# Warped product pointwise bi-slant submanifolds of Kenmotsu manifolds

Shyamal Kumar Hui∗, Joydeb Roy and Tanumoy Pal

###### Abstract.

The present paper deals with the study of warped product pointwise bi-slant
submanifolds of Kenmotsu manifolds with an example. The characterization for
such submanifold is also discussed. An inequality of such submanifold is
obtained and its equality case is also considered.

###### Key words and phrases:

Kenmotsu manifold, warped product, pointwise bi-slant submanifold.
$*$ Corresponding author.

###### 2010 Mathematics Subject Classification:

53C15, 53C42.

##  1\. Introduction

In [35] Tanno classified connected almost contact metric manifolds whose
automorphism groups possess the maximum dimension. For such a manifold, the
sectional curvature of plane sections containing $\xi$ is a constant, say $c$,
which that they could be divided into three classes: (i) homogeneous normal
contact Riemannian manifolds with $c>0$, (ii) global Riemannian products of a
line or a circle with a Kähler manifold of constant holomorphic sectional
curvature if $c=0$ and (iii) a warped product space
$\mathbb{R}\times_{f}\mathbb{C}^{n}$ if $c<0$. It is known that the manifolds
of class (i) are characterized by admitting a Sasakian structure. The
manifolds of class (ii) are characterized by a tensorial relation admitting a
cosymplectic structure. Kenmotsu [23] characterized the differential geometric
properties of the manifolds of class (iii) which are nowadays called Kenmotsu
manifolds and later studied by several authors (see [17], [18], [19] and
references therein).
In 1990, Chen [8] introduce the notion of slant submanifolds in a Hermitian
manifold. Then Lotta [28] has defined and studied slant immersions of a
Riemannian manifold into an almost contact metric manifold. As a natural
generalization of slant submanifolds Etayo [14] defined the notion of
pointwise slant subamnifolds under the name of quasi-slant submanifolds.
Pointwise slant submanifolds in almost contact metric manifold is studied in
([29],[34]). On the other hand, Carriazo [7] defined and studied bi-slant
submanifolds in almost Hermitian manifolds and simultaneously gave the notion
of pseudo-slant submanifolds. Then Khan and Khan [24] studied contact version
of pseudo-slant submanifolds. Bi-slant submanifolds of Kenmotsu space form is
studied in [33].
As a generalization of Riemannian product manifold Bishop and O’Neill [6]
defined and studied warped product manifolds. The study of warped product
submanifolds was initiated by Chen ([9], [10]). Then many authors studied
warped product submanifolds of different ambient manifold ([12], [20]-[22],
[32]). In this connection it may be mention that warped product submanifolds
of Kenmotsu manifold are studied in ([1], [2], [3], [25], [26], [31], [36],
[37], [38]).
Recently, Chen and Uddin [13] studied pointwise bi-slant submanifolds of
Kaehler manifold. Also, Khan and Shuaib [26] studied pointwise pseudo-slant
submanifolds in Kenmotsu manifolds. Motivated by the above studies the present
paper deals with the study of warped product pointwise bi-slant submanifolds
of Kenmotsu manifolds. The paper is organized as follows. Section 2 is
concerned with some preliminaries. Section 3 deals with the study of pointwise
bi-slant submanifolds of Kenmotsu manifold. In section 4, we have studied
warped product pointwise bi-slant submanifolds of Kenmotsu manifold with an
interesting example. In section 5, we have found the characterization for
warped product pointwise bi-slant submanifolds of Kenmotsu manifolds. Section
6 is concerned with an inequality on warped product pointwise bi-slant
submanifold of Kenmotsu manifold whose equality case is also considered.

##  2\. Preliminaries

An odd dimensional smooth manifold $\bar{M}^{2m+1}$ is said to be an almost
contact metric manifold [5] if it admits a $(1,1)$ tensor field $\phi$, a
vector field $\xi$, an $1$-form $\eta$ and a Riemannian metric $g$ which
satisfy

(2.1) $\phi\xi=0,\ \ \ \eta(\phi X)=0,\ \ \ \phi^{2}X=-X+\eta(X)\xi,$ (2.2)
$g(\phi X,Y)=-g(X,\phi Y),\ \ \ \eta(X)=g(X,\xi),\ \ \ \eta(\xi)=1,$ (2.3)
$g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y)$

for all vector fields $X,Y$ on $\bar{M}$.
An almost contact metric manifold $\bar{M}^{2m+1}(\phi,\xi,\eta,g)$ is said to
be Kenmotsu manifold if the following conditions hold [23]:

(2.4) $\bar{\nabla}_{X}\xi=X-\eta(X)\xi,$ (2.5)
$(\bar{\nabla}_{X}\phi)(Y)=g(\phi X,Y)\xi-\eta(Y)\phi X,$

where $\bar{\nabla}$ denotes the Riemannian connection of $g$.
Let $M$ be an $n$-dimensional submanifold of a Kenmotsu manifold $\bar{M}$.
Throughout the paper we assume that the submanifold $M$ of $\bar{M}$ is
tangent to the structure vector field $\xi$.
Let $\nabla$ and $\nabla^{\bot}$ be the induced connections on the tangent
bundle $TM$ and the normal bundle $T^{\bot}M$ of $M$ respectively. Then the
Gauss and Weingarten formulae are given by

(2.6) $\bar{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y)$

and

(2.7) $\bar{\nabla}_{X}V=-A_{V}X+\nabla_{X}^{\bot}V$

for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$, where $h$ and $A_{V}$
are second fundamental form and the shape operator (corresponding to the
normal vector field $V$) respectively for the immersion of $M$ into $\bar{M}$.
The second fundamental form $h$ and the shape operator $A_{V}$ are related by
$g(h(X,Y),V)=g(A_{V}X,Y)$ for any $X,Y\in\Gamma(TM)$ and
$V\in\Gamma(T^{\bot}M)$, where g is the Riemannian metric on $\bar{M}$ as well
as on $M$.

The mean curvature $H$ of $M$ is given by $H=\frac{1}{n}\text{trace}\ h$. A
submanifold of a Kenmotsu manifold $\bar{M}$ is said to be totally umbilical
if $h(X,Y)=g(X,Y)H$ for any $X,Y\in\Gamma(TM)$. If $h(X,Y)=0$ for all
$X,Y\in\Gamma(TM)$, then $M$ is totally geodesic and if $H=0$ then $M$ is
minimal in $\bar{M}$.

Let $\\{e_{1},\cdots,e_{n}\\}$ be an orthonormal basis of the tangent bundle
$TM$ and $\\{e_{n+1},\cdots,e_{2m+1}\\}$ an orthonormal basis of the normal
bundle $T^{\bot}M$. We put

(2.8) $h_{ij}^{r}=g(h(e_{i},e_{j}),e_{r})\ \text{and}\ \
\|h\|^{2}=g(h(e_{i},e_{j}),h(e_{i},e_{j})),$

for $r\in\\{n+1,\cdots,2m+1\\}$.
For a differentiable function $f$ on $M$, the gradient $\boldsymbol{\nabla}f$
is defined by

(2.9) $g(\boldsymbol{\nabla}f,X)=Xf,$

for any $X\in\Gamma(TM)$. As a consequence, we get

(2.10) $\|\boldsymbol{\nabla}f\|^{2}=\sum_{i=1}^{n}(e_{i}(f))^{2}.$

For any $X\in\Gamma(TM)$ and $V\in\Gamma(T^{\bot}M)$, we can write

(2.11) $\displaystyle\text{(a)}\ \phi X=PX+QX,\ \ \text{(b)}\ \phi V=bV+cV$

where $PX,\ bV$ are the tangential components and $QX,\ cV$ are the normal
components.
A submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said to be
invariant if $\phi(T_{p}M)\subseteq T_{p}M$, for every $p\in M$ and anti-
invariant if $\phi T_{p}M\subseteq T^{\bot}_{p}M$, for every $p\in M$ [4].
A submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said to be
slant if for each non-zero vector $X\in T_{p}M$, the angle $\theta$ between
$\phi X$ and $T_{p}M$ is a constant, i.e. it does not depend on the choice of
$p\in M$.

######  Definition 2.1.

[29] A submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said
to be pointwise slant if for any non-zero vector $X\in T_{p}M$ at $p\in M$,
such that $X$ is not proportional to $\xi_{p}$, the angle $\theta(X)$ between
$\phi X$ and $T_{p}^{*}M=T_{p}M-\\{0\\}$ is independent of the choice of non-
zero $X\in T_{p}^{*}M$.

For pointwise slant submanifold $\theta$ is a function on $M$, which is known
as slant function of $M$. Invariant and anti-invariant submanifolds are
particular cases of pointwise slant submanifolds with slant function
$\theta=0$ and $\frac{\pi}{2}$ respectively. Also a pointwise slant
submanifold $M$ will be slant if and only if $\theta$ is constant on $M$, Thus
a pointwise slant submanifold is proper if neither $\theta=0,\frac{\pi}{2}$
nor constant.

######  Theorem 2.1.

[29] Let $M$ be a submanifold of an almost contact metric manifold $\bar{M}$
such that $\xi\in\Gamma(TM)$. Then, $M$ is pointwise slant if and only if

(2.12) $P^{2}=\cos^{2}\theta(-I+\eta\otimes\xi),$

for some real valued function $\theta$ defined on the tangent bundle $TM$ of
$M$.

If $M$ be a pointwise slant submanifold of an almost contact metric manifold
$\bar{M}$ the following relations holds:

(2.13) $\displaystyle g(PX,PY)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta\\{g(X,Y)-\eta(X)\eta(Y)\\}$ (2.14) $\displaystyle
g(QX,QY)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta\\{g(X,Y)-\eta(X)\eta(Y)\\}$ (2.15) $\displaystyle
bQX$ $\displaystyle=$ $\displaystyle\sin^{2}\theta\\{-X+\eta(X)\xi\\},\ \ \
cQX=-QPX$

for any $X,\ Y\in\Gamma(TM)$.
We now recall the following:

######  Theorem 2.2.

[16] [Hiepko’s Theorem]. Let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be two
orthogonal distribution on a Riemannian manifold $M$. Suppose that
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ both are involutive such that
$\mathcal{D}_{1}$ is a totally geodesic foliation and $\mathcal{D}_{2}$ is a
spherical foliation. Then $M$ is locally isometric to a non-trivial warped
product $M_{1}\times_{f}M_{2}$, where $M_{1}$ and $M_{2}$ are integral
manifolds of $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, respectively.

##  3\. Pointwise bi-slant submanifolds

In this section, we define and study pointwise bi-slant submanifold of
Kenmotsu manifold.

######  Definition 3.1.

[13] A submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said
to be a pointwise bi-slant submanifold if there exists a pair of orthogonal
distribution $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ of $M$ at the point $p\in
M$ such that
(a) $TM=\mathcal{D}_{1}\oplus\mathcal{D}_{2}\oplus\\{\xi\\}$,
(b) $\phi(\mathcal{D}_{1})\bot\mathcal{D}_{2}\oplus\\{\xi\\}$,
(c) the distribution $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are pointwise
slant with slant functions $\theta_{1}$ and $\theta_{2}$, respectively.

The pair $\\{\theta_{1},\theta_{2}\\}$ of slant functions is called the bi-
slant function and if $\theta_{1},\theta_{2}\neq 0,\frac{\pi}{2}$ and both
$\theta_{1},\theta_{2}$ are not constant on $M$, then $M$ is called proper
pointwise bi-slant submanifold.
If $M$ is a pointwise bi-slant submanifold of $\bar{M}$ then for any
$X\in\Gamma(TM)$ we have

(3.1) $X=T_{1}X+T_{2}X$

where $T_{1}$ and $T_{2}$ are the projections from $TM$ onto $\mathcal{D}_{1}$
and $\mathcal{D}_{2}$, respectively.
If we put $P_{1}=T_{1}\circ P$ and $P_{2}=T_{2}\circ P$ then from (3.1), we
get

(3.2) $\phi X=P_{1}X+P_{2}X+QX,$

for $X\in\Gamma(TM)$. From Theorem 2.1, we get

(3.3) $P_{i}^{2}X=\cos^{2}\theta_{i}\\{-X+\eta(X)\xi\\},\ \ X\in\Gamma(TM),\ \
i=1,2.$

For a proper pointwise bi-slant submanifold $M$ of a Kenmotsu manifold
$\bar{M}$, the normal bundle of $M$ is decomposed as

(3.4) $T^{\bot}M=Q\mathcal{D}_{1}\oplus Q\mathcal{D}_{2}\oplus\nu,$

where $\nu$ is a $\phi$-invariant normal subbundle of $M$.
Now, we find the following lemma for a pointwise bi-slant submanifold $M$ of
Kenmotsu manifold $\bar{M}$.

######  Lemma 3.1.

Let $M$ be a pointwise bi-slant submanifold of $\bar{M}$ with pointwise slant
distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ with distinct slant
function $\theta_{1}$ and $\theta_{2}$, respectively. Then

(3.5)
$\displaystyle(\sin^{2}\theta_{1}-\sin^{2}\theta_{2})g(\nabla_{X}Y,Z)=g(A_{QP_{2}Z}Y-A_{QZ}P_{1}Y,X)$
$\displaystyle+g(A_{QP_{1}Y}Z-A_{QY}P_{2}Z,X)$

and

(3.6)
$\displaystyle(\sin^{2}\theta_{2}-\sin^{2}\theta_{1})g(\nabla_{Z}W,X)=g(A_{QP_{2}W}X-A_{QW}P_{1}X,Z)$
$\displaystyle+g(A_{QP_{1}X}W-A_{QX}P_{2}W,Z)-\eta(X)g(Z,W),$

for any $X,\ Y\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and $Z,\
W\in\Gamma(\mathcal{D}_{2})$.

