# On a metric symplectization of a contact metric manifold

Sannidhi Alape<EMAIL_ADDRESS>Department of Mathematics, Indian
Institute of Science Education and Research, Bhopal, India

###### Abstract.

In this article, we investigate metric structures on the symplectization of a
contact metric manifold and prove that there is a unique metric structure,
which we call the metric symplectization, for which each slice of the
symplectization has a natural induced contact metric structure. We then study
the curvature properties of this metric structure and use it to establish
equivalent formulations of the $(\kappa,\mu)$-nullity condition in terms of
the metric symplectization. We also prove that isomorphisms of the metric
symplectizations of $(\kappa,\mu)$-manifolds determine
$(\kappa,\mu)$-manifolds up to $\mathcal{D}$-homothetic transformations. These
classification results show that the metric symplectization provides a unified
framework to classify Sasakian manifolds, $K$-contact manifolds and
$(\kappa,\mu)$-manifolds in terms of their symplectizations.

###### Key words and phrases:

Contact metric manifold, $(\kappa,\mu)$-manifold, symplectization

###### 2020 Mathematics Subject Classification:

53C15, 53C25, 53D10

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Contact and symplectic manifolds
    2. 2.2 Contact metric manifolds
    3. 2.3 Contact metric $\boldsymbol{(\kappa,\mu)}$-manifolds
    4. 2.4 Riemannian submersions
  3. 3 The metric symplectization
  4. 4 Curvature properties of the metric symplectization
  5. 5 An application of the metric symplectization

##  1\. Introduction

The study of the interplay between contact manifolds and symplectic manifolds
is an important topic in differential geometry. Symplectic manifolds with a
boundary contact manifold, or more generally, symplectic manifolds with
contact hypersurfaces have been studied extensively (see [16]). In the contact
metric category, the existence of almost contact metric hypersurfaces in
symplectic manifolds with various metric structures has been carried out (see
[11],[15],[17]). Sufficient conditions for the existence of a contact metric
hypersurfaces have also been obtained. In this article, we present the
following result, which is the metric analogue of a well known result in
contact and symplectic topology (see Lemma 2.9).

######  Theorem 1.1.

Let $(B,\omega)$ be a symplectic manifold with an associated metric $g$.
Suppose there is a Liouville vector field $Y$ which is of unit length. Then,
any hypersurface in $B$ which is orthogonal to $Y$ is a contact metric
manifold.

We then focus our attention on metric structures on the symplectization of a
contact metric manifold $(M,\eta,g,\phi)$. The symplectization is an important
tool in the study of contact manifolds and contact metric manifolds. Many
contact invariants are defined in terms of the symplectization (see [9]). In
the contact metric category, a particular class of manifolds, called the
Sasakian manifolds, is defined using a natural almost complex structure on the
symplectization. This natural almost complex structure has also been used by
Boyer and Galicki in [5] to obtain a characterization of $K$-contact manifolds
using properties of the symplectization. However, since the natural almost
complex structure does not induce contact metric structures on the slices
$(M\times\\{t\\})$, for all $t\in\mathbb{R}$, and since the symplectization of
a contact manifold comes with a canonical choice of a Liouville vector field,
we study metric structures on the symplectization which satisfy the hypothesis
of Theorem 1.1, and present the following result which establishes uniqueness
of the same under certain reasonable assumptions.

######  Theorem 1.2.

There is a unique metric structure on the symplectization of a contact metric
manifold $(M,\eta,g,\phi)$ satisfying the following conditions.

  * •

The Liouville vector field $\partial_{t}$ is of unit length and orthogonal to
$(M\times\\{t_{0}\\})$, for every $t_{0}\in\mathbb{R}$.

  * •

The almost complex structure agrees with $\phi$ on the contact distribution
induced on the hypersurfaces $M\times\\{t\\}$.

The unique metric structure thus obtained will be called the metric
symplectization of a contact metric manifold. In addition to being contact
metric hypersurfaces in the symplectization, we also observe that the slices
$(M\times\\{t\\})$ are $\mathcal{D}$-homothetic transformations of the
underlying contact metric manifold. Using the curvature relations of
$\mathcal{D}$-homothetic transformations, and the fact that the projection map
onto the $\mathbb{R}$-coordinate is a Riemannian submersion, we obtain the
relationship between curvature of a contact metric manifold and its metric
symplectization (see Theorem 4.2). As a corollary to the curvature relations
that we obtain, we obtain the following result which specifies a constraint on
the curvature of the metric symplectization of a contact metric manifold.

######  Corollary 1.3.

The metric symplectization of a contact metric manifold cannot be of non-
negative Ricci curvature. In particular, the Ricci curvature in the direction
of the standard Liouville vector field is always $(-2n-4)$.

Next, we study the metric symplectizations of $(\kappa,\mu)$-manifolds, a
class of contact metric manifolds $(M,\eta,g,\phi)$ defined by Blair,
Koufogiorgos and Papantoniou in [2], by the condition

$R(X,Y)\xi=(\kappa I+\mu h)(\eta(Y)X-\eta(X)Y),$ (1.4)

where $\kappa,\mu\in\mathbb{R}$ are constants and $2h$ denotes the tensor
field which is the Lie derivative of $\phi$. The authors in [2] also establish
numerous interesting properties for this class of manifolds. It is clear that
$(\kappa,\mu)$-manifolds are a generalization of manifolds satisfying the
condition $R(X,Y)\xi=0$, as well as a generalization of Sasakian manifolds.
The authors prove that the condition specified in Equation (1.4) is invariant
under $\mathcal{D}$-homothetic transformations. It is also proved that the
curvature of a non-Sasakian $(\kappa,\mu)$-manifold is completely determined
by the constants $\kappa$ and $\mu$ (see Section 2.3 for more details).
Following this, $(\kappa,\mu)$-manifolds have been an important topic in
contact metric geometry as they form a rich class of examples for $H$-contact
manifolds([13]), locally $\phi$-symmetric manifolds ([3]), bilegendrian
manifolds ([7]) etc. In view of the fact that $(\kappa,\mu)$-manifolds are
classified locally up to $\mathcal{D}$-homothetic transformations ([4]) and
the fact that the metric symplectization captures the entire
$\mathcal{D}$-homothetic spectrum of a contact metric manifold, we present the
following characterization of $(\kappa,\mu)$-manifolds using curvature of
their metric symplectizations.

######  Theorem 1.5.

A contact metric manifold $(M,\eta,g,\phi)$ is a non-Sasakian
$(\kappa,\mu)$-manifold if and only if there exist
$\tilde{\kappa},\tilde{\mu}\in\mathbb{R}$ such that the metric symplectization
$(M\times\mathbb{R},\omega,\bar{g},J)$ satisfies the curvature relation

$\mathcal{V}(R(X,Y)\xi_{t})=(\tilde{\kappa}I+\tilde{\mu}h)(\eta_{t}(Y)X-\eta(X)Y),$
(1.6)

for some $t\in\mathbb{R}$ and for all $X,Y\in\mathfrak{X}(M)$. Further, if
Equation (1.6) is satisfied for some $t\in\mathbb{R}$, then it is satisfied
(for different constants $\tilde{\kappa},\tilde{\mu}$) for every
$t\in\mathbb{R}$.

