Basic Inequalities for Submanifolds of Conformal Kenmotsu Manifolds

Abstract

In this paper, we have established some basic inequalities for the submanifolds of conformal Kenmotsu manifolds. As an application, we have also derived the same inequalities for the $\theta$-slant submanifolds of conformal Kenmotsu manifolds.

1. Introduction

The famous Nash embedding theorem was motivated by the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, as late as 1985, this hope had not yet been realized. The main reason for this is due to the lack of controls for the extrinsic properties of the submanifolds by the known intrinsic invariants. In 1993, Chen made a significant contribution by introducing the renowned invariant ‘$\delta$’ and deriving optimal inequalities that relate this new intrinsic invariant to the principal extrinsic invariants for arbitrary Riemannian submanifolds [1,2].

Riemannian invariants play a central role in Riemannian geometry, as they characterize both the intrinsic and extrinsic properties of Riemannian manifolds, thereby influencing the overall behavior of the manifolds. The interplay between intrinsic and extrinsic invariants was elucidated by Chen [1], who established a connection between primary intrinsic invariants and key extrinsic invariants through a series of inequalities. Subsequently, numerous authors have explored this inequality in diverse contexts and settings. Many interesting results and inequalities involving this Chen invariant have been established in different submanifolds [3,4,5].

Another approach to addressing the connection between primary extrinsic and intrinsic invariants is through the use of inequalities concerning Casorati curvatures. Initially introduced for surfaces within Euclidean 3-space [6] as a normalized sum of squared principal curvatures, Casorati curvature has been generalized to encompass the broader scenario of submanifolds within a Riemannian manifold. Decu et al. extended its definition as the normalized square of the length of the second fundamental form of the submanifold [7,8]. These inequalities involving Casorati curvature have been investigated across various types of submanifolds.

Vilcu obtained an optimal inequality for Casorati curvature in Lagrangian submanifolds in complex space forms [9]. Brubaker and Suceava obtained the geometrical interpretation of Cauchy–Schwarz inequality in terms of Casorati curvature [10]. Lee et al. established the inequalities for Casorati curvature of submanifolds in generalized space forms endowed with semi-symmetric metric connection and Kenmotsu space forms [11,12]. The study of Casorati curvature on holomorphic statistical manifolds of constant holomorphic curvature was conducted by Decu et al. [13]. Similarly, the inequalities in slant submanifolds in metallic Riemannian space forms were established by Chaudhary and Blaga [14]. On the other side, contact geometry is one of the most fecund branches of differential geometry, with diverse applications. It has found applications in areas such as geometrical optics and in the mechanics of dynamical systems with time-dependent Hamiltonian dynamics [15]. Kenmotsu geometry, one of the newest chapters in contact geometry, was introduced by Katsui Kenmotsu in 1972 [16]. It was introduced to study the properties of the warped product in complex space with a real line, which was a natural problem, as this product is one of the three classes in Tanno’s classification of connected almost contact Riemannian manifolds with automorphism groups of maximum dimensions [17].

The conformal changes of almost metric structures were introduced by Vaisman [18]. The conformal change of the metric leads to a metric which is no more compatible with the almost contact structure, which can be corrected by a convenient change of the characteristic vector field, but this implies rather strong restrictions. A conformal Kenmotsu space form is a manifold that, under conformal deformations, forms a Kenmotsu space form. There are examples which are conformal Kenmotsu space forms but which are not Kenmotsu space forms. In this paper, we have introduced the inequalities for generalized normalized Casorati curvatures in conformal Kenmotsu space forms. The author has also introduced inequalities for the Chen invariant in the same submanifolds.

2. Preliminaries

Let M be an $(m+1)$-dimensional Riemannian submanifiold of a $(2n+1)$-dimensional Riemannian manifold $(\widehat{M},\widehat{g})$. Then, the metric tensor on M is denoted by g. Let $\{e_1,\dots ,e_{m+1}\}$ be an orthonormal basis for the tangent space $T_{p}M$ at point p and $\{e_{m+2},\dots ,e_{2n+1}\}$ be the orthonormal basis for the normal space $T_p^{\perp }M$. Let $K\left(\pi \right)$ be the sectional curvature associated with the plane section $\pi$⊂$T_{p}M$, then the scalar curvature is given as

$$\begin{array}{c}\hfill \tau \left(p\right)=\sum _{1\le i<j\le m+1}K(e_i\wedge e_j).\end{array}$$

The normalized scalar curvature $\rho$ for M is given as

$$\begin{array}{c}\hfill \rho ={\displaystyle \frac{2\tau }{m(m+1)}}.\end{array}$$

The mean curvature vector H of M in $\widehat{M}$ is given as

$$H\left(p\right)={\displaystyle \frac{1}{m+1}}\sum _{i=1}^{m+1}h(e_i,e_i),$$

where h is the second fundamental form of M in $\widehat{M}$.