###### Proof.

For any $X,\ Y\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and
$Z\in\Gamma(\mathcal{D}_{2})$, we have

$\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{X}Y,\phi Z)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}\phi Y,\phi Z)-g((\bar{\nabla}_{X}\phi)Y,Z)$
$\displaystyle=$ $\displaystyle g(\bar{\nabla}_{X}P_{1}Y,\phi
Z)+g(\bar{\nabla}_{X}QY,\phi Z)-\eta(Y)g(\phi X,\phi Z)$ $\displaystyle=$
$\displaystyle-g(\phi\bar{\nabla}_{X}P_{1}Y,Z)+g(\bar{\nabla}_{X}QY,P_{2}Z)+g(\bar{\nabla}_{X}QY,QZ)-\eta(Y)g(X,Z)$
$\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{X}P_{1}^{2}Y,Z)-g(\bar{\nabla}_{X}QP_{1}Y,Z)+g((\bar{\nabla}_{X}\phi)P_{1}Y,Z)$
$\displaystyle+g(\bar{\nabla}_{X}QY,P_{2}Z)-g(\bar{\nabla}_{X}QZ,\phi
Y)+g(\bar{\nabla}_{X}QZ,P_{1}Y)-\eta(Y)g(X,Z)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{1}g(\bar{\nabla}_{X}Y,Z)-\sin
2\theta_{1}X(\theta_{1})g(Y,Z)-g(\bar{\nabla}_{X}QP_{1}Y,Z)$
$\displaystyle+g(\bar{\nabla}_{X}QY,P_{2}Z)+g(\bar{\nabla}_{X}bQZ,Y)+g(\bar{\nabla}_{X}cQZ,Y)$
$\displaystyle-g((\bar{\nabla}_{X}\phi)QZ,Y)+g(\bar{\nabla}_{X}QZ,P_{1}Y)-\eta(Y)g(X,Z)$

Using (2.5), (2.7), (2.15), orthogonality of the distributions and symmetry of
the shape operator, the above equation reduces to

$\displaystyle g(\nabla_{X}Y,Z)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{1}g(\bar{\nabla}_{X}Y,Z)+g(A_{QP_{1}Y}Z,X)-g(A_{QY}P_{2}Z,X)$
$\displaystyle+\sin^{2}\theta_{2}g(\bar{\nabla}_{X}Y,Z)+g(A_{QP_{2}Z}Y,X)-g(A_{QZ}P_{1}Y,X),$

from which the relation (3.5) follows. Also, for $X\in\Gamma(\mathcal{D}_{1})$
and $Z,\ W\in\Gamma(\mathcal{D}_{2})$, we have

$\displaystyle g(\nabla_{Z}W,X)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{Z}W,\phi X)+\eta(X)g(\bar{\nabla}_{Z}W,\xi)$
$\displaystyle=$ $\displaystyle g(\bar{\nabla}_{Z}\phi W,\phi
X)-g((\bar{\nabla}_{Z}\phi)W,\phi X)-\eta(X)g(W,\bar{\nabla}_{Z}\xi)$
$\displaystyle=$ $\displaystyle g(\bar{\nabla}_{Z}P_{2}W,\phi
X)+g(\bar{\nabla}_{Z}QW,\phi X)-\eta(X)g(Z,W)$ $\displaystyle=$
$\displaystyle-g(\phi\bar{\nabla}_{Z}P_{2}W,X)+g(\bar{\nabla}_{Z}QW,P_{1}X)$
$\displaystyle+g(\bar{\nabla}_{Z}QW,QX)-\eta(X)g(Z,W)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{Z}P_{2}^{2}W,X)-g(\bar{\nabla}_{Z}QP_{2}W,X)+g((\bar{\nabla}_{Z}\phi)P_{2}W,X)$
$\displaystyle+g(\bar{\nabla}_{Z}QW,P_{1}X)-g(\bar{\nabla}_{Z}QX,\phi W)$
$\displaystyle+g(\bar{\nabla}_{Z}QX,P_{2}W)-\eta(X)g(Z,W)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{2}g(\bar{\nabla}_{Z}W,X)-\sin
2\theta_{2}Z(\theta_{2})g(W,X)-g(\bar{\nabla}_{Z}QP_{2}W,X)$
$\displaystyle+g(\bar{\nabla}_{Z}QW,P_{1}X)+g(\bar{\nabla}_{Z}bQX,W)+g(\bar{\nabla}_{Z}cQX,W)$
$\displaystyle-g((\bar{\nabla}_{Z}\phi)QX,W)+g(\bar{\nabla}_{Z}QX,P_{2}W)-\eta(X)g(Z,W)$

Using (2.5), (2.7), (2.15), orthogonality of the distributions and symmetry of
the shape operator, the above equation reduces to

$\displaystyle g(\nabla_{Z}W,X)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{2}g(\bar{\nabla}_{Z}W,X)+g(A_{QP_{2}W}X,Z)-g(A_{QW}P_{1}X,Z)$
$\displaystyle+\sin^{2}\theta_{1}g(\bar{\nabla}_{Z}W,X)+g(A_{QP_{1}X}W,Z)$
$\displaystyle-g(A_{QX}P_{2}W,Z)-\eta(X)g(Z,W),$

from which the relation (3.6) follows. ∎

######  Corollary 3.1.

Let $M$ be a pointwise bi-slant submanifold of $\bar{M}$ with pointwise slant
distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ having distinct slant
functions $\theta_{1}$ and $\theta_{2}$, respectively. Then the distribution
$\mathcal{D}_{1}\oplus\\{\xi\\}$ defines a totally geodesic foliation if and
only if

(3.7) $g(A_{QP_{2}Z}X-A_{QZ}P_{1}X+A_{QP_{1}Y}Z-A_{QY}P_{2}Z,Y)=0,$

for any $X,\ Y\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and
$Z\in\Gamma(D_{2})$.

###### Proof.

It follows from (3.5). ∎

######  Corollary 3.2.

Let $M$ be a pointwise bi-slant submanifold of $\bar{M}$ with pointwise slant
distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ having distinct slant
functions $\theta_{1}$ and $\theta_{2}$, respectively. Then the distribution
$\mathcal{D}_{2}$ defines a totally geodesic foliation if and only if

(3.8)
$g(A_{QP_{2}W}X-A_{QW}P_{1}X,Z)+g(A_{QP_{1}X}W-A_{QX}P_{2}W,Z)=\eta(X)g(Z,W),$

for any $X\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and $Z,\
W\in\Gamma(\mathcal{D}_{2})$.

###### Proof.

It follows from (3.6) ∎

##  4\. Warped product pointwise bi-slant submanifolds

######  Definition 4.1.

[6] Let $(N_{1},g_{1})$ and $(N_{2},g_{2})$ be two Riemannian manifolds with
Riemannian metric $g_{1}$ and $g_{2}$ respectively and $f$ be a positive
definite smooth function on $N_{1}$. The warped product of $N_{1}$ and $N_{2}$
is the Riemannian manifold $N_{1}\times_{f}N_{2}=(N_{1}\times N_{2},g)$, where

(4.1) $g=g_{1}+f^{2}g_{2}.$

A warped product manifold $N_{1}\times_{f}N_{2}$ is said to be trivial if the
warping function $f$ is constant.
Let $M=N_{1}\times_{f}N_{2}$ be a warped product manifold, then we have [6]

(4.2) $\displaystyle\nabla_{U}X=\nabla_{X}U=(X\ln f)U,$

for any $X$, $Y\in\Gamma(TN_{1})$ and $U\in\Gamma(TN_{2})$.
Now we construct an example of warped product pointwise bi-slant submanifold
of Kenmotsu manifold.

######  Example 4.1.

We consider $\bar{M}=\mathbb{R}^{13}$ with the cartesian coordinates $(X_{1},\
Y_{1},\ \cdots,$
$\ X_{6},\ Y_{6},\ t)$ and with its usual Kenmotsu structure
$(\phi,\xi,\eta,g)$, given by

$\eta=dt,\ \ \xi=\frac{\partial}{\partial t},\ \
g=\eta\otimes\eta+e^{2t}\\{(dx^{i}\otimes dx^{i}+dy^{i}\otimes dy^{i})\\}$

and

$\phi\bigg{(}\displaystyle\sum_{i=1}^{6}(X_{i}\frac{\partial}{\partial
x^{i}}+Y_{i}\frac{\partial}{\partial y^{i}})+t\frac{\partial}{\partial
t}\bigg{)}=\displaystyle\sum_{i=1}^{6}(-Y_{i}\frac{\partial}{\partial
x^{i}}+X_{i}\frac{\partial}{\partial y^{i}}).$

Now, we consider a submanifold of $\mathbb{R}^{13}$ defined by the immersion
$\chi$ as follows:

$\displaystyle\chi(u,v,\theta,\phi,\,t)=(u\cos\theta,\,v\cos\phi,\,u\sin\theta,\,v\sin\phi,\,u\cos\phi,\,v\cos\theta,$
$\displaystyle
u\sin\phi,\,u\sin\theta,3\theta+2\phi,\,2\theta+3\phi,\,0,\,0,\,t).$

Then, it is easy to see that

$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}\cos\theta\frac{\partial}{\partial
x_{1}}+\sin\theta\frac{\partial}{\partial
x_{2}}+\cos\phi\frac{\partial}{\partial
x_{3}}+\sin\phi\frac{\partial}{\partial x_{4}}\bigg{)},$ $\displaystyle Z_{2}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}\cos\phi\frac{\partial}{\partial
y_{1}}+\sin\phi\frac{\partial}{\partial
y_{2}}+\cos\theta\frac{\partial}{\partial
y_{3}}+\sin\theta\frac{\partial}{\partial y_{4}}\bigg{)},$ $\displaystyle
Z_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}-u\sin\theta\frac{\partial}{\partial
x_{1}}+u\cos\theta\frac{\partial}{\partial
x_{2}}-v\sin\theta\frac{\partial}{\partial
x_{3}}+v\cos\theta\frac{\partial}{\partial x_{4}}+3\frac{\partial}{\partial
x_{5}}+2\frac{\partial}{\partial y_{5}}\bigg{)},$ $\displaystyle Z_{4}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}-v\sin\phi\frac{\partial}{\partial
y_{1}}+v\cos\phi\frac{\partial}{\partial
y_{2}}-u\sin\phi\frac{\partial}{\partial
y_{3}}+v\cos\phi\frac{\partial}{\partial y_{4}}+2\frac{\partial}{\partial
x_{5}}+3\frac{\partial}{\partial y_{5}}\bigg{)},$ $\displaystyle Z_{5}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial t}.$

form a local orthonarmal frame of $TM$. Also, we have

$\displaystyle\phi Z_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}-\cos\theta\frac{\partial}{\partial
y_{1}}-\sin\theta\frac{\partial}{\partial
y_{2}}-\cos\phi\frac{\partial}{\partial
y_{3}}-\sin\phi\frac{\partial}{\partial y_{4}}\bigg{)},$ $\displaystyle\phi
Z_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}\cos\phi\frac{\partial}{\partial
x_{1}}+\sin\phi\frac{\partial}{\partial
x_{2}}+\cos\theta\frac{\partial}{\partial
x_{3}}+\sin\theta\frac{\partial}{\partial x_{4}}\bigg{)},$ $\displaystyle\phi
Z_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}u\sin\theta\frac{\partial}{\partial
y_{1}}-u\cos\theta\frac{\partial}{\partial
y_{2}}-v\sin\theta\frac{\partial}{\partial
y_{3}}-v\cos\theta\frac{\partial}{\partial y_{4}}-3\frac{\partial}{\partial
y_{5}}+2\frac{\partial}{\partial x_{5}}\bigg{)},$ $\displaystyle\phi Z_{4}$
$\displaystyle=$
$\displaystyle\frac{1}{e^{t}}\bigg{(}-v\sin\phi\frac{\partial}{\partial
y_{1}}+v\cos\phi\frac{\partial}{\partial
y_{2}}+u\sin\phi\frac{\partial}{\partial
y_{3}}-v\cos\phi\frac{\partial}{\partial y_{4}}-2\frac{\partial}{\partial
y_{5}}+3\frac{\partial}{\partial x_{5}}\bigg{)},$ $\displaystyle\phi Z_{5}$
$\displaystyle=$ $\displaystyle 0.$

If we define $\mathcal{D}_{1}=\\{Z_{1},\ Z_{2}\\}$ and
$\mathcal{D}_{2}=\\{Z_{3},\,Z_{4}\\}$, then by simple calculations, we obtain
$g(\phi Z_{1},Z_{2})=2\cos(\theta-\phi)$ and $g(\phi Z_{3},Z_{4})=5$ and hence
the distributions $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are pointwise slant
with slant functions $\cos^{-1}[2\cos(\theta-\phi)]$ and
$\cos^{-1}(\frac{5}{\sqrt{u^{2}+v^{2}+13}})$, respectively. Also, $\xi$ is
tangent to $\mathcal{D}_{1}$. Consequently, $M$ is a proper pointwise bi-slant
submanifold of $\mathbb{R}^{13}$. Also, it is clear that both the
distributions $\mathcal{D}_{1}\oplus\\{\xi\\}$ and $\mathcal{D}_{2}$ are
integrable. If we denote the integral manifolds of
$\mathcal{D}_{1}\oplus\\{\xi\\}$ and $\mathcal{D}_{2}$ by $M_{1}$ and $M_{2}$,
respectively then the metric tensor $g_{M}$ of $M$ is given by

$g_{M}=2(du^{2}+dv^{2})+dt^{2}+(u^{2}+v^{2}+13)(d\theta^{2}+d\phi^{2}),$

where $g_{M_{1}}=2(du^{2}+dv^{2})+dt^{2}$ is the metric tensor of $M_{1}$ and
$g_{M_{2}}=(u^{2}+v^{2}+13)(d\theta^{2}+d\phi^{2})$ is the metric tensor of
$M_{2}$. Thus $M=M_{1}\times_{f}M_{2}$ is a warped product pointwise bi-slant
submanifold of Kenmotsu manifold with warping function
$f=\sqrt{u^{2}+v^{2}+13}$.

Now we have the following lemmas:

######  Lemma 4.1.

Let $M=M_{1}\times_{f}M_{2}$ be a warped product pointwise bi-slant
submanifold of a Kenmotsu manifold $\bar{M}$ with distinct slant functions
$\theta_{1}$ and $\theta_{2}$ such that $\xi\in\Gamma(TM_{1})$. Then

(4.3) $\displaystyle g(h(X,W),QP_{2}Z)-g(h(X,P_{2}Z),QW)$ $\displaystyle=$
$\displaystyle\sin\theta_{2}X(\theta_{2})g(Z,W)$

and

(4.4) $\displaystyle g(h(X,Z),QW)-g(h(X,W),QZ)$ $\displaystyle=$
$\displaystyle\tan\theta_{2}X(\theta_{2})g(P_{2}Z,W),$

for any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$.