The curvature relations also give a geometric condition on the metric
symplectization which forces the contact metric manifold to be $\eta$-Einstein
(see Proposition 4.8). In particular, we have the following result which
specifies sufficient conditions on the metric symplectization of a
$(\kappa,\mu)$-manifold which forces it to be rigid, i.e., Sasakian.

######  Theorem 1.7.

If the metric symplectization of a $(\kappa,\mu)$-manifold $M$ satisfies the
curvature relation $\overline{Ric}=(-2n-4)dt^{2}$, then $M$ is Sasakian.

In the final section, we present an application of the metric symplectization
which is motivated by the investigation of characterization of contact
manifolds using symplectomorphic symplectizations. We prove the following
characterization for $(\kappa,\mu)$-manifolds.

######  Theorem 1.8.

Let $(M^{2n+1},\eta,g,\phi)$ be a $(\kappa,\mu)$-manifold, and $(M^{\prime
2n+1},\eta^{\prime},g^{\prime},\phi^{\prime})$ be a
$(\kappa^{\prime},\mu^{\prime})$-manifold. Then, $M$ and $M^{\prime}$ are
isomorphic up to $\mathcal{D}$-homothetic transformations if and only if their
metric symplectizations are isomorphic to each other.

######  Remark 1.9.

Theorem 1.8 is interesting because it is in contrast with the result in the
contact category where it is known (see [8]) that there exist contact
manifolds which are not contactomorphic, whose symplectizations are
symplectomorphic.

The paper is organized as follows. In Section 2, fundamental notions in
Riemannian contact and symplectic geometry relevant to the results in the
article will be recalled. In Section 3, we study contact metric hypersurfaces
in certain symplectic metric manifolds and define the metric symplectization.
In Section 4, we study the relations between the curvature of a contact metric
manifold and its metric symplectization. We establish a characterizing
condition for $(\kappa,\mu)$-manifolds using the metric symplectization. We
also prove a rigidity result for $(\kappa,\mu)$-manifolds. In Section 5, we
give an application of the metric symplectization which relates isomorphisms
of the symplectization to isomorphisms of the contact metric manifold up to
$\mathcal{D}$-homothetic transformations.

Acknowledgements: I am grateful to Atreyee Bhattacharya and Dheeraj Kulkarni
for being available for discussions and providing useful comments and
feedback. I also thank them for their detailed suggestions on the preliminary
draft of the manuscript which has significantly improved the article.

The author was supported by the CSIR grant 09/1020(0143)/2019- EMR-I, DST,
Government of India.

##  2\. Preliminaries

###  2.1. Contact and symplectic manifolds

######  Definition 2.1 (Contact manifolds).

A differential $1$-form $\eta$ on a manifold $M$ of dimension $(2n+1)$ is said
to be a contact form if

$\eta_{p}\wedge(d\eta_{p})^{n}\neq 0\text{ for all }p\in M.$ (2.2)

In this case, the distribution $Ker(\eta)$ is called a contact distribution on
the manifold $M$. A contact manifold is a pair $(M,\mathcal{D}=Ker(\eta))$
comprising an odd-dimensional manifold $M$ and a contact distribution
$\mathcal{D}$.

For a detailed treatment of contact manifolds, we refer the reader to [10].

######  Remark 2.3.

If $\mathcal{D}=Ker(\eta)$ for some contact form $\eta$, then
$\mathcal{D}=Ker(f\eta)$ for any non-vanishing function $f\in C^{\infty}(M)$.
Thus, a contact distribution can be given by various contact forms. We will
use the notation $(M,\eta)$ for a contact manifold, where $\eta$ is a contact
form on $M$.

######  Definition 2.4 (Reeb vector field).

Given any contact manifold $(M,\eta)$, the following conditions determine a
unique non-vanishing vector field $\xi$.

  * •

$\eta(\xi)=1$, and

  * •

$d\eta(\xi,X)=0$, for any $X\in\mathfrak{X}(M)$.

This vector field is called the Reeb vector field associated to the contact
form $\eta$.

Throught the article, $\xi$ will be used to denote the Reeb vector field
associated to the contact manifold in consideration.

The even-dimensional counterparts of contact manifolds are symplectic
manifolds, which we now define.

######  Definition 2.5.

A differential $2$-form $\omega$ on a manifold $B$ of dimension is called a
symplectic form if it is closed ($d\omega=0$) and

$\omega_{b}^{n}\neq 0\text{ for all }b\in B.$ (2.6)

A symplectic manifold is a pair $(B,\omega)$ comprising an even-dimensional
manifold $B$ and a symplectic form $\omega$ on $B$.

Equivalence of contact (resp. symplectic) manifolds are given in terms of the
contact (resp. symplectic) forms as follows.

######  Definition 2.7 (Equivalence of Contact and Symplectic manifolds).

  * •

Contact manifolds $(M_{1},\eta_{1})$ and $(M_{2},\eta_{2})$ are said to be
contactomorphic to each other if there is a diffeomorphism $f:M_{1}\rightarrow
M_{2}$ such that $f^{*}\eta_{2}=\eta_{1}$.

  * •

Symplectic manifolds $(B_{1},\omega_{1})$ and $(B_{2},\omega_{2})$ are said to
be symplectomorphic to each other if there is a diffeomorphism
$f:B_{1}\rightarrow B_{2}$ such that $f^{*}\omega_{2}=\omega_{1}$.

The presence of certain types of vector fields on a symplectic manifold can be
used to detect the existence of hypersurfaces (embedded submanifolds of
codimension 1) which support an induced contact structure.

######  Definition 2.8.

Let $(B,\omega)$ be a symplectic manifold. A vector field $Y$ on $B$ is said
to be a Liouville vector field if $\mathcal{L}_{Y}\omega=\omega$.

######  Lemma 2.9.

Let $(B,\omega)$ be a symplectic manifold with a Liouville vector field $Y$.
Then, any hypersurface which is transverse to $Y$ is a contact manifold with
the induced contact form given by $\eta:=i_{Y}\omega=\omega(Y,-)$.

The interplay between symplectic manifolds and contact manifolds is a very
prominent theme in the field and the above result is one instance of the same.
The following is a way to associate a symplectic manifold to any contact
manifold.

######  Definition 2.10.

Let $(M,\eta)$ be a contact manifold. On the product manifold
$(M\times\mathbb{R})$, the $2$-form $\omega:=d(e^{2t}\eta)$ is a symplectic
form, where $t$ denotes the coordinate on $\mathbb{R}$. The symplectic
manifold $(M\times\mathbb{R},\omega)$ thus obtained is called the
symplectization of the contact manifold $(M,\eta)$.

It can be verified that the coordinate vector field $\partial_{t}$ is a
Liouville vector field in the symplectization. For every $t_{0}\in\mathbb{R}$,
the slice $(M\times\\{t_{0}\\})$ is a hypersurface of $(M\times\mathbb{R})$
which is transverse to the Liouville vector field $\partial_{t}$. In view of
Lemma 2.9, the slice at $t_{0}$ is a contact manifold with the contact form
$e^{2t}\eta$. In this article, we establish the analogue of Lemma 2.9 in the
contact metric category.

