respectively, the fibres) is non-
positive, and $\lambda\leq 0$.

  2. (2)

$A=0$, parallel $T$, the scalar curvature of the base (respectively, the
fibres) is non-negative, and $\lambda\geq 0$.

In both cases, $A$ and $T$ vanish. Additionally, if the scalar curvature of
$(M,g)$ has a particular sign, then $\pi_{f}$ is rigid.

######  Remark 4.9.

Assuming the scalar curvature of the total space has a particular sign is not
a strong condition since any closed Riemannian manifold admits a metric in its
conformal class that has constant scalar curvature due to Yamabe theorem (see
[2]).

The following example shows that $dim(F)=dim(B)$ is necessary in Theorem 4.8.

######  Example 4.10.

Let $M=S^{1}\times S^{3}$ with metric $g=g_{S^{1}}+e^{2f}g_{S^{3}}$ where $f$
is a non-constant function on $S^{1}$. Consider the map
$\pi_{f}\colon(M,g)\rightarrow(S^{3},g_{3})$ defined as $\pi_{f}(x,y)=y$. One
can verify by direct computation that $(M,g)$ is locally conformally flat, and
all the conditions of Theorem 4.8 are satisfied by $\pi_{f}$ (note that here,
the scalar curvature of $(M,g)$ is constant only up to conformality) except
the dimension condition, but $\pi_{f}$ is not rigid.

Next, we discuss the rigidity of conformal submersions with some conditions on
mixed scalar curvature (see [24] for a details on mixed scalar curvatures).

######  Lemma 4.11.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
vanishing $T$, then $\pi_{f}$ is rigid if and only if
$R_{mix}=\sum_{i,j}K(X_{i},U_{j})=|A|^{2}$ where $\\{X_{i}\\}$ and
$\\{U_{j}\\}$ form an orthonormal basis of horizontal and vertical
distributions, respectively.

###### Proof.

By Equation 2.2.1,

(4.19) $\displaystyle R_{mix}=$
$\displaystyle|A|^{2}-dim(F)div(\mathcal{H}\nabla
f)-dim(B)div(\mathcal{V}\nabla f)$
$\displaystyle+dim(F)(dim(F)-1)|\mathcal{H}(\nabla
f)|^{2}+dim(B)(dim(B)-1)|\mathcal{V}(\nabla f)|^{2}.$

The proof follows by employing the divergence theorem and the weak maximal
principle in Equation 4.19. ∎

The following result is an immediate consequence of Lemma 4.11.

######  Theorem 4.12.

If $\pi_{f}\colon(M,g)\rightarrow(B,g_{B})$ is a conformal submersion with
vanishing $T$ and $R_{mix}\leq 0$, then $R_{mix}=0$, $(M,g)$ is locally a
Riemannian product, and $\pi_{f}$ is rigid.

##  5\. An Application: Rigidity of quasi-Einstein manifolds

In this section, we conclude rigidity results for certain quasi-Einstein
manifolds that can be realized as the total space, the fibre or the base of
suitable conformal submersions. All the quasi-Einstein manifolds mentioned
here are closed and orientable. The first result is about a quasi-Einstein
manifold realized as the fibre or the base of a conformal submersion.

######  Theorem 5.1.

Let $(M,g,f,m,\lambda)$ be a quasi-Einstein manifold with $\lambda\geq 0$ and
$m\geq 1$. Then $(M,g,f)$ is rigid if $(M,g)$ is the fibre (respectively, the
base) of a conformal submersion with an integrable horizontal distribution and
parallel $T$ satisfying the following conditions.

  1. (1)

The total space of the conformal submersion is locally conformally flat and
its dimension is twice the dimension of $M$.

  2. (2)

The base (respectively, the fibre) has non-negative scalar curvature.

###### Proof.

Proposition $3.6$ of [4] says that‘If $(M,g,f,m,\lambda)$ is quasi-Einstein
with $m\geq 1$, $\lambda>0$ and $M$ is compact, then the scalar curvature of
$(M,g)$ is bounded below by $\frac{n(n-1)\lambda}{m+n-1}$’. It readily follows
from the above result that the scalar curvature of $(M,g)$ (as in hypothesis)
is non-negative if $\lambda>0$. Using the same idea of Proposition $3.6$ of
[4], it can further be checked that the scalar curvature of $(M,g)$ is non-
negative whenever $\lambda=0$. Using Theorem 4.8, we conclude the result. ∎

The next result is about a locally conformally flat quasi-Einstein manifold
realized as the total space of a conformal submersion.

######  Theorem 5.2.

If $(M,g,f,m,\lambda)$ is a locally conformally flat quasi-Einstein manifold
of dimension at least 4 with $m\geq 1$ and $\lambda\geq 0$, then $(M,g,f)$ is
a flat space form if it is the total space of a conformal submersion with an
integrable horizontal distribution and parallel $N$ whose base and fibre have
the same dimension.

###### Proof.

Again, by using Proposition $3.6$ of [4], the scalar curvature of $(M,g)$ is
non-negative. Finally, Theorem 4.5 concludes the result. ∎

######  Example 5.3.

Let $(M_{1}\times M_{2},g)$ be a locally conformally flat manifold with
$dim(M_{1})=dim(M_{2})\geq 2$, and $g$ be the product metric. Then
$(M_{1}\times M_{2},g)$ is a flat space form if there exist functions $f,h\in
C^{\infty}(M_{1}\times M_{2})$ such that $(M_{1}\times
M_{2},e^{2f}g,h,m,\lambda)$ is quasi-Einstein with $m\geq 1$ and $\lambda\geq
0$.