###### Proof.

For any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$, we have from (4.2)
that

(4.5) $g(\bar{\nabla}_{X}Z,W)=g(\nabla_{X}Z,W)=(X\ln f)g(Z,W).$

Also, we have

$\displaystyle g(\bar{\nabla}_{X}Z,W)$ $\displaystyle=$ $\displaystyle
g(\phi\bar{\nabla}_{X}Z,\phi W)+\eta(W)g(\bar{\nabla}_{X}Z,\xi)$
$\displaystyle=$ $\displaystyle g(\bar{\nabla}_{X}\phi Z,\phi
W)-g((\bar{\nabla}_{X}\phi)Z,W)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}P_{2}Z,P_{2}W)+g(\bar{\nabla}_{X}P_{2}Z,QW)+g(\bar{\nabla}_{X}QZ,\phi
W)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}P_{2}Z,P_{2}W)+g(\bar{\nabla}_{X}P_{2}Z,QW)-g(\bar{\nabla}_{X}bQZ,W)$
$\displaystyle-g(\bar{\nabla}_{X}cQZ,W)+g(\bar{\nabla}_{X}QP_{2}Z,W).$

Using (2.5)-(2.7) and (2.15), the above equation yields

$\displaystyle
g(\bar{\nabla}_{X}Z,W)=g(\bar{\nabla}_{X}P_{2}Z,P_{2}W)+g(h(X,P_{2}Z),QW)$
$\displaystyle+\sin^{2}\theta_{2}g(\bar{\nabla}_{X}Z,W)+\sin\theta_{2}X(\theta_{2})g(Z,W)+g(h(X,W),QP_{2}Z).$

Using (4.2) in the above equation, we obtain

(4.6) $\displaystyle g(\bar{\nabla}_{X}Z,W)=(X\ln
f)g(P_{2}Z,P_{2}W)+g(h(X,P_{2}Z),QW)$ $\displaystyle+\sin^{2}\theta_{2}(X\ln
f)g(Z,W)+\sin\theta_{2}X(\theta_{2})g(Z,W)+g(h(X,W),QP_{2}Z).$

Using (2.12) and (4.5), (4.6) yields

(4.7) $\displaystyle(X\ln f)g(Z,W)=\cos^{2}\theta_{2}X\ln
fg(Z,W)+g(h(X,P_{2}Z),QW)$ $\displaystyle+\sin^{2}\theta_{2}(X\ln
f)g(Z,W)+\sin\theta_{2}X(\theta_{2})g(Z,W)+g(h(X,W),QP_{2}Z).$

The relation (4.3) follows from (4.7) and (4.4) obtained from (4.3) by
interchanging $Z$ by $P_{2}Z$. ∎

######  Lemma 4.2.

Let $M=M_{1}\times_{f}M_{2}$ be a warped product pointwise bi-slant
submanifold of a Kenmotsu manifold $\bar{M}$ such that $\xi\in\Gamma(TM_{1})$
with distinct slant functions $\theta_{1},\theta_{2}$. Then

(4.8) $\displaystyle g(h(X,W),QP_{2}Z)-g(h(X,P_{2}Z),QW)=$ $\displaystyle
2\cos^{2}\theta_{2}\bigg{\\{}(X\ln f)-\eta(X)\bigg{\\}}g(Z,W),$

for any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$.

###### Proof.

For any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$, we have

$\displaystyle g(h(X,Z),QW)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}X,QW)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}X,\phi W)-g(\bar{\nabla}_{Z}X,P_{2}W)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{Z}\phi
X,W)+g((\bar{\nabla}_{Z}\phi)X,W)-g(\bar{\nabla}_{Z}X,P_{2}W)$
$\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{Z}P_{1}X,W)-g(\bar{\nabla}_{Z}QX,W)$
$\displaystyle-\eta(X)g(\phi Z,W)-g(\bar{\nabla}_{Z}X,P_{2}W)$

Using (4.2) in the above equation, we get

$\displaystyle g(h(X,Z),QW)$ $\displaystyle=$ $\displaystyle-(P_{1}X\ln
f)g(Z,W)+g(h(Z,W),QX)$ $\displaystyle+\bigg{\\{}(X\ln
f)-\eta(X)\bigg{\\}}g(P_{2}Z,W).$

By polarization, we obtain

$\displaystyle g(h(X,W),QZ)$ $\displaystyle=$ $\displaystyle-(P_{1}X\ln
f)g(W,Z)+g(h(Z,W),QX)$ $\displaystyle+\bigg{\\{}(X\ln
f)-\eta(X)\bigg{\\}}g(P_{2}W,Z).$

Subtracting (4) from (4), we get

(4.11) $\displaystyle g(h(X,Z),QW)-g(h(X,W),QZ)$ $\displaystyle=2[(X\ln
f)-\eta(X)]g(P_{2}Z,W).$

Interchanging $Z$ by $P_{2}Z$ in (4.11), we get (4.8). ∎

######  Theorem 4.2.

There exists a proper warped product pointwiase bi-slant submanifold
$M=M_{1}\times_{f}M_{2}$ of a Kenmotsu manifold $\bar{M}$, with distinct slant
functions $\theta_{1}$ and $\theta_{2}$, if and only if
$\tan\theta_{2}X(\theta_{2})-\eta(X)\neq 0$.

###### Proof.

For any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$ from (4.3) and (4.8)
, we have

(4.12) $2\cos^{2}\theta_{2}[(X\ln f)-\eta(X)]g(Z,W)=-\sin
2\theta_{2}X(\theta_{2})g(Z,W).$

Since $M$ is proper so $\theta_{2}\neq\frac{\pi}{2}$ and hence from (4.12), we
get $[(X\ln f)-\eta(X)+\tan\theta_{2}X(\theta_{2})]g(Z,W)=0$, which implies
that $(X\ln f)=\eta(X)-\tan(\theta_{2})X(\theta_{2})$. This proves the
theorem. ∎

A warped product submanifold $M=M_{1}\times_{f}M_{2}$ of a Kenmotsu manifold
$\bar{M}$ is known to be mixed totally geodesic if $h(X,Z)=0$, for any
$X\in\Gamma(TM_{1})$ and $Z\in\Gamma(TM_{2})$.

######  Theorem 4.3.

Let $M=M_{1}\times_{f}M_{2}$ be a warped product pointwise bi-slant
submanifold of a Kenmotsu manifold $\bar{M}$, withd istinct slant functions
$\theta_{1}$ and $\theta_{2}$. $M$ is a mixed totally geodesic warped product
submanifold, then one of the following two cases holds:
(i) $\theta_{2}=\frac{\pi}{2}$, i.e., $M$ is a warped product pointwise pseudo
slant submanifold of the form $M_{1}\times_{f}M_{\bot}$, where $M_{\bot}$ is
an anti-invariant submanifold $\bar{M}$.
(ii) or $(X\ln f)=\eta(X)$.

######  Corollary 4.1.

For a proper warped product mixed totally geodesic pointwise bi-slant
submanifold $M=M_{1}\times fM_{2}$ of a Kenmotsu manifold $\bar{M}$ such that
$\xi\in\Gamma(TM_{1})$, $(\xi\ln f)=1$.

###### Proof.

If $M$ be mixed totally geodesic then from (4.8) we get
$\cos^{2}\theta_{2}[(X\ln f)-\eta(X)]g(Z,W)=0$. From which we get either
$\cos^{2}\theta_{2}=0$ i.e., $\theta_{2}=\frac{\pi}{2}$ or $(X\ln f)=\eta(X)$
and hence the theorem is proved. ∎

######  Lemma 4.3.

Let $M=M_{1}\times_{f}M_{2}$ be a warped product pointwise bi-slant
submanifold of a Kenmotsu manifold $\bar{M}$, such that $\xi$ is tangent to
$M_{1}$, with slant functions $\theta_{1}$ and $\theta_{2}$. Then we have

(4.13) $g(h(X,Y),QZ)=g(h(X,Z),QY),$ (4.14) $\displaystyle
g(h(Z,W),QX)-g(h(X,Z),QW)$ $\displaystyle=(P_{1}X\ln f)g(Z,W)+[(X\ln
f)-\eta(X)]g(Z,P_{2}W),$ (4.15) $\displaystyle
g(h(Z,W),QP_{1}X)-g(h(P_{1}X,Z),QW)$ $\displaystyle=(P_{1}X\ln
f)g(Z,P_{2}W)-\cos^{2}\theta_{1}[(X\ln f)-\eta(X)]g(Z,W),$ (4.16)
$\displaystyle g(h(Z,P_{2}W),QX)-g(h(X,Z),QP_{2}W)$ $\displaystyle=(P_{1}X\ln
f)g(Z,P_{2}W)-\cos^{2}\theta_{2}[(X\ln f)-\eta(X)]g(Z,W),$ (4.17)
$\displaystyle g(h(Z,W),QP_{1}X)-g(h(P_{1}X,Z),QW)+g(h(X,Z),QP_{2}W)$
$\displaystyle-g(h(Z,P_{2}W),QX)=(\cos^{2}\theta_{2}-\cos^{2}\theta_{1})[(X\ln
f)-\eta(X)]g(Z,W).$

for any $X,\ Y\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$.

###### Proof.

For any $X,\ Y\in\Gamma(TM_{1})$ and $Z\in\Gamma(TM_{2})$, we have

$\displaystyle g(h(X,Y),QZ)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{X}Y,\phi Z)-g(\bar{\nabla}_{X}Y,P_{2}Z)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{X}\phi
Y,Z)+g((\bar{\nabla}_{X}\phi)Y,Z)+g(\bar{\nabla}_{X}P_{2}Z,Y).$

By virtue of (2.5), (3.2) yields

$\displaystyle g(h(X,Y),QZ)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{X}P_{1}Y,Z)-g(\bar{\nabla}_{X}QY,Z)-\eta(Y)g(\phi
X,Z)$ $\displaystyle+g(\bar{\nabla}_{X}P_{2}Z,Y)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{X}P_{1}Y,Z)+g(h(X,Z),QY)-\eta(Y)g(\phi X,Z)$
$\displaystyle+g(\bar{\nabla}_{X}P_{2}Z,Y)$

Using (4.2) in the above equation, we get

$\displaystyle g(h(X,Y),QZ)$ $\displaystyle=$ $\displaystyle(X\ln
f)g(P_{1}Y,Z)+g(h(X,Z),QY)$ $\displaystyle-\eta(Y)g(P_{1}X,Z)+(X\ln
f)g(Y,P_{2}Z),$

from which the relation (4.13) follows.
Also for any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$, we have

$\displaystyle g(h(Z,W),QX)$ $\displaystyle=$ $\displaystyle
g(\bar{\nabla}_{Z}W,\phi X)-g(\bar{\nabla}_{Z}W,P_{1}X)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{Z}\phi
W,X)+g((\bar{\nabla}_{Z}\phi)W,X)+g(\bar{\nabla}_{Z}P_{1}X,W)$

By virtue of (2.5) and (3.2), the above equation yields

$\displaystyle g(h(Z,W),QX)$ $\displaystyle=$
$\displaystyle-g(\bar{\nabla}_{Z}X,P_{2}W)-g(\bar{\nabla}_{Z}QW,X)$
$\displaystyle+\eta(X)g(\phi Z,W)+g(\bar{\nabla}_{Z}P_{1}X,W).$

Using (2.7) and (4.2) in (4), we get (4.14). The relations (4.15) and (4.16)
can be derived from (4.14) by replacing $X$ by $P_{1}X$ and $W$ by $P_{2}W$,
respectively. Subtracting (4.16) from (4.15), we get (4.17). ∎

Now, interchanging $W$ by $P_{2}W$ in (4.15), we can derive

(4.20) $\displaystyle g(h(Z,P_{2}W),QP_{1}X)-g(h(P_{1}X,Z),QP_{2}W)$
$\displaystyle=-\cos^{2}\theta_{2}(P_{1}X\ln
f)g(Z,W)-\cos^{2}\theta_{1}\bigg{\\{}(X\ln f)-\eta(X)\bigg{\\}}g(Z,P_{2}W).$

If we interchange $Z$ by $P_{2}Z$ in (4.14) and (4.15) then, we obtain

(4.21) $\displaystyle g(h(P_{2}Z,W),QX)-g(h(X,P_{2}Z),QW)$
$\displaystyle=(P_{1}X\ln f)g(P_{2}Z,W)+\cos^{2}\theta_{2}[(X\ln
f)-\eta(X)]g(Z,W)$

and

(4.22) $\displaystyle g(h(P_{2}Z,W),QP_{1}X)-g(h(P_{1}X,P_{2}Z),QW)$
$\displaystyle=\cos^{2}\theta_{2}(P_{1}X\ln f)g(Z,W)-\cos^{2}\theta_{1}[(X\ln
f)-\eta(X)]g(P_{2}Z,W).$

Interchanging $W$ by $P_{2}W$ in (4.21) and (4.22), we get

(4.23) $\displaystyle g(h(P_{2}Z,P_{2}W),QX)-g(h(X,P_{2}Z),QP_{2}W)$
$\displaystyle\cos^{2}\theta_{2}(P_{1}X\ln f)g(Z,W)-\cos^{2}\theta_{2}[(X\ln
f)-\eta(X)]g(P_{2}Z,W).$

and

$\displaystyle g(h(P_{2}Z,P_{2}W),QP_{1}X)-g(h(P_{1}X,P_{2}Z),QP_{2}W)$
$\displaystyle=\cos^{2}\theta_{2}(P_{1}X\ln
f)g(Z,P_{2}W)-\cos^{2}\theta_{1}\cos^{2}\theta_{2}[(X\ln f)-\eta(X)]g(Z,W).$

##  5\. characterization of warped product pointwise bi-slant submanifold

In this section we found the characterization for a warped product pointwise
bi-slant submanifold of Kenmotsu manifold.