###  2.2. Contact metric manifolds

Riemannian geometry of contact (resp. symplectic manifolds) lies in the
intersection of Riemannian geometry and contact (resp. symplectic geometry).
In this subsection, we recall the notions from contact and symplectic metric
geometry which will be used in the later sections. For a comprehensive
treatment of Riemannian Geometry of Contact and Symplectic manifolds, see [1].

######  Definition 2.11 (Symplectic metric manifold).

Let $(B,\omega)$ be a symplectic manifold. A Riemannian metric $g$ is said to
be compatible with $\omega$ if there exists an almost complex structure $J$ (a
$(1,1)$-tensor such that $J^{2}=-I$) which satisfies

$g(X,JY)=\omega(X,Y),$ (2.12)

for all $X,Y\in\mathfrak{X}(B)$. In this case, the tuple $(B,\omega,g,J)$ will
be called a symplectic metric manifold.

######  Remark 2.13.

For a symplectic metric manifold $(B,\omega,g,J)$, any two of the structure
tensors $\omega,g$ and $J$ determine the other structure tensor. Equivalent
definitions for symplectic metric manifolds can be given in terms of
compatibility between any two of the structure tensors.

It is known that a compatible Riemannian metric always exists for any
symplectic manifold $(B,\omega)$. Next, we recall the notion of compatibility
between a contact form and a Riemannian manifold.

######  Definition 2.14 (Contact metric manifold).

Let $(M,\eta)$ be a contact manifold with Reeb vector field $\xi$. A
Riemannian metric $g$ is said to be compatible with $\eta$ if

  * •

$g(X,\xi)=\eta(X)$, for all $X\in\mathfrak{X}(M)$,

  * •

There exists a $(1,1)$-tensor field $\phi$ on $M$ which satisfies
$\phi^{2}=-I+\eta\otimes\xi$ and $d\eta(X,Y)=g(X,\phi Y)$, for all
$X,Y\in\mathfrak{X}(M)$.

In this case, the tuple $(M,\eta,g,\phi)$ will be called a contact metric
manifold.

######  Remark 2.15.

The notion of equivalence for the metric versions of contact (resp.
symplectic) manifolds is given by contactomorphisms (resp. symplectomorphisms)
which are isometries of the underlying Riemannian manifold. Equivalent contact
or symplectic metric manifolds will be called isomorphic to each other.

On the symplectization of a contact metric manifold $(M,\eta,g,\phi)$, there
is a natural almost complex structure $J$ defined by

$J(X,f\partial_{t})=(\phi X-f\xi,\eta(X)\partial_{t}),$ (2.16)

for all $X\in\mathfrak{X}(M)$ and $f\in C^{\infty}(M\times\mathbb{R})$. It can
be checked that $\bar{g}(\bar{X},\bar{Y})=\omega(J\bar{X},\bar{Y})$, for all
$\bar{X},\bar{Y}\in\mathfrak{X}(M\times\mathbb{R})$ defines a Riemannian
metric on the symplectization which is compatible with the symplectic form.
This symplectic metric structure, which will be referred to as the natural
symplectic metric structure on the symplectization, has been used to define a
particular class of contact metric manifolds by Shigeo Sasaki (see [14]).

######  Definition 2.17 (Sasakian manifold).

A contact metric manifold is said to be Sasakian if the natural almost complex
structure on its symplectization is integrable.

As is evident from the definition, Sasakian manifolds are closely related to
symplectic metric manifolds with integrable almost complex structures, which
are more popularly known as Kähler manifolds. The relationship between
Sasakian and Kähler manifolds has been studied and it is known that a Sasakian
manifold is naturally sandwiched between two Kähler manifolds. We refer the
reader to [6] for a detailed treatment of the same.

We now recall the definition of another class of contact metric manifolds,
which generalize Sasakian manifolds.

######  Definition 2.18 ($K$-contact manifold).

A contact metric manifold for which the Reeb vector field is a Killing vector
field (the Reeb flow is by isometries) is called a $K$-contact manifold.

######  Remark 2.19.

Given a contact metric manifold $(M,\eta,g,\phi)$, consider the tensor field
$h:=\frac{1}{2}\mathcal{L}_{\xi}\phi$. It can be checked that the contact
metric manifold is $K$-contact if and only if $h\equiv 0$. The tensor $h$
plays a very important role in the study of contact metric manifolds which are
not $K$-contact.

In [5], Boyer and Galicki give a formulation of the $K$-contact condition
using the natural symplectic structure on the symplectization
$(M\times\mathbb{R}^{+},d(t\eta))$.

######  Theorem 2.20 ([5]).

A compact contact metric manifold $(M,\eta,g,\phi)$ is a $K$-contact manifold
if and only if its natural symplectization $(\bar{M},\omega,\bar{g},J)$ is
almost Kähler with $t\partial_{t}-i\xi$ pseudo-holomorphic.

It is clear, from the above illustrations, that contact metric structures are
closely related to the symplectic metric structures on the symplectization. In
this paper, we define an alternative symplectic metric structure on the
symplectization in view of Theorem 1.1 and study the influence of geometric
properties of the metric symplectization on the contact metric manifold and
vice versa.

###  2.3. Contact metric $\boldsymbol{(\kappa,\mu)}$-manifolds

In [2], Blair, Koufogiorgos and Papantoniou define and study a class of
manifolds for which the Reeb vector field satisfies the nullity condition

$R(X,Y)\xi=(\kappa I+\mu h)(\eta(Y)X-\eta(X)Y)$ (2.21)

This class of manifolds contains the class of manifolds satisfying the
condition $R(X,Y)\xi=0$, which has been an important class of contact metric
manifolds, since it contains tangent sphere bundles of flat Riemannian
manifolds as examples. However, the manifolds satisfying the condition
$R(X,Y)\xi=0$ are not stable under $\mathcal{D}$-homothetic transformations.
By a $\mathcal{D}_{a}$-homothetic transformation of a contact metric manifold
$(M,\eta,g,\phi)$, for a real number $a>0$, we mean a transformation of the
structure tensors given by

$\eta^{\prime}=a\eta,\qquad
g^{\prime}=ag+a(a-1)\eta\otimes\eta,\qquad\phi^{\prime}=\phi.$ (2.22)

It can be checked that the tuple $(M,\eta^{\prime},g^{\prime},\phi^{\prime})$
is also a contact metric manifold with

$\xi^{\prime}=\frac{\xi}{a},\qquad h^{\prime}=\frac{h}{a}.$ (2.23)

It can also be checked that the $\mathcal{D}_{a}$-homothetic transformation of
a $(\kappa,\mu)$-manifold is a $(\kappa^{\prime},\mu^{\prime})$-manifold,
where

$\kappa^{\prime}=\frac{\kappa+a^{2}-1}{a^{2}},\qquad\mu^{\prime}=\frac{\mu+2a-2}{a}.$
(2.24)

Many interesting properties of $(\kappa,\mu)$-manifolds have been given in
[2], some of which will be recalled here.