Furthermore, set

$$\begin{array}{c}\hfill h_{ij}^{\alpha }=g(h(e_i,e_j),e_{\alpha }),\end{array}$$

where $i,j$ ∈ $\{1,\dots ,m+1\}$ and $\alpha$ ∈ $\{m+2,\dots ,2n+1\}$.

The Casorati curvature for the submanifold M is given as

$$\begin{array}{c}\hfill C={\displaystyle \frac{1}{m+1}}\sum _{\alpha =m+2}^{2n+1}\sum _{i,j=1}^{m+1}{\left(h_{ij}^{\alpha }\right)}^2.\end{array}$$

If V is an r-dimensional subspace of $T_{p}M$, and let $\{e_1,\dots ,e_r\}$ be an orthonormal basis of V, then we have

$$\begin{array}{c}\hfill C\left(V\right)={\displaystyle \frac{1}{r}}\sum _{\alpha =m+2}^{2n+1}\sum _{i,j=1}^r{\left(h_{ij}^{\alpha }\right)}^2.\end{array}$$

The generalized normalized $\delta$-Casorati curvatures ${\delta }_C(r;m)$ and ${\widehat{\delta }}_C(r;m)$ for any positive real number r are given as [8]

$${\left[{\delta }_C(r;m)\right]}_p=r{C}_p+{\displaystyle \frac{m(m+1+r)(m^2+m-r)}{r(m+1)}}inf\{C\left(V\right):V\mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},$$

if $0<r<m^2+m$.

$${\left[{\widehat{\delta }}_C(r;m)\right]}_p=r{C}_p-{\displaystyle \frac{m(m+1+r)(r-m^2-m)}{r(m+1)}}sup\{C\left(V\right):V\mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},$$

if $r>m^2+m$.

The normalized $\delta$-Casorati curvatures ${\delta }_C\left(m\right)$ and ${\widehat{\delta }}_C\left(m\right)$ are given as

$$\begin{array}{c}\hfill {\left[{\delta }_C\left(m\right)\right]}_p={\displaystyle \frac{1}{2}}{C}_p+{\displaystyle \frac{m+2}{2(m+1)}}inf\{C\left(V\right):V\mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\},\end{array}$$

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_C\left(m\right)\right]}_p=2C_p-{\displaystyle \frac{2m+1}{2(m+1)}}sup\{C\left(V\right):V\mathrm{is}\mathrm{a}\mathrm{hyperplane}\mathrm{of}{T}_{p}M\}.\end{array}$$

The above equations imply that normalized $\delta$-Casorati curvature equalities can be obtained from generalized normalized equalities by providing a suitable value to the positive number r [19]. A $2n+1$-dimensional differentiable manifold $\widehat{M}$ is said to be an almost contact metric manifold if it admits an almost contact metric structure $(\varphi ,\widehat{\zeta },\widehat{\eta },\widehat{g})$ consisting of a tensor field $\varphi$ of type $(1,1)$, a vector field $\widehat{\zeta }$, a $1-form$$\widehat{\eta }$ and a Riemannian metric $\widehat{g}$ which is compatible with $(\varphi ,\widehat{\zeta },\widehat{\eta },\widehat{g})$ and satisfies the following properties:

$$\begin{array}{c}\hfill {\varphi }^2=-Id+\widehat{\eta }\otimes \widehat{\zeta },\widehat{\eta }\left(\widehat{\zeta }\right)=1,\\ \hfill \varphi \widehat{\zeta }=0,\widehat{\eta }o\varphi =0,\\ \hfill \widehat{g}(\varphi X,\varphi Y)=\widehat{g}(X,Y)-\widehat{\eta }\left(X\right)\widehat{\eta }\left(Y\right),\\ \hfill \widehat{\eta }\left(X\right)=\widehat{g}(X,\widehat{\xi }).\end{array}$$

An almost contact manifold $({\widehat{M}}^{2m+1},\varphi ,\widehat{\zeta },\widehat{\eta },\widehat{g})$ is said to be a Kenmotsu manifold if [16]

$$\left({\widehat{\nabla }}_X\varphi \right)Y=-\widehat{g}(X,\varphi Y)\widehat{\zeta }-\widehat{\eta }\left(Y\right)\varphi X.$$

From the foregoing equation, we obtain

$${\widehat{\nabla }}_X\widehat{\zeta }=X-\widehat{\eta }\left(X\right)\widehat{\zeta },$$

where $\widehat{\nabla }$ denotes the Riemannian connection of $\widehat{g}$ and X and Y are vector fields on $\widehat{M}$.