######  Theorem 5.1.

Let $M$ be a proper pointwise bi-slant submanifold of a Kenmotsu manifold
$\bar{M}$ with pointwise slant distributions $\mathcal{D}_{1}\oplus\\{\xi\\}$
and $\mathcal{D}_{2}$ . Then $M$ is locally a wraprd product submanifold of
the form $M_{1}\times_{f}M_{2}$, where $M_{1}$ and $M_{2}$ are pointwise slant
submanifolds with distinct slant functions $\theta_{1}$, $\theta_{2}$, if and
only if the shape operator of $M$ satisfies

(5.1) $\displaystyle A_{QP_{1}X}Z-A_{QZ}P_{1}X+A_{QP_{2}Z}X-A_{QX}P_{2}Z$
$\displaystyle=(\cos^{2}\theta_{2}-\cos^{2}\theta_{1})[(X\mu)-\eta(X)]Z$

for any $X\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and
$Z\in\Gamma(\mathcal{D}_{2})$ and for some function $\mu$ on $M$ satisfying
$(W\mu)=0$, for any $W\in\Gamma(\mathcal{D}_{2})$.

###### Proof.

Let $M=M_{1}\times_{f}M_{2}$ be a warped product submanifold of $\bar{M}$.
Then for any $X,\ Y\in\Gamma(TM_{1})$ and $Z\in\Gamma(TM_{2})$, we have from
(4.13) that

(5.2) $g(A_{QY}Z-A_{QZ}Y,X)=0.$

Interchanging $Y$ by $P_{1}Y$ in (5.2), we obtain

(5.3) $g(A_{QP_{1}Y}Z-A_{QZ}P_{1}Y,X)=0$

Again interchanging $Z$ by $P_{2}Z$, (5.2) yields

(5.4) $g(A_{QY}P_{2}Z-A_{QP_{2}Z}Y,X)=0.$

Subtracting (5.4) from (5.3), we get

(5.5) $\displaystyle
g(A_{QP_{1}X}Z-A_{QZ}P_{1}X+A_{QP_{2}Z}X-A_{QX}P_{2}Z,X)=0.$

From (4.16) and (5.5), we get (5.1)
Conversely, let $M$ be a proper pointwise bi-slant subamnifold of $\bar{M}$.
Then for any $X,\ Y\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and
$Z\in\Gamma(\mathcal{D}_{2})$, we have from (3.5) and (5.1) that

(5.6) $\displaystyle(\sin^{2}\theta_{1}-\sin^{2}\theta_{)}g(\nabla_{X}Y,Z)=$
$\displaystyle(\cos^{2}\theta_{2}-\cos^{2}\theta_{1})[(X\mu)-\eta(X)]g(X,Z)=0.$

Since $\theta_{1}\neq\theta_{2}$ , the leaves of the distribution
$\mathcal{D}_{1}\oplus\\{\xi\\}$ are totally geodesic in $M$. Also, for any
$X\in\Gamma(\mathcal{D}_{1}\oplus\\{\xi\\})$ and $Z,\
W\in\Gamma(\mathcal{D}_{2})$, we have from (3.6) and (5.1) that

(5.7)
$\displaystyle(\sin^{2}\theta_{2}-\sin^{2}\theta_{1})g(\nabla_{Z}W,X)=(\cos^{2}\theta_{2}-\cos^{2}\theta_{1})$
$\displaystyle\\{(X\mu)-\eta(X)\\}g(Z,W)-\eta(X)g(Z,W)$

Using trigonometric identities on (5.7), we get

(5.8) $g(\nabla_{Z}W,X)=-(X\mu)g(Z,W).$

By polarization, we find

(5.9) $g(\nabla_{W}Z,X)=-(X\mu)g(Z,W).$

From (5.8) and (5.9), we obtain $g([Z,W],X)=0$ and hence the distribution
$\mathcal{D}_{2}$ is integrable. We consider a leaf $M_{2}$ of
$\mathcal{D}_{2}$ and $h_{2}$ be the second fundamental form of $M_{2}$ in
$M$. Then from (5.8), we obtain

(5.10) $g(h_{2}(Z,W),X)=g(\nabla_{Z}W,X)=-(X\mu)g(Z,W).$

Therefore, we get $h_{2}(Z,W)=-\boldsymbol{\nabla}\mu g(Z,W)$, where
$\boldsymbol{\nabla}\mu$ is the gradient of $\mu$ and hence the leaf $M_{2}$
is totally umbilical in $M$ with mean curvature vector
$H_{2}=-\boldsymbol{\nabla}\mu$. Since $W(\mu)=0$ for any
$W\in\Gamma(\mathcal{D}_{2})$, we can easily obtain $H_{2}$ is parallel
corresponding to the normal connection $D^{\theta_{2}}$ of $M_{2}$ in $M$
[36]. Thus $M_{2}$ is an extrinsic sphere in $M$. Therefore by Theorem 2.2, we
conclude that $M$ is locally a warped product submanifold. Thus the proof is
complete. ∎

##  6\. An inequality for warped product pointwise bi-slant submanifold

In this section, we establish a general sharp geometric inequality for a mixed
totally geodesic proper pointwise bi-slant warped product submanifold of the
form $M_{1}\times_{f}M_{2}$ such that $\xi\in\Gamma(TM_{1})$ of a Kenmotsu
manifold $\bar{M}$. Let $M_{1}\times_{f}M_{2}$ be a warped product mixed
totally geodesic proper pointwise bi-slant submanifold of a Kenmotsu manifold
$\bar{M}$ such that $\xi\in\Gamma(TM_{1})$. Put $dim\bar{M}=2m+1$, $\dim
M_{1}=2p+1$, $dimM_{2}=2q$ and $dimM=n=2p+2q+1$. Let $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ be the tangent bundles of $M_{1}$ and $M_{2}$, respectively.
Assume that
$\\{e_{1},\cdots,e_{p},e_{p+1}=\sec\theta_{1}P_{1}e_{1},\cdots,e_{2p}=\sec\theta_{1}P_{1}e_{p},e_{2p+1}=\xi\\}$
is a local orthonormal frame of $\mathcal{D}_{1}$,
$\\{e_{2p+2}=e_{1}^{*},\cdots,e_{2p+q+1}=e_{q}^{*},e_{2p+q+2}=e_{q+1}^{*}=\sec\theta_{2}P_{2}e_{1}^{*},\cdots,e_{2p+2q+1}=e_{2q}^{*}=\sec\theta_{2}P_{2}e_{q}^{*}\\}$
is a local orthonormal frame of $\mathcal{D}_{2}$. Then
$\\{e_{n+1}=\hat{e}_{1}=\csc\theta_{1}Qe_{1},\cdots,e_{n+p}=\hat{e}_{p}=\csc\theta_{1}Qe_{p},\\\
e_{n+p+1}=\hat{e}_{p+1}=\csc\theta_{1}\sec\theta_{1}QP_{1}e_{1},\cdots,e_{n+2p}=\hat{e}_{2p}=\csc\theta_{1}\sec\theta_{1}QP_{1}e_{p}\\}$,
$\\{e_{n+2p+1}=\tilde{e}_{1}=\csc\theta_{2}Qe^{*}_{1},\cdots,e_{n+2p+q}=\tilde{e}_{q}=\csc\theta_{2}Qe^{*}_{q},e_{n+2p+q+1}=\tilde{e}_{q+1}=\csc\theta_{2}\sec\theta_{2}QP_{2}e^{*}_{1},\cdots,e_{2n-1}=\tilde{e}_{2q}=\csc\theta_{2}\sec\theta_{2}QP_{2}e^{*}_{1}\\}$
and $\\{e_{2n},\cdots,e_{2m+1}\\}$ are local orthonormal frames of
$Q\mathcal{D}_{1},\ Q\mathcal{D}_{2}$ and $\nu$ respectively.

######  Theorem 6.1.

Let $M_{1}\times_{f}M_{2}$ be a warped product mixed totally geodesic proper
pointwise bi-slant submanifold of a Kenmotsu manifold $\bar{M}$. Then we have
(i) The squared norm of the second fundamental form $h$ of $M$ satisfies

(6.1) $\displaystyle\qquad\|h\|^{2}\geq
2q\csc^{2}\theta_{1}\bigg{(}\cos^{2}\theta_{1}+\cos^{2}\theta_{2}\bigg{)}\bigg{\\{}\|\boldsymbol{\nabla}^{\theta_{1}}lnf\|^{2}-1-\sum_{r=1}^{p}(e_{r}lnf)^{2}\bigg{\\}},$

where $\boldsymbol{\nabla}^{\theta_{1}}\ln f$ denotes the gradient of $\ln f$
along $M_{1}$, $2q=$ dim $M_{2}$ and $\theta_{1},\ \theta_{2}$ are the slant
angles of $M_{1}$ and $M_{2}$, respectively.
(ii) If the equality sign of (6.1) holds, then $M_{1}$ is totally geodesic and
$M_{2}$ is totally umbilical in $\bar{M}$.

###### Proof.

From the definition of $h$, we have

$\|h\|^{2}=\sum_{i,j=1}^{n}g(h(e_{i},e_{j}),h(e_{i},e_{j}))=\sum_{r=n+1}^{2m+1}\sum_{i,j=1}^{n}g(h(e_{i},e_{j}),e_{r})^{2}.$

Now we decompose the above relation for the frames of $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$, as follows

$\displaystyle\|h\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{r=n+1}^{2m+1}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),e_{r})^{2}+2\sum_{r=n+1}^{2m+1}\sum_{i=1}^{2p+1}\sum_{j=1}^{2q}g(h(e_{i},e^{*}_{j}),e_{r})^{2}$
$\displaystyle+\sum_{r=n+1}^{2m}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),e_{r})^{2}$

Since $M$ is mixed totally geodesic, the second term in the right hand side of
(6) is zero and decomposing the remaining terms according to (3.4), we obtain

$\displaystyle\|h\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{r=n+1}^{n+2q}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),e_{r})^{2}+\sum_{r=n+2q+1}^{2n-1}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),e_{r})^{2}$
$\displaystyle+\sum_{r=2n}^{2m+1}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),e_{r})^{2}+\sum_{r=n+1}^{n+2p}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),e_{r})^{2}$
$\displaystyle+\sum_{r=n+2p+1}^{2n-1}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),e_{r})^{2}+\sum_{r=2n}^{2m+1}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),e_{r})^{2}.$

Next by removing the $\nu$ components in (6), we have

$\displaystyle\|h\|^{2}\geq$
$\displaystyle\sum_{r=1}^{2q}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),\widetilde{e}_{r})^{2}+\sum_{r=1}^{2p}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),\hat{e}_{r})^{2}$
$\displaystyle+\sum_{r=1}^{2p}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),\widetilde{e}_{r})^{2}+\sum_{r=1}^{2q}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),\hat{e}_{r}).$

Since we could not find any relation for $g(h(e_{i},e_{j}),\hat{e}_{r})$ for
any $i,j=1,2,\cdots,2p+1$, $r=1,2,\cdots,2p$. And
$g(h(e^{*}_{i},e^{*}_{j}),\widetilde{e}_{r})$, for any $i,j,r=1,2,\cdots,2q$
therefore, we leave the $2$nd and $4$th terms and get

(6.5)
$\|h\|^{2}\geq\sum_{r=1}^{2q}\sum_{i,j=1}^{2p+1}g(h(e_{i},e_{j}),\widetilde{e}_{r})^{2}+\sum_{r=1}^{2p}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),\hat{e}_{r})^{2}.$

In view of (4.13), (6.5) yields

(6.6)
$\|h\|^{2}\geq\sum_{r=1}^{2q}\sum_{i,j=1}^{2p+1}g(h(e_{i},e^{*}_{j}),\tilde{e}_{r})^{2}+\sum_{r=1}^{2p}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),\hat{e}_{r})^{2}.$

Since $M$ is mixed totally geodesic, we have

(6.7) $g(h(e_{i},e_{j}),e_{r})=0,$

for every $i,\ j=1,\cdots,2p+1,\ j=n+1,\cdots,2q$.
By virtue of (6.7),we have from (6.6) that

(6.8)
$\|h\|^{2}\geq\sum_{r=1}^{2p}\sum_{i,j=1}^{2q}g(h(e^{*}_{i},e^{*}_{j}),\hat{e}_{r})^{2}.$

Thus, by using the orthonormal frame fields of $Q\mathcal{D}_{1}$ and
$Q\mathcal{D}_{2}$, we derive

(6.9)
$\displaystyle\|h\|^{2}\geq\csc^{2}\theta_{1}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(e^{*}_{i},e^{*}_{j}),Qe_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}{\theta}_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(P_{2}e^{*}_{i},e^{*}_{j}),Qe_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}{\theta}_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(e^{*}_{i},P_{2}e^{*}_{j}),Qe_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{4}{\theta}_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(P_{2}e^{*}_{i},P_{2}e^{*}_{j}),Qe_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}{\theta}_{1}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(e^{*}_{i},e^{*}_{j}),QP_{1}e_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}{\theta}_{1}\sec^{2}{\theta}_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(P_{2}e^{*}_{i},e^{*}_{j}),QP_{1}e_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}{\theta}_{1}\sec^{2}{\theta}_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(Pe^{*}_{i},P_{2}e^{*}_{j}),QP_{1}e_{r})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}{\theta}_{1}\sec^{4}{\theta}_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}g(h(P_{2}e^{*}_{i},P_{2}e^{*}_{j}),QP_{1}e_{r})^{2}.$

Using (4.14)-(4.16), (4.20)-(4) and (6.7) in (6.9), we get

$\displaystyle\|h\|^{2}\geq$
$\displaystyle\csc^{2}\theta_{1}\sum_{r=1}^{p}\sum_{i,j=1}^{q}(P_{1}e_{r}lnf)^{2}g(e^{*}_{i},e^{*}_{j})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sum_{r=1}^{p}\sum_{i,j=1}^{q}(P_{1}e_{r}lnf)^{2}g(e^{*}_{i},e^{*}_{j})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}\theta_{1}\cos^{2}\theta_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}(P_{1}e_{r}lnf)^{2}g(e^{*}_{i},e^{*}_{j})^{2}$
$\displaystyle+\csc^{2}\theta_{1}\sec^{2}\theta_{1}\cos^{2}\theta_{2}\sum_{r=1}^{p}\sum_{i,j=1}^{q}(P_{1}e_{r}lnf)^{2}g(e^{*}_{i},e^{*}_{j})^{2}$
$\displaystyle=$ $\displaystyle
2q\csc^{2}\theta_{1}[1+\sec^{2}\theta_{1}\cos^{2}\theta_{2}]\sum_{r=1}^{p}(P_{1}e_{r}lnf)^{2}.$