######  Theorem 2.25 ([2]).

Let $(M,\eta,g,\phi)$ be a $(\kappa,\mu)$-manifold. Then,

$h^{2}=-(1-\kappa)\phi^{2}.$

Thus, $\kappa\leq 1$. Moreover, $\kappa=1$ if and only if $M$ is Sasakian. If
$M$ is non-Sasakian, then $h$ has three eigenvalues 0, $\lambda$ and
$-\lambda$, where $\lambda=\sqrt{1-\kappa}$. The tangent bundle of $M$ admits
a decomposition into three mutually orthogonal and integrable distributions
$\mathcal{D}_{\lambda},\mathcal{D}_{-\lambda}$ and $\mathcal{D}_{0}$ given by
the eigenspaces of $h$ corresponding to the eigenvalues $\lambda,\,-\lambda$
and 0, respectively.

The following result establishes the fact that the curvature of a non-Sasakian
$(\kappa,\mu)$-manifold is completely determined by the constants $\kappa$ and
$\mu$.

######  Theorem 2.26 ([2]).

Let $(M^{2n+1},\eta,g,\phi)$ be a non-Sasakian $(\kappa,\mu)$-manifold. For
$X_{\lambda},Y_{\lambda},Z_{\lambda}\in\mathcal{D}_{\lambda}$ and
$X_{-\lambda},Y_{-\lambda},Z_{-\lambda}\in\mathcal{D}_{-\lambda}$, we have

$\displaystyle R(X_{\lambda},Y_{\lambda})Z_{-\lambda}=$
$\displaystyle(\kappa-\mu)\left[g(\phi Y_{\lambda},Z_{-\lambda})\phi
X_{\lambda}-g(\phi X_{\lambda},Z_{-\lambda})\phi Y_{\lambda}\right]$
$\displaystyle R(X_{-\lambda},Y_{-\lambda})Z_{\lambda}=$
$\displaystyle(\kappa-\mu)\left[g(\phi Y_{-\lambda},Z_{\lambda})\phi
X_{-\lambda}-g(\phi X_{-\lambda},Z_{\lambda})\phi Y_{-\lambda}\right]$
$\displaystyle R(X_{\lambda},Y_{-\lambda})Z_{-\lambda}=$ $\displaystyle\kappa
g(\phi X_{\lambda},Z_{-\lambda})\phi Y_{-\lambda}+\mu g(\phi
X_{\lambda},Y_{-\lambda})\phi Z_{-\lambda}$ $\displaystyle
R(X_{\lambda},Y_{-\lambda})Z_{\lambda}=$ $\displaystyle-\kappa g(\phi
Y_{-\lambda},Z_{\lambda})\phi X_{\lambda}-\mu g(\phi
Y_{-\lambda},X_{\lambda})\phi Z_{\lambda}$ $\displaystyle
R(X_{\lambda},Y_{\lambda})Z_{\lambda}=$
$\displaystyle\left[2(1+\lambda)-\mu\right]\left[g(Y_{\lambda},Z_{\lambda})X_{\lambda}-g(X_{\lambda},Z_{\lambda})Y_{\lambda}\right]$
$\displaystyle R(X_{-\lambda},Y_{-\lambda})Z_{-\lambda}=$
$\displaystyle\left[2(1-\lambda)-\mu\right]\left[g(Y_{-\lambda},Z_{-\lambda})X_{-\lambda}-g(X_{-\lambda},Z_{-\lambda})Y_{-\lambda}\right]$
(2.27)

A classification of $3$-dimensional $(\kappa,\mu)$-manifolds was also given
(see [2, Theorem 3]). However, the local classification for all dimensions was
given in [4], where Boeckx defines a quantity called the index of a
$(\kappa,\mu)$-manifold $M$, which is denoted by $I_{M}$ and is given by

$I_{M}=\frac{1-\frac{\mu}{2}}{\sqrt{1-\kappa}}.$ (2.28)

It was shown that the index is invariant under $\mathcal{D}$-homothetic
transformations and that the index determines $(\kappa,\mu)$-manifolds
locally, in the following sense.

######  Theorem 2.29 ([4]).

Let $(M_{i},\eta_{i},g_{i},\phi_{i})$ be $(\kappa_{i},\mu_{i})$-manifolds for
$i\in\\{1,2\\}$. Then, $I_{M_{1}}=I_{M_{2}}$ if and only if the two spaces are
locally isometric up to a $\mathcal{D}$-homothetic transformation. In
particular, if the spaces are simply connected and complete, then they are
isometric up to a $\mathcal{D}$-homothetic transformation.

###  2.4. Riemannian submersions

The main results of the article involve curvature computations using
Riemannian submersions. Here, we briefly recall the notions associated with
the same. Let $\pi:M\rightarrow N$ be a smooth submersion. The Kernel of the
differential of $\pi$ defines a distribution, which is known as the vertical
distribution and is denoted by $\mathcal{V}$. If $M$ is a Riemannian manifold,
then the orthogonal complement of $\mathcal{V}$ is called the horizontal
distribution and is denoted by $\mathcal{H}$. In [12], O’Neill defined and
studied Riemannian submersions.

######  Definition 2.30 ([12]).

A smooth submersion $\pi:(M,\overline{g})\rightarrow(N,g)$ is said to be a
Riemannian submersion if $d\pi$ is an isometry from
$(\mathcal{H}_{x},\overline{g}_{x})$ to $(T_{\pi x}N,g_{\pi x})$, for every
$x\in M$.

######  Notation 2.31.

Throughout the article, in the context of a Riemannian submersion
$\pi:(M,\bar{g})\rightarrow(N,g)$, $\mathcal{H}(V)$ and $\mathcal{V}(V)$ will
be used to denote the horizontal and vertical components of a vector field
$V$. The quantities associated with the manifold $M$ will have an overhead
line, the quantities associated with the fiber over an element $y\in N$ will
have a superscript $y$. For example, $\overline{R},R^{y}$ and $R$ will denote
the curvature of the manifold $M$, the curvature of the fiber $\pi^{-1}(y)$,
and the curvature of $N$, respectively.

The author in [12] also defines the following tensor fields associated to a
Riemannian submersion.

######  Definition 2.32 ([12]).

Given a Riemannian submersion $\pi:(M,\overline{g})\rightarrow(N,g)$, define
(2,1)-tensors $T$ and $A$ by the formulae

$\displaystyle T_{E_{1}}E_{2}$
$\displaystyle=\mathcal{H}\overline{\nabla}_{\mathcal{V}E_{1}}\mathcal{V}E_{2}+\mathcal{V}\overline{\nabla}_{\mathcal{V}E_{1}}\mathcal{H}E_{2}$
(2.33) $\displaystyle A_{E_{1}}E_{2}$
$\displaystyle=\mathcal{H}\overline{\nabla}_{\mathcal{H}E_{1}}\mathcal{V}E_{2}+\mathcal{V}\overline{\nabla}_{\mathcal{H}E_{1}}\mathcal{H}E_{2}$
(2.34)

These tensor fields are called the fundamental tensor fields associated with
the Riemannian submersion $\pi$.

It is known that the tensor $T$ is identically zero if and only if each fibre
is totally geodesic. On the other hand, the tensor $A$ is identically zero if
and only if the horizontal distribution is integrable. Further, the following
formulae can be used to compute $T$ and $A$.