Finally, we obtain the following rigidity result as a consequence of Theorem
3.12, Remark 3.13 and Equation 2.12.

######  Theorem 5.4.

A quasi-Einstein manifold $(M,g,f,m,\lambda)$ is rigid if it can be realized
as the fibre of a conformal submersion with an integrable horizontal
distribution and vanishing $T$, where

  1. (1)

the total space is a homogeneous Einstein manifold and

  2. (2)

the base is Einstein with a non-positive Einstein constant.

## References

  * [1] Sannidhi Alape, Atreyee Bhattacharya, and Dheeraj Kulkarni, _On certain rigidity results of compact regular $(\kappa,\mu)$-manifolds_, arXiv:2308.01576 (2023).
  * [2] Arthur L. Besse, _Einstein manifolds_ , Classics in Mathematics, Springer-Verlag, Berlin, 2008, Reprint of the 1987 edition. MR 2371700
  * [3] Christoph Böhm, _Unstable Einstein metrics_ , Math. Z. 250 (2005), no. 2, 279–286. MR 2178786
  * [4] Jeffrey Case, Yu-Jen Shu, and Guofang Wei, _Rigidity of quasi-Einstein metrics_ , Differential Geom. Appl. 29 (2011), no. 1, 93–100. MR 2784291
  * [5] Giovanni Catino, Carlo Mantegazza, Lorenzo Mazzieri, and Michele Rimoldi, _Locally conformally flat quasi-Einstein manifolds_ , J. Reine Angew. Math. 675 (2013), 181–189. MR 3021450
  * [6] Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, _Riemannian geometry_ , third ed., Universitext, Springer-Verlag, Berlin, 2004. MR 2088027
  * [7] Chenxu He, Peter Petersen, and William Wylie, _Warped product Einstein metrics over spaces with constant scalar curvature_ , Asian J. Math. 18 (2014), no. 1, 159–189. MR 3215345
  * [8] Guangyue Huang and Yong Wei, _The classification of $(m,\rho)$-quasi-Einstein manifolds_, Ann. Global Anal. Geom. 44 (2013), no. 3, 269–282. MR 3101859
  * [9] Thomas Ivey, _Ricci solitons on compact three-manifolds_ , Differential Geom. Appl. 3 (1993), no. 4, 301–307. MR 1249376
  * [10] Dong-Soo Kim and Young Ho Kim, _Compact Einstein warped product spaces with nonpositive scalar curvature_ , Proc. Amer. Math. Soc. 131 (2003), no. 8, 2573–2576. MR 1974657
  * [11] Norihito Koiso and Yusuke Sakane, _Nonhomogeneous Kähler-Einstein metrics on compact complex manifolds. II_ , Osaka J. Math. 25 (1988), no. 4, 933–959. MR 983813
  * [12] John M. Lee, _Introduction to Riemannian manifolds_ , second ed., Graduate Texts in Mathematics, vol. 176, Springer, Cham, 2018. MR 3887684
  * [13] TV Mahesh and KS Subrahamanian Moosath, _Harmonicity of conformally-projectively equivalent statistical manifolds and conformal statistical submersions_ , Geometric Science of Information: 5th International Conference, GSI 2021, Paris, France, July 21–23, 2021, Proceedings 5, Springer, 2021, pp. 397–404.
  * [14] MT Mustafa, _Applications of harmonic morphisms to gravity_ , Journal of Mathematical Physics 41 (2000), no. 10, 6918–6929.
  * [15] Aaron Naber, _Noncompact shrinking four solitons with nonnegative curvature_ , J. Reine Angew. Math. 645 (2010), 125–153. MR 2673425
  * [16] Peter Petersen, _Riemannian geometry_ , second ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
  * [17] Peter Petersen and William Wylie, _On gradient Ricci solitons with symmetry_ , Proc. Amer. Math. Soc. 137 (2009), no. 6, 2085–2092. MR 2480290
  * [18] by same author, _Rigidity of gradient Ricci solitons_ , Pacific J. Math. 241 (2009), no. 2, 329–345. MR 2507581
  * [19] by same author, _On the classification of gradient Ricci solitons_ , Geom. Topol. 14 (2010), no. 4, 2277–2300. MR 2740647
  * [20] Vladimir Rovenski and Tomasz Zawadzki, _The einstein-hilbert type action on foliated pseudo-riemannian manifolds_ , arXiv preprint arXiv:1604.00985 (2016).
  * [21] Yusuke Sakane, _Examples of compact Einstein-Kähler manifolds with positive Ricci tensor_ , no. 576, 1985, Riemannian manifolds and Lie groups (Japanese) (Kyoto, 1985), pp. 154–176. MR 860230
  * [22] McKenzie Y. Wang and Wolfgang Ziller, _Einstein metrics on principal torus bundles_ , J. Differential Geom. 31 (1990), no. 1, 215–248. MR 1030671
  * [23] William Wylie, _Rigidity of compact static near-horizon geometries with negative cosmological constant_ , Lett. Math. Phys. 113 (2023), no. 2, Paper No. 29, 5. MR 4557955
  * [24] Tomasz Zawadzki, _Existence conditions for conformal submersions with totally umbilical fibers_ , Differential Geom. Appl. 35 (2014), 69–85. MR 3254292