Now,

$\displaystyle\|\boldsymbol{\nabla}^{\theta_{1}}\ln f\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{2p+1}(e_{r}lnf)^{2}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{p}(e_{r}lnf)^{2}+\sum_{r=1}^{p}(\sec\theta_{1}P_{1}e_{r}lnf)^{2}+(\xi
lnf)^{2}$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{p}(e_{r}lnf)^{2}+\sec^{2}\theta_{1}\sum_{r=1}^{p}(P_{1}e_{r}lnf)^{2}+1.$
$\displaystyle\ \ \text{as}\ \ (\xi lnf)=1$

From the above relation, we get

(6.11)
$\sum_{r=1}^{p}(P_{1}e_{r}lnf)^{2}=\cos^{2}\theta_{1}\left[\|\boldsymbol{\nabla}^{\theta_{1}}lnf\|^{2}-1-\sum_{r=1}^{p}(e_{r}lnf)^{2}\right].$

Using (6.11) in (6), we get (6.1).
If equality of (6.1) holds, then from the leaving third term of (6), we get
$g(h(\mathcal{D}_{1},\mathcal{D}_{1}),\nu)=0$, which implies that
$h(\mathcal{D}_{1},\mathcal{D}_{1})\bot\nu$, i.e.,

(6.12) $h(\mathcal{D}_{1},\mathcal{D}_{1})\subset Q\mathcal{D}_{1}\oplus
Q\mathcal{D}_{2}.$

and from the leaving second term in (6), we find
$g(h(\mathcal{D}_{1},\mathcal{D}_{1}),Q\mathcal{D}_{1})=0$, which implies that
$h(\mathcal{D}_{1},\mathcal{D}_{1})\bot Q\mathcal{D}_{1}$, i.e.,

(6.13) $h(\mathcal{D}_{1},\mathcal{D}_{1})\subset Q\mathcal{D}_{2}\oplus\nu.$

Also, from (4.13) and (6.7), we get $h(\mathcal{D}_{1},\mathcal{D}_{1})\bot
Q\mathcal{D}_{2}$, i.e.,

(6.14) $h(\mathcal{D}_{1},\mathcal{D}_{1})\subset Q\mathcal{D}_{1}\oplus\nu.$

From (6.12)-(6.14), we conclude that

(6.15) $h(\mathcal{D}_{1},\mathcal{D}_{1})=0.$

Since $M_{1}$ is totally geodesic in $M$ ([6], [8]), from (6.15), we can say
that $M_{1}$ is totally geodesic in $\bar{M}$.
Similarly, for the leaving sixth term in (6), we get
$g(h(\mathcal{D}_{2},\mathcal{D}_{2}),\nu)=0$, which implies that
$h(\mathcal{D}_{2},\mathcal{D}_{2})\bot\nu$, i.e.,

(6.16) $h(\mathcal{D}_{2},\mathcal{D}_{2})\subset Q\mathcal{D}_{2}\oplus\nu.$

Also, for the leaving fourth term in (6), we obtain
$g(h(\mathcal{D}_{2},\mathcal{D}_{2}),Q\mathcal{D}_{2})=0$, which implies that
$h(\mathcal{D}_{2},\mathcal{D}_{2})\bot Q\mathcal{D}_{2}$ i.e.,

(6.17) $h(\mathcal{D}_{2},\mathcal{D}_{2})\subset Q\mathcal{D}_{1}.$

From (6.16) and (6.17), we get

(6.18) $h(\mathcal{D}_{2},\mathcal{D}_{2})\subset Q\mathcal{D}_{1}.$

Moreover, using (6.7) in (6.15), we obtain

(6.19) $g(h(Z,W),QX)=(P_{1}X\ln f)g(Z,W)+\\{(X\ln f)-\eta(X)\\}g(Z,P_{2}W),$

for any $X\in\Gamma(TM_{1})$ and $Z,\ W\in\Gamma(TM_{2})$. By Polarization of
(6.19), we obtain

(6.20) $g(h(Z,W),QX)=(P_{1}X\ln f)g(Z,W)+[(X\ln f)-\eta(X)]g(W,P_{2}Z).$

Subtracting (6.20) from (6.19), we get

(6.21) $g(h(Z,W),QX)=(P_{1}X\ln f)g(Z,W).$

From (6.18), (6.21) and the fact that $M_{2}$ is totally umbilical in $M$
([6], [8]), we can conclude that $M_{2}$ is totally umbilical in $\bar{M}$.
Thus the proof of the theorem is complete. ∎

Acknowledgement: The first two authors (S. K. Hui and J. Roy) gratefully
acknowledges to the SERB (Project No: EMR/2015/002302), Govt. of India for
financial assistance of the work.

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Shyamal Kumar Hui
Department of Mathematics, The University of Burdwan,
Burdwan, 713104, West Bengal, India.
E-mail<EMAIL_ADDRESS>

Joydeb Roy
Department of Mathematics, The University of Burdwan,
Burdwan, 713104, West Bengal, India.
<EMAIL_ADDRESS>

Tanumoy Pal
Department of Mathematics, The University of Burdwan,
Burdwan, 713104, West Bengal, India.
E-mail<EMAIL_ADDRESS>


By the $\mathbb{R}$-local bounded geometry of $\tilde{\phi}_{l}$ and the
compactness of $I$, there are $C_{4},C_{5}>0$ such that, for all $t\in I$,

(7.3.3) $\big{|}\tilde{\phi}_{l,z}^{t*}\big{|}\leq C_{4}\;,\quad|f(t)|\leq
C_{5}\;.$

Assume $I\subset\mathbb{R}^{\times}$ and $I\cap\mathcal{P}_{l}=\emptyset$ to
prove i. Thus

$\\{\,(p,\phi_{l}^{t}(p))\mid p\in M^{1}_{l,\epsilon^{\prime\prime}},\ t\in
I\,\\}$

is a compact subset of $(M^{1}_{l})^{2}\setminus\Delta_{l}$. By Lemma 3.1.1
ii, there is some $C_{6}>0$ such that
$d_{\mathcal{F}^{1}_{l}}(\phi_{l}^{t}(p),p)\geq C_{6}$ for all $p\in
M^{1}_{l,\epsilon^{\prime\prime}}$ and $t\in I$. So, for all
$\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$ and
$\gamma\in\Gamma_{l}$ with $-h_{l}(\gamma)\in I$,

(7.3.4)
$d_{\widetilde{\mathcal{F}}^{1}_{l}}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\geq
C_{6}\;.$

Take some $C_{7}>0$ such that, for all $\gamma\in\Gamma_{l}$ with
$-h_{l}(\gamma)\in I$,

$C_{7}|\gamma|\leq\begin{cases}C_{3}^{-1}|\gamma|-R&\text{if
$|\gamma|>C_{3}R$}\\\ C_{6}&\text{if $|\gamma|\leq C_{3}R$}\;.\end{cases}$

Since $\tilde{d}_{l}\leq d_{\widetilde{\mathcal{F}}^{1}_{l}}$ (Section 3.1.6),
it follows from (7.3.2) and (7.3.4) that, for all
$\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$ and
$\gamma\in\Gamma_{l}$ with $-h_{l}(\gamma)\in I$,

(7.3.5)
$d_{\widetilde{\mathcal{F}}^{1}_{l}}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\geq
C_{7}|\gamma|\;.$

By (2.9.14) and (7.3.5), and since the leaves of
$\widetilde{\mathcal{F}}^{1}_{l}$ are of equi-bounded geometry, there are
$C_{1},C_{2},u_{0}>0$ such that, for all $0<u\leq u_{0}$,
$\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$ and
$\gamma\in\Gamma_{l}$ with $-h_{l}(\gamma)\in I$,

(7.3.6)
$\big{|}\tilde{k}_{l,z,u}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\big{|}\leq
C_{1}u^{(n-1)/2}e^{-C_{2}C_{7}^{2}|\gamma|^{2}/u}\;.$

Hence, by (7.3.1) and (7.3.3), for all $0<u\leq u_{0}$ and $p\in
M^{1}_{l,\epsilon^{\prime\prime}}$,

(7.3.7) $\big{|}\mathring{k}_{l,z,u}(p,p)\big{|}\leq
C_{4}C_{5}C_{1}u^{(n-1)/2}\sum_{\gamma\in\Gamma_{l}}e^{-C_{2}C_{7}^{2}|\gamma|^{2}/u}\;,$

which converges to zero as $u\downarrow 0$ because $\Gamma_{l}$ is of
polynomial growth. Since $M^{1}_{l,\epsilon^{\prime\prime}}$ is compact, we
get

$\lim_{u\downarrow 0}\int_{p\in
M^{1}_{l,\epsilon^{\prime\prime}}}\operatorname{str}\mathring{k}_{l,z,u}(p,p)=0\;,$

and therefore i follows by Corollaries 7.2.8 and 2.5.10.

Now assume $I\subset\mathbb{R}^{\times}$ and $I\cap\mathcal{P}=\\{t_{0}\\}$ to
prove ii, and let $\mathcal{C}_{l,t_{0}}=\\{c_{1},\dots,c_{m}\\}$. Then the
following properties hold (Section 4.3.4):

  1. (N)

There is a unique $\gamma_{0}\in\Gamma_{l}$ such that
$t_{0}=-h_{l}(\gamma_{0})$.

  2. (O)

We have $k_{j}:=t_{0}/\ell(c_{j})\in\mathbb{Z}$ ($j=1,\dots,m$).

  3. (P)

There is some $y_{j}\in L_{l}$ such that
$\pi_{l}:\mathbb{R}\times\\{y_{j}\\}\to c_{j}$ is a $C^{\infty}$ covering map
with fundamental domain $[0,\ell(c_{j})]\times\\{y_{j}\\}$.

  4. (Q)

For all $\tilde{p}\in\mathbb{R}\times\\{y_{j}\\}$, we have
$\gamma_{0}\cdot\tilde{\phi}_{l}^{t_{0}}(\tilde{p})=\tilde{p}$.

  5. (R)

For all $x\in\mathbb{R}$, every $y_{j}$ is a simple fixed point of the
diffeomorphism $T_{\gamma_{0}}\tilde{\phi}_{l,x}^{t_{0}}$ of $L_{l}$ with
$\epsilon_{y_{j}}(T_{\gamma_{0}}\tilde{\phi}_{l,x}^{t_{0}})=\epsilon_{c_{j}}(k_{j},\phi)=\epsilon_{c_{j}}(k_{j})$.

In particular, there are no other fixed points of
$T_{\gamma_{0}}\tilde{\phi}_{l,x_{j}}^{t_{0}}$ in some open neighborhood
$W_{j}$ of $y_{j}$ in $L_{l}$. Then $\pi_{l}([0,\ell(c_{j})]\times W_{j})$ is
a neighborhood of $c_{j}$, whose interior is denoted by $V_{j}$, which does
not intersect other closed orbits with period in $I$. Note that
$\pi_{l}:(0,\ell(c_{j}))\times W_{j}\to V_{j}$ is a $C^{\infty}$ embedding and
$V_{j}\setminus\pi_{l}((0,\ell(c_{j}))\times W_{j})=\pi_{l}(\\{0\\}\times
W_{j})$ is of measure zero. For every $p\in V_{j}$, let $\tilde{p}$ be the
unique point in $[0,\ell(c_{j}))\times W_{j}$ with $\pi_{l}(\tilde{p})=p$. We
have

$\int_{V_{j}}\operatorname{str}\big{(}\tilde{\phi}_{l,z}^{t_{0}*}\,T_{\gamma_{0}}^{*}\,\tilde{k}_{l,z,u}\big{(}T_{\gamma_{0}}\tilde{\phi}_{l}^{t_{0}}(\tilde{p}),\tilde{p}\big{)}\big{)}\,f(t_{0})\,|\omega_{\text{\rm
b},l}|(p)\\\ \begin{aligned} &=\int_{[0,\ell(c_{j})]\times
W_{j}}\operatorname{str}\big{(}\tilde{\phi}_{l,z}^{t_{0}*}\,T_{\gamma_{0}}^{*}\,\tilde{k}_{l,z,u}\big{(}T_{\gamma_{0}}\tilde{\phi}_{l}^{t_{0}}(\tilde{p}),\tilde{p}\big{)}\big{)}\,f(t_{0})\,\big{|}\tilde{\omega}_{\text{\rm
b},l}\big{|}(\tilde{p})\\\
&=f(t_{0})\int_{0}^{\ell(c_{j})}\int_{W_{j}}\operatorname{str}\big{(}\tilde{\phi}_{l,z}^{t_{0}*}\,T_{\gamma_{0}}^{*}\,\tilde{k}_{l,z,u}\big{(}\big{(}x,T_{\gamma_{0}}\tilde{\phi}_{l,x}^{t_{0}}(y)\big{)},(x,y)\big{)}\,|dx|\\\
&=f(t_{0})\int_{0}^{\ell(c_{j})}\int_{W_{j}}\operatorname{str}\big{(}\tilde{\phi}_{l,x,z}^{t_{0}*}\,T_{\gamma_{0}}^{*}\,\tilde{k}_{l,x,u,z}\big{(}T_{\gamma_{0}}\tilde{\phi}_{l,x}^{t_{0}}(y),y\big{)}\,|dx|\;.\end{aligned}$

But, by Proposition 2.9.6,

$\lim_{u\downarrow
0}\int_{W_{j}}\operatorname{str}\big{(}\tilde{\phi}_{l,x,z}^{t_{0}*}\,T_{\gamma_{0}}^{*}\,\tilde{k}_{l,x,u,z}\big{(}T_{\gamma_{0}}\tilde{\phi}_{l,x}^{t_{0}}(y),y\big{)}\big{)}=\epsilon_{y_{j}}\big{(}T_{\gamma_{0}}\tilde{\phi}_{l,x}^{t_{0}}\big{)}=\epsilon_{c_{j}}(k_{j})\;.$