######  Proposition 2.35.

Suppose $\pi:(M,\overline{g})\rightarrow(N,g)$ is a Riemannian submersion. For
horizontal vector fields $X,Y$ and vertical vector fields $U,V$, we have

$\displaystyle T_{X}U=T_{X}Y=0$ (2.36) $\displaystyle
T_{V}U=T_{U}V=\mathcal{H}\overline{\nabla}_{U}V,\quad
T_{U}X=\mathcal{V}\overline{\nabla}_{U}X$ (2.37) $\displaystyle
A_{U}X=A_{U}V=0$ (2.38) $\displaystyle
A_{X}Y=-A_{Y}X=\mathcal{V}\overline{\nabla}_{X}Y,\quad
A_{X}U=\mathcal{H}\overline{\nabla}_{X}U$ (2.39)

The relationship between the curvature of the domain and range of a Riemannian
submersion can be formulated in terms of the fundamental tensors $T$ and $A$.
Since the Riemannian submersion that we consider in this article has a one-
dimensional base space, and integrable horizontal distribution, we recollect
the formulae for curvature only for this specific case. For the complete set
of relations in the general case, we refer the reader to [12].

######  Theorem 2.40 (O’Neill’s formulae for curvature ([12])).

Let $\pi:(M,\bar{g})\rightarrow(N,g)$ be a Riemannian submersion such that
$\dim(N)=1$ and the horizontal distribution $\mathcal{H}$ is integrable. Then,
for horizontal vector fields $X_{i}$ and vertical vector fields $V_{i}$ (for
$i\in\\{1,2,3,4\\}$, and $y\in N$,

$\displaystyle(V_{1},V_{2},V_{3},V_{4})=$
$\displaystyle\bar{g}(R^{y}(V_{1},V_{2})V_{3},V_{4})+\bar{g}(T_{V_{1}}V_{3},T_{V_{2}}V_{4})-\bar{g}(T_{V_{2}}V_{3},T_{V_{1}}V_{4}),$
$\displaystyle(V_{1},V_{2},V_{3},X_{1})=$
$\displaystyle\bar{g}((\bar{\nabla}_{V_{1}}T)_{V_{2}}V_{3},X_{1})-\bar{g}((\bar{\nabla}_{V_{2}}T)_{V_{1}}V_{3},X_{1})$
$\displaystyle(X_{1},V_{1},X_{2},V_{2})=$
$\displaystyle\bar{g}(T_{V_{1}}X_{1},T_{V_{2}}X_{2})-\bar{g}((\bar{\nabla}_{X_{1}}T)_{V_{1}}V_{2},X_{2})$
$\displaystyle(V_{1},V_{2},X_{1},X_{2})=$
$\displaystyle\bar{g}(T_{V_{1}}X_{1},T_{V_{2}}X_{2})-\bar{g}(T_{V_{2}}X_{1},T_{V_{1}}X_{2})$
(2.41)

The Ricci curvature relations and the sectional curvature relations can be
derived from this (see [12]).

##  3\. The metric symplectization

In this section, we will define the metric symplectization of a contact metric
manifold. We begin with the proof of Theorem 1.1, which is the analogue of
Lemma 2.9 in the contact metric category.

###### Proof of Theorem 1.1.

Since $\omega$ is a symplectic form on $B$, there is an isomorphism

$\displaystyle\Phi:TB$ $\displaystyle\rightarrow T^{*}B$ $\displaystyle X$
$\displaystyle\mapsto(Y\mapsto\omega(Y,X))$

Let $M$ be a hypersurface in $B$ which is orthogonal to $Y$. Let $A_{Y}$
denote the covector field which takes value $1$ when evaluated on $Y$ and
vanishes on vector fields in $M$. Using the fact that $Y$ is a Liouville
vector field, it can be checked that the vector field $\Phi^{-1}(A_{Y})$ is
the Reeb vector field for the induced contact structure $\eta:=i_{Y}\omega$ on
$M$. This vector field will henceforth be denoted by $\xi$. Let $J$ denote the
almost complex structure on $B$ given by the compatibility of $g$ and
$\omega$. In order to compute $J\xi$, note that, for any vector field $X$ in
$M$,

$g(X,J\xi)=\omega(X,\xi)=A_{Y}(X)=0$

Thus, $J\xi$ lies along the line field which is orthogonal to $M$. By our
assumption of orthogonality, we must have that $J\xi$ lies along the direction
of the vector field $Y$. Therefore, $J\xi=cY$, for some
$c\in\mathbb{R}\setminus\\{0\\}$. In particular,

$g(\xi,\xi)=g(J\xi,J\xi)=c^{2}$

On the other hand, the compatibility of the structure tensors of the
symplectic metric structures gives us

$g(\xi,\xi)=-\omega(\xi,J\xi)=A_{Y}(J\xi)=c$

From the equations above, we must have $c^{2}=c$. Since $c$ cannot be zero, we
have $c=1$. Consequently, $J\xi=Y$. Since $J$ is an almost complex structure,
we have $JY=-\xi$. For any vector field $X$ in the contact distribution, note
that

$\displaystyle g(Y,JX)=\omega(Y,X)=i_{Y}\omega(X)=\eta(X)=0$ $\displaystyle
g(\xi,JX)=-\omega(X,\xi)=-A_{Y}(X)=0$

Thus, we can conclude that $J$ maps the contact distribution of $M$ to itself.
Therefore, we can define a $(1,1)$-tensor field $\phi$ on $M$ which is equal
to $J$ on the contact distribution and vanishes on the line field determined
by $\xi$. We claim that $(M,\eta,\phi,g)$ is a contact metric structure. In
order to see this, we need to verify the conditions specified in Definition
2.14. We have,

$g(X,\xi)=-\omega(\xi,JX)=A_{Y}(JX)$

Using the values for $J$ obtained above, we can check that $g(X,\xi)=\eta(X)$
for all vector fields $X$ in $M$. The condition $\phi^{2}=-I+\eta\otimes\xi$
follows directly from the fact that $J$ is an almost complex structure and
$\phi\xi=0$. We also have, for $X,X^{\prime}\in\mathfrak{X}(M)$,

$d\eta(X,X^{\prime})=(d\circ
i_{Y}\omega)(X,X^{\prime})=\omega(X,X^{\prime})=g(X,JX^{\prime})=g(X,\phi
X^{\prime})$

Therefore, $(M,\eta,\phi,g)$ is a contact metric manifold. ∎

######  Remark 3.1.

A standard example of contact hypersurfaces transverse to Liouville vector
fields in symplectic manifolds is given in the context of the symplectization.
Given the symplectization $(M\times\mathbb{R},d(e^{2t}\eta))$, it is known
that the coordinate vector field $\partial_{t}$ on $\mathbb{R}$ is a Liouville
vector field. Consequently, the hypersurfaces $(M\times\\{t_{0}\\})$ are of
contact type, with contact form $i_{\partial_{t}}\omega$. However, in the
metric category, the natural almost complex structure on the symplectization
(see Equation (2.16)) does not induce contact metric structures on the
hypersurfaces $(M\times\\{t_{0}\\})$, for $t_{0}\neq 0$.