So

(7.3.8) $\lim_{u\downarrow
0}\int_{V_{j}}\operatorname{str}\big{(}\tilde{\phi}_{l,z}^{t_{0}*}\,T_{\gamma_{0}}^{*}\,\tilde{k}_{l,z,u}\big{(}T_{\gamma_{0}}\tilde{\phi}_{l}^{t_{0}}(\tilde{p}),\tilde{p}\big{)}\big{)}\,f(t_{0})\,|\omega_{\text{\rm
b},l}|(p)\\\ =f(t_{0})\ell(c_{j})\epsilon_{c_{j}}(k_{j})\;.$

By i, we can assume the length of $I$ is as small as desired. By the
$\mathbb{R}$-local bounded geometry of $\tilde{\phi}_{l}$, if the length of
$I$ is small enough, there is some $0<r<C_{3}^{-1}/2$ such that
$d_{l}(\phi_{l}^{t}(\tilde{p}),\phi_{l}^{s}(\tilde{p}))\leq r$ for all
$\tilde{p}\in\widetilde{M}^{1}_{l}$ and $t,s\in I$. So, by (4.3.5), N and Q,
for all $p\in c_{j}$ and $\gamma\in\Gamma_{l}\setminus\\{\gamma_{0}\\}$ with
$-h_{l}(\gamma)\in I$,

$d_{l}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\\\
\begin{aligned} &\geq
d_{l}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\gamma_{0}\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p})\big{)}\\\
&\phantom{=\text{}}\text{}-d_{l}\big{(}\gamma_{0}\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\gamma_{0}\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma_{0})}(\tilde{p})\big{)}-d_{l}\big{(}\gamma_{0}\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma_{0})}(\tilde{p}),\tilde{p}\big{)}\\\
&=d_{l}\big{(}\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\gamma^{-1}\gamma_{0}\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p})\big{)}-d_{l}\big{(}\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{\phi}_{l}^{-h_{l}(\gamma_{0})}(\tilde{p})\big{)}\\\
&\geq C_{3}^{-1}|\gamma^{-1}\gamma_{0}|-r\;.\end{aligned}$

Thus, by continuity, the neighborhood $W_{j}$ of every $y_{j}$ can be chosen
so small that, for all $p\in V_{j}$ and
$\gamma\in\Gamma_{l}\setminus\\{\gamma_{0}\\}$ with $-h_{l}(\gamma)\in I$,

$d_{l}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\geq
C_{3}^{-1}|\gamma^{-1}\gamma_{0}|-2r\geq C_{3}^{-1}-2r>0\;.$

Hence, by (2.9.14) and since the leaves of $\widetilde{\mathcal{F}}^{1}_{l}$
are of equi-bounded geometry, there are $C_{1},C_{2},u_{0}>0$ such that, for
all $0<u\leq u_{0}$, $p\in V_{j}$ and
$\gamma\in\Gamma_{l}\setminus\\{\gamma_{0}\\}$ with $-h_{l}(\gamma)\in I$,

$\big{|}\tilde{k}_{l,z,u}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\big{|}\leq
C_{1}u^{(n-1)/2}e^{-C_{2}(C_{3}^{-1}|\gamma^{-1}\gamma_{0}|-2r)^{2}/u}\;.$

Then, by (7.3.3), for all $0<u\leq u_{0}$ and $p\in V_{j}$,

$\bigg{|}\sum_{\gamma\in\Gamma_{l}\setminus\\{\gamma_{0}\\}}\tilde{\phi}_{l,z}^{-h_{l}(\gamma)*}\,T_{\gamma}^{*}\,\tilde{k}_{l,z,u}\big{(}T_{\gamma}\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\,f(-h_{l}(\gamma))\,|\omega_{\text{\rm
b},l}|(p)\bigg{|}\\\ \leq
C_{4}C_{5}C_{1}u^{(n-1)/2}\sum_{\gamma\in\Gamma_{l}\setminus\\{\gamma_{0}\\}}e^{-C_{2}(C_{3}^{-1}|\gamma^{-1}\gamma_{0}|-2r)^{2}/u}\;,$

which converges to zero as $u\downarrow 0$ because $\Gamma_{l}$ is of
polynomial growth. So, by (7.3.1) and (7.3.8),

(7.3.9) $\lim_{u\downarrow 0}\int_{p\in
V_{j}}\operatorname{str}\mathring{k}_{l,z,u}(p,p)=f(t_{0})\ell(c_{j})\epsilon_{c_{j}}(k_{j})\;.$

On the other hand, since $\phi$ has no closed orbits in $T^{1}_{l,\epsilon}$
(Section 4.3.3), we can assume $V_{j}\subset
M^{1}_{l,\epsilon^{\prime\prime}}$. Let
$\widetilde{V}_{j}=\pi_{l}^{-1}(V_{j})\subset\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$.
If $p\in M^{1}_{l,\epsilon^{\prime\prime}}\setminus(V_{1}\cup\dots\cup V_{m})$
and $t\in I$, then $\phi^{t}(p)\neq p$. Hence, like in the proof of i, there
are $C_{7},C_{1},C_{2},u_{0}>0$ such that (7.3.5) and (7.3.6) hold for all
$0<u\leq u_{0}$,
$\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}\setminus(\widetilde{V}_{1}\cup\dots\cup\widetilde{V}_{m})$
and $\gamma\in\Gamma_{l}$ with $-h_{l}(\gamma)\in I$. Thus (7.3.7) holds for
all $p$ in the compact space
$M^{1}_{l,\epsilon^{\prime\prime}}\setminus(V_{1}\cup\dots\cup V_{m})$,
yielding

$\lim_{u\downarrow 0}\int_{p\in
M^{1}_{l,\epsilon^{\prime\prime}}\setminus(V_{1}\cup\dots\cup
V_{m})}\operatorname{str}\mathring{k}_{l,z,u}(p,p)=0\;.$

So ii is true by (7.3.9) and Corollaries 2.5.10 and 7.2.8.

Finally, assume $0\in I$ and $I\cap\mathcal{P}_{l}=\emptyset$ to prove iii. By
i, we can suppose again that the length of $I$ is as small as desired. By
(7.3.2), there are finitely many elements $\gamma\in\Gamma_{l}$ such that
$-h_{l}(\gamma)\in I$ and, for all $\tilde{p}\in\widetilde{M}^{1}_{l}$,

(7.3.10)
$d_{\widetilde{\mathcal{F}}^{1}_{l}}\big{(}\gamma\cdot\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}>1\;.$

Thus, if $I$ is small enough, we can assume (7.3.10) is true for all
$\tilde{p}\in\widetilde{M}^{1}_{l}$ and $\gamma\in\Gamma_{l}\setminus\\{e\\}$
with $-h_{l}(\gamma)\in I$. Then, like in the proof of i, there are
$C_{7},C_{1},C_{2},u_{0}>0$ such that (7.3.5) and (7.3.6) hold for all
$0<u\leq u_{0}$, $\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$
and $\gamma\in\Gamma_{l}\setminus\\{e\\}$ with $-h_{l}(\gamma)\in I$. Hence,
by (7.3.3), for all $0<u\leq u_{0}$ and $p\in
M^{1}_{l,\epsilon^{\prime\prime}}$,

$\bigg{|}\sum_{\gamma\in\Gamma_{l}\setminus\\{e\\}}\tilde{\phi}_{l,z}^{-h_{l}(\gamma)*}\,T_{\gamma}^{*}\,\tilde{k}_{l,z,u}\big{(}T_{\gamma}\tilde{\phi}_{l}^{-h_{l}(\gamma)}(\tilde{p}),\tilde{p}\big{)}\,f(-h_{l}(\gamma))\,|\omega_{\text{\rm
b},l}|(p)\bigg{|}\\\ \leq
C_{4}C_{5}C_{1}u^{(n-1)/2}\sum_{\gamma\in\Gamma_{l}}e^{-C_{2}C_{7}^{2}|\gamma|^{2}/u}\;,$

which converges to zero as $u\downarrow 0$ because $\Gamma_{l}$ is of
polynomial growth. On the other hand, by (2.9.22) and Theorem 2.9.4,

$\lim_{u\downarrow
0}\operatorname{str}\tilde{k}_{l,z,u}(\tilde{p},\tilde{p})=\begin{cases}e(\widetilde{\mathcal{F}}_{l},g_{\widetilde{\mathcal{F}}_{l}})(\tilde{p})\equiv
e(\mathcal{F}_{l},g_{\mathcal{F}_{l}})([\tilde{p}])&\text{if $n-1$ is even}\\\
0&\text{if $n-1$ is odd}\;,\end{cases}$

uniformly on $\tilde{p}\in\widetilde{M}^{1}_{l,\epsilon^{\prime\prime}}$. So,
by (7.3.1),

$\lim_{u\downarrow
0}\operatorname{str}\kappa_{l,u}=f(0)\,e(\mathcal{F}_{l},g_{\mathcal{F}_{l}})\,|\omega_{\text{\rm
b},l}|$

uniformly on $\Delta_{\text{\rm b},l}\cap\beta_{\text{\rm
b}}^{-1}(M_{l,\epsilon^{\prime\prime}}^{2})\equiv
M_{l,\epsilon^{\prime\prime}}$. Therefore iii follows using Corollaries
2.5.10, 7.2.9 and 2.5.11. ∎

######  Remark 7.3.3.

The simpler argument given in [ÁLK02, ÁLK08] for the case of Theorem 7.3.1
with no preserved leaves cannot be applied here because now ${}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{u})$ depends on $u$.

Theorem 1.3.8 is a restatement of Theorem 7.3.1.

###  7.4. The limit of ${}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{\mu,u})$ as $u\uparrow+\infty$ and
$\mu\to\pm\infty$

####  7.4.1. An expression of ${}^{\text{\rm
b}}\\!\operatorname{Tr}([d_{\boldsymbol{\mathcal{F}},\mu},\boldsymbol{P}_{\mu}\mathsf{w}])$

From now on, we will only consider $\boldsymbol{P}_{z}$ for
$z=\mu\in\mathbb{R}$; written $\boldsymbol{P}_{\mu}$. We keep the notation
$z=\mu+i\lambda$ for any other $\lambda\in\mathbb{R}$ ($i=\sqrt{-1}$). In the
following, $L$ runs in $\pi_{0}M^{0}$. Recall that $\eta_{0}=\eta$ around
$M^{0}$. For $\psi\in\mathcal{A}$, $\mu\in\mathbb{R}$ and $f\in
C^{\infty}_{\text{\rm c}}(\mathbb{R})$, let

$\displaystyle S_{L,\mu}$
$\displaystyle=-\frac{1}{2\pi}\int_{-\infty}^{+\infty}{\eta_{L}\wedge}\,\psi(D_{L,z})\hat{f}(-\varkappa_{L}\lambda)\,d\lambda$
(7.4.1)
$\displaystyle=-\frac{1}{2\pi|\varkappa_{L}|}\int_{-\infty}^{+\infty}{\eta_{L}\wedge}\,\psi(D_{L,z})\widehat{f_{L}}(\lambda)\,d\lambda\;,$

where $f_{L}(\lambda)=f(-\lambda/\varkappa_{L})$. Again, we may also add the
subscript “$\psi$” or “$f$” to the notation $S_{L,\mu}$ if needed. Recall also
that $\mathsf{w}$ denotes the degree involution. Observe that
$\mathsf{w}d_{\boldsymbol{\mathcal{F}},z}=-d_{\boldsymbol{\mathcal{F}},z}\mathsf{w}$
and
$I_{\boldsymbol{\nu}}(\boldsymbol{P}_{\mu}\mathsf{w},\lambda)=I_{\boldsymbol{\nu}}(\boldsymbol{P}_{\mu},\lambda)\,\mathsf{w}$
by (2.5.63).

######  Lemma 7.4.1.

We have

${}^{\text{\rm
b}}\\!\operatorname{Tr}([d_{\boldsymbol{\mathcal{F}},\mu},\boldsymbol{P}_{\mu}\mathsf{w}])=2\sum_{L}\operatorname{Str}(S_{L,\mu})\;.$

###### Proof.

By the version of (2.5.65) with a b-differential operator and a
b-pseudodifferential operator of order $-\infty$, Corollaries 7.2.6 and
7.2.10,

$\displaystyle{}^{\text{\rm
b}}\\!\operatorname{Tr}([d_{\boldsymbol{\mathcal{F}},\mu},\boldsymbol{P}_{\mu}\mathsf{w}])$
$\displaystyle=-\frac{1}{2\pi
i}\int_{-\infty}^{+\infty}\operatorname{Tr}(\partial_{\lambda}I_{\boldsymbol{\nu}}(d_{\boldsymbol{\mathcal{F}},\mu},\lambda)\,I_{\boldsymbol{\nu}}(\boldsymbol{P}_{\mu},\lambda)\mathsf{w})\,d\lambda$
$\displaystyle=-\frac{1}{\pi}\sum_{L}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\operatorname{Tr}({\eta_{L}\wedge}\,\psi(D_{L,z})\mathsf{w})\,e^{i\lambda\varkappa_{L}t}f(t)\,dt\,d\lambda$
$\displaystyle=2\sum_{L}\operatorname{Str}(S_{L,\mu})\;.\qed$

####  7.4.2. Variation of ${}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{\mu,u})$ with respect to $u$

For any $\psi\in\mathcal{A}$ and $u>0$, let $\psi_{u}\in\mathcal{A}$ be
defined by $\psi_{u}(x)=\psi(\sqrt{u}x)$, and consider the corresponding
operator $\boldsymbol{P}_{\mu,u}$. Recall that $\boldsymbol{P}_{\mu,u}$ is the
operator $\boldsymbol{P}_{\mu,u,f}$ of Section 1.3.4 if $\psi(x)=e^{-x^{2}}$.