This motivates us to look for metric structures on the symplectization which
satisfy the hypothesis of Theorem 1.1. To this end, we have Theorem 1.2, which
is proved below.

###### Proof of Theorem 1.2.

Let $(M\times\mathbb{R},\omega=d(e^{2t}\eta))$ be the symplectization of the
contact manifold $(M,\eta)$. We know that the Liouville vector field
$\partial_{t}$ induces a contact form $\eta_{t_{0}}=i_{\partial_{t}}\omega$ on
$(M\times\\{t_{0}\\})$ for every $t_{0}\in\mathbb{R}$. For
$X\in\mathfrak{X}(M)$,

$i_{\partial_{t}}\omega(X)=\omega(\partial_{t},X)=e^{2t_{0}}\eta(X)$

Therefore, we have the contact form $\eta_{t}:=e^{2t}\eta$ on the hypersurface
$M\times\\{t\\}$. The corresponding Reeb vector field is given by
$\xi_{t}:=e^{-2t}\xi$.

Suppose $J$ is an almost complex structure on $(M\times\mathbb{R})$ which is
compatible with the symplectic form $\omega$ and induces a Riemannian metric
$\bar{g}$ which satisfies the conditions specified in the statement of the
theorem. Let $J\partial_{t}=X_{0}+f_{1}\xi+f_{2}\partial_{t}$, for some
$X_{0}\in Ker(\eta_{t})$ and $f_{1},f_{2}\in C^{\infty}(M\times\mathbb{R})$.
By the assumption of ortogonality, we get

$0=\bar{g}(\partial_{t},\xi)=\bar{g}(J\partial_{t},J\xi)=\omega(J\partial_{t},\xi)=e^{2t}dt(J\partial_{t})=f_{2}$

On the other hand, for any vector field $X$ in the contact distribution,

$0=\bar{g}(\partial_{t},X)=\bar{g}(J\partial_{t},JX)=\omega(J\partial_{t},X)=e^{2t}d\eta(J\partial_{t},X)=e^{2t}d\eta(X_{0},X)$

Since the form $d\eta$ is non-degenerate on the contact distribution, we have
$X_{0}=0$. Thus, $J\partial_{t}=f_{1}\xi$ for some non-vanishing function
$f_{1}\in C^{\infty}(M)$. Since $J$ is an almost complex structure, this also
means that $J\xi=\frac{-\partial_{t}}{f_{1}}$. We now use the fact that
$\partial_{t}$ is a Liouville vector field of unit length.

$1=\bar{g}(\partial_{t},\partial_{t})=\bar{g}(J\partial_{t},J\partial_{t})=\omega(J\partial_{t},\partial_{t})=f_{1}\omega(\xi,\partial_{t})=-f_{1}e^{2t}$

Hence, we can conclude that the only possible almost complex structure $J$
satisfying the required conditions is given by

$JX=\phi X\text{, for }X\in Ker(\eta_{t}),\qquad J\xi_{t}=\partial_{t},\qquad
J\partial_{t}=-\xi_{t}.$

It is easy to check the almost complex structure defined above is compatible
with the symplectic form $\omega$ and thus defines a symplectic metric
structure on $(M\times\mathbb{R},\omega)$. ∎

The compatibility of the almost complex structure and the symplectic form
uniquely determines the Riemannian metric $\bar{g}$ on the symplectization.
Since the vector field $\partial_{t}$ is of unit length and orthogonal to the
slices $(M\times\\{t\\})$, we have $\bar{g}=g_{t}+dt^{2}$, for a Riemannian
metirc $g_{t}$ on the slice $(M\times\\{t\\})$. By Theorem 1.1, we can
conclude that $(M\times\\{t\\},\eta_{t},g_{t},\phi)$ is a contact metric
manifold for every $t\in\mathbb{R}$. Using the compatibility conditions, we
obtain

$g_{t}=e^{2t}g+e^{2t}(e^{2t}-1)\eta\otimes\eta.$ (3.2)

Throughout the article $(M_{t},eta_{t},g_{t},phi)$ will be used to denote the
induced contact metric structure on $(M\times\\{t\\})$.

######  Definition 3.3.

Let $(M,\eta,g,\phi)$ be a contact metric manifold. The symplectic metric
manifold $(M\times\mathbb{R},\omega,\bar{g},\phi)$ as defined above is called
the metric symplectization of the given contact metric manifold.

Next, we note that for $t\in\mathbb{R}$, the contact metric hypersurface
$M\times\\{t\\}$ in the metric symplectization has structure tensors which is
related to the original contact metric structure (or equivalently, the contact
metric structure on $M\times\\{0\\}$) by

$\eta_{t}=e^{2t}\eta,\qquad g_{t}=e^{2t}g+e^{2t}(e^{2t}-1)\eta\otimes\eta.$
(3.4)

This proves that the slice at $t$ is a $\mathcal{D}_{e^{2t}}$-homothetic
transformation of the original contact metric manifold (or the slice at $0$).

######  Remark 3.5.

More generally, for $t_{1},t_{2}\in\mathbb{R}$, the slice at $t_{2}$ is a
$\mathcal{D}$-homothetic transformation of the slice at $t_{1}$. Thus, the
metric symplectization of a contact metric manifold captures the entire
spectrum of its $\mathcal{D}$-homothetic transformations.

Since $(\kappa,\mu)$-manifolds in particular are locally classified up to
$\mathcal{D}$-homothetic transformations, we now focus on this class of
contact metric manifolds and their metric symplectizations.

##  4\. Curvature properties of the metric symplectization

In this section, we will derive the relationship between the curvature of a
contact metric manifold and the curvature of its metric symplectization. In
particular, since the curvature of non-Sasakian $(\kappa,\mu)$-manifolds are
completely determined by the constants $\kappa$ and $\mu$ (see Theorem 2.26),
we obtain a characterization of the $(\kappa,\mu)$ condition in terms of
structure tensors of its metric symplectization. As an application of the
curvature relations, we also present a rigidity result for
$(\kappa,\mu)$-manifolds under certain geometric constraints on the metric
symplectization.

From the description of the Riemannian metric on the symplectization
$\bar{g}=g_{t}+dt^{2}$, it is clear that the projection map
$\pi_{2}:(M\times\mathbb{R},\bar{g})\rightarrow(\mathbb{R},dt^{2})$ is a
Riemannian submersion. In this case, the vertical distribution at a point
$(x,t)$ is given by the tangent space $T_{x}M$ of $M$ and the horizontal
distribution is given by the line field determined by the coordinate vector
$\partial_{t}$. Using O’Neill’s formulae (see [12]), we can compute the
curvature of the total space $M\times\mathbb{R}$ in terms of the curvature of
the horizontal distribution, the curvature of the vertical distribution, and
the fundamental tensors $T$ and $A$ associated with the Riemannian submersion
$\pi_{2}$. The curvature of the horizontal distribution is zero since it is a
one-dimensional space. Also, since the horizontal distribution is integrable
for the Riemannian submersion $\pi_{2}$ under consideration, we have $A=0$.

######  Lemma 4.1.

The fundamental tensor $T$ associated with the Riemannian submersion $\pi_{2}$
satisfies the following relations. For $X,Y,Z\in\mathfrak{X}(M)$,

  * •

$T_{X}Y=-(\bar{g}(X,Y)+\eta_{t}(X)\eta_{t}(Y))\partial_{t}$,

  * •

$T_{X}\partial_{t}=X+\eta_{t}(X)\xi_{t}$

The tensor $T$ is fully determined by the above result and the properties of
$T$ stated in Proposition 2.35. Using the values for obtained above and the
curvature relations in Theorem 2.40, we obtain the following result.

######  Theorem 4.2.

The curvatures of the metric symplectization and the contact metric manifold
satisfy the following relations.

$\displaystyle\mathcal{V}(\bar{R}(X,Y)Z)=$ $\displaystyle
R^{t}(X,Y)Z+g_{t}(X+\eta_{t}(X)\xi_{t},Z)Y$ $\displaystyle-
g_{t}(Y+\eta_{t}(Y)\xi_{t},Z)X+g_{t}(\eta_{t}(Y)X-\eta_{t}(X)Y,Z)\xi_{t}$
(4.3) $\displaystyle\mathcal{H}(\bar{R}(X,Y)Z)=$ $\displaystyle-g_{t}(\phi
Z,\eta_{t}(Y)X-\eta_{t}(X)Y+\eta_{t}(Y)hX-\eta_{t}(X)hY)$
$\displaystyle+2\eta_{t}(Z)g_{t}(Y,\phi X)$ (4.4)
$\displaystyle\bar{g}(\bar{R}(\partial_{t},X)\partial_{t},Y)=$ $\displaystyle
g_{t}(X,Y)+3\eta_{t}(X)\eta_{t}(Y)$ (4.5)
$\displaystyle\bar{g}(\bar{R}(X,Y)\partial_{t},\partial_{t})=$ $\displaystyle
0$ (4.6)

The proof involves lengthy but straightforward computations using O’Neill’s
formulae and will be omitted.

In particular, for non-Sasakian $(\kappa,\mu)$-manifolds, we use the
$(\kappa,\mu)$ condition (1.4) to prove Theorem 1.5.

###### Proof of Theorem 1.5.

Suppose $(M,\eta,g,\phi)$ is a contact metric $(\kappa,\mu)$-manifold.
Substituting $Z=\xi_{t}$ in Equation (4.3), we see that

$\displaystyle\mathcal{V}(\bar{R}(X,Y)\xi_{t})=$ $\displaystyle
R^{t}(X,Y)\xi_{t}+g_{t}(X+\eta_{t}(X)\xi_{t},\xi_{t})Y$ $\displaystyle-
g_{t}(Y+\eta_{t}(Y)\xi_{t},\xi_{t})X+g_{t}(\eta_{t}(Y)X-\eta_{t}(X)Y,\xi_{t})\xi_{t}$
$\displaystyle=$
$\displaystyle(\kappa_{t}I+\mu_{t}h)(\eta_{t}(Y)X-\eta_{t}(X)Y)+2\eta_{t}(X)Y-2\eta_{t}(Y)X$
$\displaystyle=$
$\displaystyle((\kappa_{t}-2)I+\mu_{t}h)(\eta_{t}(Y)X-\eta_{t}(X)Y)$

Thus, Equation (1.6) holds for all $t\in\mathbb{R}$ and
$X,Y\in\mathfrak{X}(M)$. For the converse, note that if Equation (4.3) holds
for the slice at $t$, then a computation similar to the one above shows that

$R^{t}(X,Y)\xi_{t}=((\tilde{\kappa}+2)I+\tilde{\mu}h)(\eta_{t}(Y)X-\eta_{t}(X)Y)$

In other words, the slice at $t$ is a
$(\tilde{\kappa}+2,\tilde{\mu})$-manifold. Since we have seen that every slice
of the metric symplectization is a $\mathcal{D}$-homothetic transformation of
every other slice, we can conclude that the slice at $t=0$ is a
$(\kappa,\mu)$-manifold. But the slice at $t=0$ is isomorphic to the contact
metric manifold $M$. Therefore, $M$ is a $(\kappa,\mu)$-manifold.

The last statement is also a consequence of the observation that any two
slices of the metric symplectization are $\mathcal{D}$-homothetic
transformations of each other, and the fact that $\mathcal{D}$-homothetic
transformation of a $(\kappa,\mu)$-manifold is a
$(\kappa^{\prime},\mu^{\prime})$-manifold, for some
$\kappa^{\prime},\mu^{\prime}\in\mathbb{R}$. ∎

As a corollary to Proposition 4.2, we obtain the relations between the Ricci
curvature of a contact metric manifold and its metric symplectization.

######  Corollary 4.7.

Let $(M^{2n+1},\eta,g,\phi)$ be a contact metric manifold and
$(M\times\mathbb{R},\omega,\bar{g},J)$ be its metric symplectizaition.
Let$\\{e_{1},e_{2},\dots,e_{2n}\\}$ be an orthonormal basis of the contact
distribution of $M$. Then,

$\displaystyle\overline{Ric}(e_{i},e_{j})=Ric^{t}(e_{i},e_{i})-(2n+4)\delta_{ij},$
$\displaystyle\overline{Ric}(e_{i},\xi_{t})=Ric^{t}(e_{i},\xi_{t}),\quad\overline{Ric}(e_{i},\partial_{t})=0$
$\displaystyle\overline{Ric}(\xi_{t},\partial_{t})=0,\quad\overline{Ric}(\xi_{t},\xi_{t})=Ric^{t}(\xi_{t},\xi_{t})-4n-4$
$\displaystyle\overline{Ric}(\partial_{t},\partial_{t})=-2n-4$

Irrespective of the curvature of the contact metric manifold $M$, we notice
that the Ricci curvature in the direction of $\partial_{t}$ is always equal to
$(-2n-4)$, which proves Corollary 1.3.

Since the Ricci curvature in the $\partial_{t}$ direction is constant, we
investigate the case in which the metric on the metric symplectization is
Ricci flat in all other directions. Note that a contact metric manifold
$(M,\eta,g,\phi)$ is said to be $\eta$-Einstein iff there exist
$\alpha,\beta\in C^{\infty}(M)$ such that $Ric=\alpha g+\beta\eta\otimes\eta$.

######  Proposition 4.8.

Let $(M,\eta,g,\phi)$ be a contact metric manifold. If the metric
symplectization $(M\times\mathbb{R},\omega,\bar{g},J)$ satisfies the curvature
relation $\overline{Ric}=(-2n-4)dt^{2}$, then $M\times\\{t\\}$ is
$\eta_{t}$-Einstein, for all $t\in\mathbb{R}$. In particular, $M$ is
$\eta$-Einstein.