######  Proposition 7.4.2.

If $\psi\in\mathcal{A}$ is even, then

$\frac{d}{du}{}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{\mu,u})=-\frac{1}{\sqrt{u}}\sum_{L}\operatorname{Str}(S_{L,(\psi^{\prime})_{u},\mu})\;.$

###### Proof.

This result follows like in the heat equation proof of the usual Lefschetz
trace formula [AB67, Gil95, Roe98], but the stated derivative does not vanish
because the b-trace of commutators may not be zero. To simplify the arguments,
consider the change of variables $v=\sqrt{u}$, and let
$\psi^{v}(x)=\psi_{u}(x)=\psi(vx)$ and
$\boldsymbol{P}^{v}_{\mu}=\boldsymbol{P}_{\psi^{v},\mu,f}$. By Lemma 7.4.1 and
since $\psi^{\prime}$ is odd,

${}^{\text{\rm
b}}\\!\operatorname{Str}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}d_{\boldsymbol{\mathcal{F}},\mu}\,\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})\,f(t)\,dt\right)\\\
\begin{aligned} &={}^{\text{\rm
b}}\\!\operatorname{Tr}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}d_{\boldsymbol{\mathcal{F}},\mu}\,\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})\,\mathsf{w}\,f(t)\,dt\right)\\\
&={}^{\text{\rm
b}}\\!\operatorname{Tr}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})\,\mathsf{w}\,d_{\boldsymbol{\mathcal{F}},\mu}f(t)\,dt\right)+2\sum_{L}\operatorname{Str}(S_{L,(\psi^{\prime})^{v},\mu})\\\
&=-{}^{\text{\rm
b}}\\!\operatorname{Tr}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}\delta_{\boldsymbol{\mathcal{F}},\mu}\,\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})\,\mathsf{w}\,f(t)\,dt\right)+2\sum_{L}\operatorname{Str}(S_{L,(\psi^{\prime})^{v},\mu})\\\
&=-{}^{\text{\rm
b}}\\!\operatorname{Str}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}\delta_{\boldsymbol{\mathcal{F}},\mu}\,\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})\,f(t)\,dt\right)+2\sum_{L}\operatorname{Str}(S_{L,(\psi^{\prime})^{v},\mu})\;.\end{aligned}$

So

$\displaystyle\frac{d}{dv}{}^{\text{\rm b}}\\!\operatorname{Tr}^{\text{\rm
s}}\boldsymbol{P}^{v}_{\mu}$ $\displaystyle={}^{\text{\rm
b}}\\!\operatorname{Str}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}D_{\boldsymbol{\mathcal{F}},\mu}\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})f(t)\,dt\right)$
$\displaystyle={}^{\text{\rm
b}}\\!\operatorname{Str}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}d_{\boldsymbol{\mathcal{F}},\mu}\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})f(t)\,dt\right)$
$\displaystyle\phantom{=\text{}}\text{}+{}^{\text{\rm
b}}\\!\operatorname{Str}\left(\int_{-\infty}^{+\infty}\phi^{t*}_{\mu}\delta_{\boldsymbol{\mathcal{F}},\mu}\psi^{\prime}(vD_{\boldsymbol{\mathcal{F}},\mu})f(t)\,dt\right)$
$\displaystyle=2\sum_{L}\operatorname{Str}(S_{L,(\psi^{\prime})^{v},\mu})\;.$

Now apply the chain rule. ∎

####  7.4.3. The limit of ${}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{\mu,u})$ as $u\uparrow+\infty$ and
$\mu\to\pm\infty$

Now take $\psi(x)=e^{-x^{2}}$. Hence $\psi_{u}(x)=e^{-ux^{2}}$ and
$(\psi^{\prime})_{u}(x)=-2\sqrt{u}xe^{-ux^{2}}$. Thus, by (7.4.1),

(7.4.2)
$\operatorname{Str}(S_{L,(\psi^{\prime})_{u},\mu})=-\frac{1}{\pi|\varkappa_{L}|}\int_{-\infty}^{+\infty}\operatorname{Str}\big{(}{\eta_{L}\wedge}\,\delta_{z}e^{-u\Delta_{L,z}}\big{)}\widehat{f_{L}}(\lambda)\,d\lambda\;.$

######  Theorem 7.4.3.

For all $\tau\gg 0$, we can choose every $\eta_{L}$ and $g_{L}$
($L\in\pi_{0}M^{0}$) so that

$\lim_{\mu\uparrow+\infty}\Big{(}\lim_{u_{1}\uparrow+\infty}\boldsymbol{P}_{\mu,u_{1}}-\lim_{u_{0}\downarrow
0}\boldsymbol{P}_{\mu,u_{0}}\Big{)}=\tau f(0)\;.$

If $n-1$ is even, this is true for all $\tau\in\mathbb{R}$ and as
$\mu\to\pm\infty$.

###### Proof.

By Theorem 2.9.7, if $\tau\gg 0$, we can choose every $\eta_{L}$ and $g_{L}$
so that (2.9.25) defines a tempered distribution
$Z_{L,\mu}:=Z(L,g_{L},\eta_{L})\in\SS^{\prime}$ for $|\mu|\gg 0$, and
$Z_{L,\mu}\to\tau\delta_{0}$ in $\SS^{\prime}$ as $\mu\to\infty$. If $n-1=\dim
L$ is even, then this is true for all $\tau\in\mathbb{R}$ and as
$\mu\to\pm\infty$. Then the result follows because, by (2.9.25), (7.4.2) and
Proposition 7.4.2,

$\displaystyle\lim_{u_{1}\uparrow+\infty}\boldsymbol{P}_{\mu,u_{1}}-\lim_{u_{0}\downarrow
0}\boldsymbol{P}_{\mu,u_{0}}$
$\displaystyle=-\frac{1}{\sqrt{u}}\sum_{L}\int_{0}^{\infty}\operatorname{Str}(S_{L,(\psi^{\prime})_{u},\mu})\,du$
$\displaystyle=-\frac{1}{\pi}\sum_{L}\frac{1}{|\varkappa_{L}|}\langle
Z_{L,\mu},f_{L}\rangle\;.\qed$

Corollary 7.4.4 gives Theorem 1.3.9 by taking $\tau=0$ when $n-1$ is even.

######  Corollary 7.4.4.

For all $\tau\gg 0$, we can choose every $\eta_{L}$ and $g_{L}$
($L\in\pi_{0}M^{0}$) so that

$\lim_{\mu\uparrow+\infty}\lim_{u\uparrow+\infty}{}^{\text{\rm
b}}\\!\operatorname{Str}(\boldsymbol{P}_{\mu,u})=\big{(}{}^{\text{\rm
b}}\\!\chi_{|\boldsymbol{\omega}_{b}|}(\mathring{\boldsymbol{\mathcal{F}}})+\tau\big{)}\,f(0)+\sum_{c\in\mathcal{C}}\ell(c)\sum_{k\in\mathbb{Z}^{\times}}\epsilon_{c}(k)\,f(k\ell(c))\;.$

If $n$ is even, this is true for all $\tau\in\mathbb{R}$ and as
$\mu\to\pm\infty$.

###### Proof.

Apply Theorems 7.3.1 and 7.4.4. ∎

Theorem 1.3.9 follows taking $\tau=0$ in Corollary 7.4.4.

Like in[ÁLK02, ÁLKL20], by (5.4.6) and (6.4.6), the distributions

$f\mapsto\lim_{u\uparrow+\infty}{}^{\text{\rm
b}}\\!\operatorname{Str}\big{(}\boldsymbol{P}_{m+\frac{1}{2},u,f}\big{)}\;,\quad
f\mapsto\lim_{u\uparrow+\infty}{}^{\text{\rm
b}}\\!\operatorname{Str}\big{(}\boldsymbol{P}_{m-\frac{1}{2},u,f}\big{)}$

can be considered as a distributional supertraces of the action
$\boldsymbol{\phi}^{*}$ of $\mathbb{R}$ on
$\bar{H}^{\bullet}J^{m}(\mathcal{F})$ and
$\bar{H}^{\bullet}J^{\prime\,m}(\mathcal{F})$. So, by the analogs of (5.1.2)
and (6.1.2) for $J(\mathcal{F})$ and $J^{\prime}(\mathcal{F})$, and using
(5.4.6) and (6.4.6), the distributions

$f\mapsto\lim_{m\downarrow-\infty}\lim_{u\uparrow+\infty}{}^{\text{\rm
b}}\\!\operatorname{Str}\big{(}\boldsymbol{P}_{m+\frac{1}{2},u,f}\big{)}\;,\quad
f\mapsto\lim_{m\downarrow\infty}\lim_{u\uparrow+\infty}{}^{\text{\rm
b}}\\!\operatorname{Str}\big{(}\boldsymbol{P}_{m-\frac{1}{2},u,f}\big{)}$

can be considered as a distributional supertraces of the action
$\boldsymbol{\phi}^{*}$ of $\mathbb{R}$ on $\bar{H}^{\bullet}J(\mathcal{F})$
and $\bar{H}^{\bullet}J^{\prime}(\mathcal{F})$, as indicated in Section 1.3.6.

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## Index

  * $[\alpha]_{s}$ §5.1
  * $\widehat{H}^{\bullet}I^{\prime}(\mathcal{F})$ §6.2
  * $\hat{\jmath}^{\prime}_{s}$ §6.2
  * $\hat{\jmath}_{s*}$ §5.1, §6.1
  * $\widehat{R}^{\prime}_{*}$ §6.2
  * $\hat{\iota}^{\prime}_{*}$ §6.2
  * $\overline{[\alpha]}_{s}$ §5.1
  * $\bar{B}I^{(s)}(\mathcal{F})$ §5.1
  * $\bar{H}^{\bullet}(\mathcal{F})$ §3.2.1
  * $\bar{H}^{\bullet}I(\mathcal{F})$ §5.1
  * $\bar{H}^{\bullet}I^{(s)}(\mathcal{F})$ §5.1
  * $\bar{H}^{\bullet}I^{\prime}(\mathcal{F})$ §6.1
  * $\bar{H}^{\bullet}I^{\prime\,(s)}(\mathcal{F})$ §6.1
  * $\bar{H}^{\bullet}J(\mathcal{F})$ §5.1
  * $\bar{H}^{\bullet}J^{\prime}(\mathcal{F})$ §6.1
  * $\overline{\operatorname{Pen}}_{\mathcal{F}}(S,r)$ §3.1.6
  * $\overline{\mathfrak{X}}(M,\mathcal{F})$ §3.1.3
  * $\breve{M}$ §2.5.2
  * $\dot{\mathcal{A}}(M)$ §2.5.8
  * $\dot{\mathcal{A}}^{(s)}(M)$ §2.5.8
  * $\dot{\mathcal{A}}^{\prime}(M)$ §2.5.21
  * $\dot{\mathcal{A}}^{\prime\,(s)}(M)$ §2.5.21
  * $\dot{\mathcal{A}}^{m}(M)$ §2.5.12
  * $\dot{C}^{-\infty}(M)$ §2.5.3
  * $\dot{C}^{-\infty}_{\partial M}(M)$ §2.5.5, §2.5.5
  * $\dot{C}^{\prime-k}(M)$ §2.5.3
  * $\dot{C}^{\prime\,-k}_{\partial M}(M)$ §2.5.5
  * $\dot{C}^{\infty}(M)$ §2.5.2
  * $\dot{H}^{s}(M)$ §2.5.4
  * $\dot{H}^{s}_{\partial M}(M)$ §2.5.5
  * $\check{L}$ §2.6.16
  * $\check{T}$ §2.6.16
  * $\check{\pi}$ §2.6.16
  * $\check{\sigma}$ §2.6.16
  * $\mathring{\nabla}$ §3.4.1
  * $\widetilde{A}$ §2.6.2
  * $\widetilde{H}^{\bullet}I^{\prime}(\mathcal{F})$ §6.2
  * $\tilde{\jmath}^{\prime}_{s}$ §6.2
  * $\tilde{\jmath}_{s*}$ §5.1, §6.1
  * $\widetilde{M}$ §2.6.1
  * $\star_{\perp}$ §3.2.6
  * $\star_{\mathcal{F}}$ §3.2.6
  * $\mathcal{A}$ §2.9.8, §3.4.1
  * $\mathcal{A}(M)$ §2.5.8
  * $\mathcal{A}^{(s)}(M)$ §2.5.8
  * $\mathcal{A}^{\prime}(M)$ §2.5.21
  * $\mathcal{A}^{\prime\,(s)}(M)$ §2.5.21
  * acyclic §2.1.1
  * almost without holonomy foliation §3.1.12
  * $\mathcal{A}^{m}(M)$ §2.5.10
  * b-Connes-Euler characteristic §1.3.1
  * b-diagonal §2.5.26
  * b-differential operator §2.5.7
  * b-elliptic §2.5.27
  * b-integral §2.5.29
  * b-metric §2.5.1
  * b-pseudodifferential operator §2.5.27
  * b-Sobolev space §2.5.17
  * b-stretched product §2.5.26
  * b-supertrace §2.5.30
  * b-trace §2.5.30
  * b-vector field §2.5.7
  * basic complex §3.2.4
  * $BI^{(s)}(\mathcal{F})$ §5.1
  * boundary-defining function §2.5.1
  * bounded geometry §2.4.2
  * ${}^{\text{\rm b}}\\!\operatorname{Str}$ §2.5.30
  * ${}^{\text{\rm b}}T^{*}M$ §2.5.1
  * ${}^{\text{\rm b}}TM$ §2.5.1
  * ${}^{\text{\rm b}}\\!\operatorname{Tr}$ §2.5.30
  * bundle-like metric §3.1.9
  * ${}^{\text{\rm b}}\\!\sigma_{m}$ §2.5.27
  * $C^{-\infty}(M,L)$ §2.6.3
  * $C^{-\infty}_{L}(M)$ §2.6.4
  * $\mathcal{C}(\phi)$ §4.1.1
  * $C^{\prime\,-k}(M,L)$ §2.6.3
  * $C^{\prime\,-k}_{{\cdot}/{\text{\rm c}}}(M)$ §2.1.4
  * $C^{\prime\,-k}_{L}(M)$ §2.6.4
  * $C^{\pm\infty}_{{\cdot}/{\text{\rm c}}}(M;E)$ §2.1.4
  * $C^{\pm\infty}_{\text{\rm cv}}(M;E)$ §2.1.4
  * $C^{\pm\infty}_{S}(M;E)$ §2.1.4
  * $C^{\infty}(M,L)$ §2.6.3
  * $C^{\infty}_{\text{\rm ub}}(M;E)$ §2.4.3
  * $\mathcal{C}_{I}(\phi)$ §4.1.1
  * $C^{k}(M,L)$ §2.6.3
  * $\mathcal{C}_{l,t_{0}}$ §7.3
  * compactly retractive §2.1.1
  * conormal distribution §2.2.2
  * conormal sequence §2.6.12
  * conormal sequence at the boundary §2.5.13
  * $D_{0,z,z^{\prime}}$ §3.3.3
  * defining function §2.6.1
  * developing map §3.1.9
  * $d_{\mathcal{F}}$ §3.2.1
  * $D_{\mathcal{F},z}$ §3.3.2
  * $d_{\mathcal{F},z}$ §3.3.2
  * $D_{\mathcal{F},z,z^{\prime}}$ §3.3.3
  * $d_{i,1-i}$ §3.2.3
  * $\operatorname{Diff}(\mathcal{F})$ §3.1.8
  * $\operatorname{Diff}(M,L)$ §2.2.1
  * $\operatorname{Diff}_{\text{\rm b}}(M)$ §2.5.7
  * $\operatorname{Diff}^{m}(M;E,F)$ §2.1.7
  * $\operatorname{Diff}_{\text{\rm ub}}(\mathcal{F};E)$ §3.4.1
  * $\operatorname{Diff}_{\text{\rm ub}}(M;E)$ §2.4.4
  * dual-conormal distribution §2.3.1
  * dual-conormal sequence §2.7.4
  * dual-conormal sequence at the boundary §2.5.22
  * $D_{z,\perp}$ §3.3.1
  * $D_{z,0}$ §3.3.1
  * $d_{z,i,1-i}$ §3.3.1
  * $e(\mathcal{F},g_{\mathcal{F}})$ §3.2.7
  * $E^{\prime}_{m}$ §6.5.2
  * $E_{m}$ Proposition 2.5.1
  * $E_{m,\pm}$ §5.6.2
  * $E_{m,T}$ §5.5.1
  * $\operatorname{End}(X)$ §2.1.1
  * $\mathring{\exp}$ §3.4.1
  * extendible conormal distribution §2.5.8
  * extendible distribution §2.5.3
  * extendible dual-conormal distributions §2.5.21
  * extendible function §2.5.2
  * extendible Sobolev space §2.5.4
  * $F^{\prime}_{m}$ §6.5.1
  * $\operatorname{ff}$ §2.5.26
  * $\operatorname{Fix}(\phi)$ §2.9.11
  * flow leafwise homotopy §3.1.7
  * $F_{m}$ §5.5.2
  * foliated flow §3.1.7
  * foliated map §3.1.7
  * foliation §3.1.1
  * front face §2.5.26
  * ${}^{\mathcal{F}}\\!\sigma_{m}$ §3.1.8
  * $\mathfrak{G}$ §3.1.4
  * germinal holonomy §3.1.2
  * $\mathcal{H}$ §3.1.2
  * $H^{\prime\,s}(M,L)$ §2.6.3
  * $H^{\bullet}(\mathcal{F})$ §3.2.1
  * $H^{\bullet}I(\mathcal{F})$ §5.1
  * $H^{\bullet}I^{(s)}(\mathcal{F})$ §5.1
  * $H^{\bullet}I^{\prime}(\mathcal{F})$ §6.1
  * $H^{\bullet}I^{\prime\,(s)}(\mathcal{F})$ §6.1
  * $H^{\bullet}J(\mathcal{F})$ §5.1
  * $H^{\bullet}J^{\prime}(\mathcal{F})$ §6.1
  * $H^{\bullet}K(\mathcal{F})$ §5.1
  * $H^{\bullet}K^{\prime}(\mathcal{F})$ §6.1
  * $H_{\text{\rm b}}^{m}(M;{}^{\text{\rm b}}\Omega^{\frac{1}{2}})$ §2.5.17
  * $hH^{s}_{\text{\rm c/loc}}(M;E)$ §2.1.12
  * Hilbertian space §2.1.1
  * $hL^{\infty}(M;E)$ §2.1.12
  * holonomy cover §3.1.2
  * holonomy groupoid §3.1.4
  * holonomy homomorphism §3.1.9
  * holonomy pseudogroup §3.1.2
  * horizontal subbundle §3.4.1
  * $H^{r}(M,L)$ §2.6.3
  * $H^{s}(M;E)$ §2.1.11
  * $H^{s}_{L}(M)$ §2.6.4
  * $I^{-\infty}(M,L)$ §2.2.3
  * $I^{(\infty)}(M,L)$ §2.2.2, §2.2.3
  * $I(\mathcal{F})$ §5.1
  * $I(M,L)$ §2.2.2
  * $I^{(s)}(\mathcal{F})$ §5.1
  * $I^{(s)}_{{\cdot}/{\text{\rm c}}}(M,L)$ §2.2.2
  * $I^{\prime}(\mathcal{F})$ §6.1
  * $I^{\prime}(M,L)$ §2.3.1
  * $I^{\prime\,m}_{{\cdot}/{\text{\rm c}}}(M,L)$ §2.3.2
  * $I_{{\cdot}/{\text{\rm c}}}(M,L)$ §2.2.2
  * $I_{\text{\rm cv}}(M;E)$ §2.2.4, §2.3.2
  * $I^{m}(M,L)$ §2.2.3
  * indicial family §2.5.28
  * infinitesimal holonomy §3.1.2
  * $I_{\nu}(A,\lambda)$ §2.5.28
  * $J^{(\infty)}(M,L)$ §2.6.7
  * $J(\mathcal{F})$ §5.1
  * $J(M,L)$ §2.6.7
  * $J^{(s)}(M,L)$ §2.6.7
  * $J^{\prime}(M,L)$ §2.7.1
  * $J^{\prime\,(s)}(M,L)$ §2.7.1
  * $J^{\prime\,m}(M,L)$ §2.7.1
  * $J^{m}(M,L)$ §2.6.7
  * $K(\mathcal{F})$ §5.1
  * $K(M,L)$ §2.6.10
  * $\mathcal{K}(M)$ §2.5.13
  * $K^{(s)}(M,L)$ §2.6.10
  * $\mathcal{K}^{(s)}(M)$ §2.5.13
  * $K^{\prime}(M,L)$ §2.7.1
  * $\mathcal{K}^{\prime}(M)$ §2.5.21
  * $K^{\prime\,(s)}(M,L)$ §2.7.1
  * $\mathcal{K}^{\prime\,(s)}(M)$ §2.5.21
  * $\mathring{K}^{\prime}_{L,l}$ §7.2
  * $K^{\prime\,m}(M,L)$ §2.7.1
  * $\mathcal{K}^{\prime\,m}(M)$ §2.5.21
  * $\mathring{K}_{\pm}$ §7.1
  * $\mathring{K}_{\pm,\gamma}$ §7.1
  * $K_{A}$ §2.1.5
  * $\mathring{K}_{l}$ §7.2
  * $K^{m}(M,L)$ §2.6.10
  * $\mathcal{K}^{m}(M)$ §2.5.13
  * $\ell(c)$ §4.1.1
  * $L(X,Y)$ §2.1.1
  * $L^{\infty}(M;E)$ §2.1.10
  * $L^{2}(M;E)$ §2.1.10
  * $\operatorname{lb}$ §2.5.26
  * LCHS §2.1.1
  * LCS §2.1.1
  * leafwise cohomology §3.2.1
  * leafwise complex §3.2.1
  * leafwise currents §3.2.8
  * leafwise differential complex §3.1.8
  * leafwise differential operator §3.1.8
  * leafwise distance §3.1.6
  * leafwise Euler form §3.2.7
  * leafwise form §3.2.1
  * leafwise homotopy §3.1.7
  * leafwise homotopy operator §3.2.16
  * leafwise metric §3.1.6
  * leafwise penumbra §3.1.6
  * leafwise principal symbol §3.1.8
  * leafwise-closed form §3.2.1
  * leafwise-exact form §3.2.1
  * left boundary §2.5.26
  * Lie foliation §3.1.9
  * limit subspace §2.1.1
  * $\mathcal{L}_{X,i,-i}$ §3.2.12
  * $\boldsymbol{M}$ §2.6.1
  * $M^{0}$ §4.1.2
  * $M^{1}$ §4.1.2
  * $M^{2}_{\text{\rm b}}$ §2.5.26
  * Novikov Betti number §2.9.1
  * $o(E)$ §2.1.3
  * $o(M)$ §2.1.3
  * oriented foliation §3.2.6
  * $\mathcal{P}(\phi)$ §4.1.1
  * $\mathring{P}^{\prime}_{L,l}$ §7.2
  * $\mathring{P}_{\pm}$ §7.1
  * $P_{\infty}$ §3.4.6
  * partial extension map §2.5.15
  * $\operatorname{Pen}_{\mathcal{F}}(S,r)$ §3.1.6
  * $\mathring{P}_{l}$ §7.2
  * positive injectivity bi-radius §3.4.1
  * principal b-symbol §2.5.27
  * $P_{u}$ §3.4.6
  * $R$ §2.1.5, §2.4.5
  * $R^{\prime}$ §6.1
  * $R^{\prime}_{s}$ §6.1
  * $\operatorname{rb}$ §2.5.26
  * regular §2.1.1
  * Riemannian foliation §3.1.9
  * Riemannian foliation of bounded geometry §3.4.1
  * right boundary §2.5.26
  * $R_{s}$ §5.1
  * $S^{(m)}(U\times\mathbb{R}^{l})$ §2.1.8
  * Schwartz kernel §2.1.5
  * simple flow §4.1.1
  * $\SS$ §2.9.12
  * $\operatorname{Str}$ §2.5.30
  * supported conormal distributions §2.5.8
  * supported distribution §2.5.3
  * supported dual-conormal distributions §2.5.21
  * supported function §2.5.2
  * supported Sobolev space §2.5.4
  * $\mathsf{T}$ §3.4.1
  * TC foliation §3.1.9
  * TP foliation §3.1.9
  * transitive foliation §3.1.9
  * transverse structure §3.1.9
  * transversely elliptic §3.1.8
  * transversely oriented foliation §3.1.9
  * transversely simple foliated flow §4.1.2
  * TVS §2.1.1
  * uniformly elliptic §2.4.4
  * uniformly leafwise elliptic §3.4.1
  * weakly simple foliated flow §4.1.2
  * Witten’s operators §2.9.1
  * $\mathfrak{X}(\mathcal{F})$ §3.1.3
  * $\mathfrak{X}(M,\mathcal{F})$ §3.1.3
  * $\mathfrak{X}(M,L)$ §2.2.1
  * $\mathfrak{X}(\Sigma,\mathcal{H})$ §3.1.3
  * $x^{a}H_{\text{\rm b}}^{m}(M;{}^{\text{\rm b}}\Omega^{1/2})$ §2.5.18
  * $\mathfrak{X}_{\text{\rm b}}(M)$ §2.5.7
  * $\mathfrak{X}_{\text{\rm com}}(M,\mathcal{F})$ §3.1.7
  * $\mathfrak{X}_{\text{\rm ub}}(\mathcal{F})$ §3.4.1
  * $\mathfrak{X}_{\text{\rm ub}}(M,\mathcal{F})$ §3.4.1
  * $Z(M,g,\eta)$ §2.9.12
  * $ZI^{(s)}(\mathcal{F})$ §5.1
  * $\beta_{\text{\rm No}}^{k}$ §2.9.1
  * $\beta_{z}^{k}$ §2.9.1
  * $\delta_{-i,i-1}$ §3.2.5
  * $\delta$-section §2.2.6
  * $\Delta_{0,z,z^{\prime}}$ §3.3.3
  * $\Delta_{\text{\rm b}}$ §2.5.26
  * $\delta_{\mathcal{F}}$ §3.2.5
  * $\Delta_{\mathcal{F},z}$ §3.3.2
  * $\delta_{\mathcal{F},z}$ §3.3.2
  * $\Delta_{\mathcal{F},z,z^{\prime}}$ §3.3.3
  * $\delta_{L}^{u}$ §2.2.6
  * $\Delta_{z,0}$ §3.3.1
  * $\delta_{z,i,i-1}$ §3.3.1
  * $\epsilon_{c}(k)$ §4.1.1
  * $\epsilon_{p}(\phi)$ §2.9.11
  * $\Theta_{X}$ §3.2.12
  * $\iota$ §2.1.5
  * $\iota^{\prime}$ §6.1
  * $\iota^{\prime}_{s}$ §6.1
  * $\iota_{s}$ §5.1
  * $\kappa^{\prime}_{L,l}$ §7.2
  * $\mathring{\kappa}^{\prime}_{L,l}$ §7.2
  * $\mathring{\kappa}_{\pm}$ §7.1
  * $\mathring{\kappa}_{\pm,\gamma}$ §7.1
  * $\Lambda E$ §2.1.3
  * $\Lambda\mathcal{F}$ §3.1.3
  * $\Lambda M$ §2.1.3
  * $\Lambda N\mathcal{F}$ §3.1.3
  * $\Lambda^{\text{\rm top}}E$ §2.1.3
  * $\Lambda^{u,v}M$ §3.2.2
  * $\\!\\!\\!{\phantom{\int}}^{\nu}\\!\\!\\!\int$ §2.5.29
  * $\Xi_{X}$ §3.4.2
  * $\boldsymbol{\pi}$ §2.6.1
  * $\Sigma$ §3.1.2
  * $\phi^{*}_{i,-i}$ §3.2.11
  * $\phi^{*}_{z,i,-i}$ §3.3.4
  * $\phi^{t*}_{z}$ §2.9.4
  * $\Psi^{m}(M;E)$ §2.1.9
  * $\Omega^{a}E$ §2.1.3
  * $\Omega^{a}M$ §2.1.3
  * $\Omega E$ §2.1.3
  * $\Omega_{\text{\rm fiber}}$ §2.1.6
  * $\Omega M$ §2.1.3