###### Proof.

Using relations obtained in Corollary 4.7 and the basis, we have, for all
$t\in\mathbb{R}$,

$Ric^{t}(e_{i},e_{j})=(2n+4)\delta_{ij},Ric^{t}(e_{i},\xi_{t})=0,Ric^{t}(\xi_{t},\xi_{t})=(4n+4)$
(4.9)

Since the basis $\\{e_{i}\\})_{i=1}^{n}$ is assumed to be orthonormal, we get

$Ric^{t}=(2n+4)g_{t}+(4n+4)\eta_{t}\otimes\eta_{t}.$ (4.10)

Thus, the contact metric structure on the slice at $t$ of the metric
symplectization is an $\eta_{t}$-Einstein manifold. ∎

In particular, in the class of $(\kappa,\mu)$-manifolds, we have the rigidity
result given by Theorem 1.7.

###### Proof.

It is known that a non-Sasakian $(\kappa,\mu)$-manifold is $\eta$-Einstein if
and only if $\mu=2-2n$. Since Proposition 4.8 forces the manifold $M$ and each
of its $\mathcal{D}$-homothetic transformations to be $\eta_{t}$-Einstein, we
need $\mu_{t}=2-2n$ for all $t\in\mathbb{R}$. But it is also easy to check
that the constant $\mu$ is invariant under $\mathcal{D}$-homothetic
transformations if and only if $\mu=2$. Thus, we must have $n=0$, which is not
possible. Hence, the $(\kappa,\mu)$-manifold is Sasakian. ∎

##  5\. An application of the metric symplectization

In this section, we prove a result which characterizes
$(\kappa,\mu)$-manifolds using the metric symplectizations. For contact
manifolds, the following result shows that characterization is not possible
using symplectization.

######  Theorem 5.1 ([8]).

Let $M_{1}$ and $M_{2}$ be closed contact manifolds of same (odd) dimension
greater than or equal to $5$ such that $\mathbb{R}\times M_{1}$ and
$\mathbb{R}\times M_{2}$ are diffeomorphic. Then, for every contact structure
$\mathcal{D}_{1}$ on $M_{1}$, there is a contact structure $\mathcal{D}_{2}$
on $M_{2}$ such that the symplectizations of $(M_{1},\mathcal{D}_{1})$ and
$(M_{2},\mathcal{D}_{2})$ are symplectomorphic.

However, in the case of $(\kappa,\mu)$-manifolds, we observe more rigidity,
which is formulated in Theorem 1.8 and proved below.

###### Proof of Theorem 1.8.

If $M$ and $M^{\prime}$ are $\mathcal{D}$-homothetic transformations of each
other, then by the property 3.5, we can conclude that a copy of $M$ is
isomorphic to some slice $M^{\prime}\times\\{t^{\prime}\\}$ in the metric
symplectization of $M^{\prime}$. Consequently, the translation map
$(x,t)\mapsto(x,t+t^{\prime})$ can be checked to be a symplectomorphism of the
metric symplectizations.

For the converse, let $f:M^{\prime}\times\mathbb{R}\rightarrow
M\times\mathbb{R}$ be an isomorphism of the metric symplectizations. Let
$\partial_{t^{\prime}}$ and $\partial_{t}$ denote the natural Liouville vector
fields on the domain and codomain, respectively. Suppose
$df(\partial_{t^{\prime}})=Y$, where $Y\in\mathfrak{X}(M\times\mathbb{R})$.
Since $f$ is an isomorphism and
$\overline{Ric^{\prime}}(\partial_{t^{\prime}},\partial_{t^{\prime}})=-2n-4$,
we must have $\overline{Ric}(Y,Y)=-2n-4$. The vector field $Y$ has an
orthogonal decomposition

$Y=Y_{\lambda_{t}}+Y_{-\lambda_{t}}+\alpha\xi_{t}+\beta\partial_{t},$ (5.2)

for
$Y_{\lambda_{t}}\in\mathcal{D}_{\lambda_{t}},Y_{-\lambda_{t}}\in\mathcal{D}_{-\lambda_{t}}$,
and $\alpha,\beta\in C^{\infty}(M\times\mathbb{R})$. Using the formulae
obtained for $\overline{Ric}$ in Corollary 4.7 and the fact that $Y$ is a
vector field of unit length, we obtain

$\displaystyle\overline{Ric}(Y,Y)=$
$\displaystyle\|Y_{\lambda_{t}}\|^{2}(Ric^{t}(\hat{Y}_{\lambda_{t}},\hat{Y}_{\lambda_{t}})+\|Y_{-\lambda_{t}}\|^{2}(Ric^{t}(\hat{Y}_{-\lambda_{t}},\hat{Y}_{-\lambda_{t}})$
$\displaystyle+\alpha^{2}(Ric^{t}(\xi_{t},\xi_{t})-2n)-2n-4$

We can simplify the above equation using the Ricci curvature formulae for a
$(\kappa_{t},\mu_{t})$-manifold, and substituting
$\mu_{t}=2(1-\lambda_{t}I_{M_{t}})$ to get

$\begin{matrix}\|Y_{\lambda_{t}}\|^{2}(\lambda_{t}^{2}I_{M}-\lambda_{t}n(1+I_{M})+1)\\\
+\|Y_{-\lambda_{t}}\|^{2}(-\lambda_{t}^{2}+\lambda_{t}n(1-I_{M})+1)+\alpha^{2}n\lambda_{t}^{2}\end{matrix}=0$
(5.3)

The quadratic expressions $(\lambda_{t}^{2}I_{M}-\lambda_{t}n(1+I_{M})+1)$ and
$(-\lambda_{t}^{2}+\lambda_{t}n(1-I_{M})+1)$ are positive for small values of
$\lambda_{t}$. Since $\lambda_{t}=\frac{\lambda}{e^{2t}}$, we see that
$\lambda_{t}$ is small for large values of $t$. In other words, there exists a
$t_{0}\in\mathbb{R}$ such that both of the quadratic expressions are positive
for $t>t_{0}$. Therefore, for $t>t_{0}$, Equation (5.3) can be satisfied only
if $\|Y_{\lambda_{t}}\|=\|Y_{-\lambda_{t}}\|=\alpha=0$. In this case, we will
have $Y=\partial_{t}$. In this case, there exists $t^{\prime}\in\mathbb{R}$
such that $f$ maps $M^{\prime}\times\\{t^{\prime}\\}$ to $M\times\\{t\\}$.
Using the properties of the Liouville vector field, we can check that
$f^{*}(\eta_{t})=\eta_{t^{\prime}}$. This shows that $f$ is a contactomorphism
from $M^{\prime}\times\\{t^{\prime}\\}$ to $M\times\\{t\\}$, which is also a
Riemannian isometry by definition. Hence, we can conclude that $f$, when
restricted to $M^{\prime}\times\\{t^{\prime}\\}$, is an isomorphism onto
$M\times\\{t\\}$. Since $M$ and $M^{\prime}$ are $\mathcal{D}$-homothetic
transformations of these slices, we have the required result. ∎

######  Remark 5.4.

One can check that a contact metric manifold is Sasakian if and only if the
almost complex structure corresponding to its metric symplectization is
integrable. One can also check that the classification of $K$-contact
condition using the natural almost complex structure given in [5, Theorem 6.3]
has an equivalent formulation in terms of the metric symplectization. Thus,
the metric symplectization is a tool that can be used to classify Sasakian
manifolds, $K$-contact manifolds, and non-Sasakian $(\kappa,\mu)$-manifolds.

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