For any $(m+1)$-dimensional submanifold with the induced metric g of $\widehat{M}$, we have the following decompositions for any vector field X tangent to M and N normal to M:

$$\varphi X=PX+FX,$$

$$\varphi N=tN+fN.$$

where $PX\left(FX\right)$ denotes the tangential (normal) component of $\varphi X$ and $tN\left(fN\right)$ denotes the tangential (normal) component of $\varphi N$. Here it is worth mentioning that P is an endomorphism of the tangent bundle $TM$, while F is a normal bundle valued 1-form on $TM$. Given a local frame $\{e_1,\dots ,e_{m+1}\}$ on M, the squared norms of P and F are given as

$${\left|\right|P\left|\right|}^2=\sum _{i,j=1}^{m+1}g{(e_i,P{e}_j)}^2,$$

$${\left|\right|F\left|\right|}^2=\sum _{i,j=1}^{m+1}\left|\right|F{e}_i{\left|\right|}^2,$$

where ${\left|\right|P\left|\right|}^2$ and ${\left|\right|F\left|\right|}^2$ are independent of the choice of orthonormal frame.

Let us suppose that $(\widehat{M},\varphi ,\widehat{\zeta },\widehat{\eta },\widehat{g})$ is a Kenmotsu manifold. This $\widehat{M}$ is said to be a Kenmotsu space form if its $\varphi$-sectional curvature of $\varphi$-holomorphic plane $[X,\varphi X]$ is only dependent on the point p, but not on the $\varphi$-holomorphic plane. It is usually denoted by $\widehat{M}\left(c\right)$. The Kenmotsu manifold has constant $\varphi$-sectional curvature c at a point if and only if the curvature tensor $\widehat{R}$ follows the equation [16]:

$$\begin{array}{ccc}\hfill \widehat{R}(X,Y)Z& =& {\displaystyle \frac{c-3}{4}}\{\widehat{g}(Y,Z)X-\widehat{g}(X,Z)Y\}+{\displaystyle \frac{c+1}{4}}\{\eta \left(X\right)\eta \left(Z\right)Y\hfill \\ & -& \eta \left(Y\right)\eta \left(Z\right)X+\eta \left(Y\right)\widehat{g}(X,Z)\zeta -\eta \left(X\right)\widehat{g}(Y,Z)\zeta \hfill \\ & -& \widehat{g}(\varphi X,Z)\varphi Y+\widehat{g}(\varphi Y,Z)\varphi X+2\widehat{g}(X,\varphi Y)\varphi Z\},\hfill \end{array}$$

for all $X,Y,Z$ on $\widehat{M}$.

A smooth manifold $\widehat{M}$ with an almost contact metric structure $(\varphi ,\zeta ,\eta ,g)$ is said to be a conformal Kenmotsu manifold if there exists a positive function $f:\widehat{M}\to R$, such that [20]

$$\begin{array}{c}\hfill \widehat{g}=exp\left(f\right)g,\widehat{\zeta }={\left(exp(-f)\right)}^{{\displaystyle \frac{1}{2}}}\zeta ,\widehat{\eta }={\left(exp\left(f\right)\right)}^{{\displaystyle \frac{1}{2}}}\eta ,\widehat{\varphi }=\varphi .\end{array}$$

Manifold $({\widehat{M}}^{2n+1},\varphi ,\zeta ,\eta ,g)$ is called a conformal Kenmotsu space form if ${\widehat{M}}^{2n+1}$ with this new almost contact metric structure, $(\varphi ,\widehat{\zeta },\widehat{\eta },\widehat{g})$ is a Kenmotsu space form and is denoted by $\widehat{M}\left(c\right)$.

Let $\widehat{R}$ and R denote the Riemannian curvature tensors on $({\widehat{M}}^{2n+1},\varphi ,\widehat{\zeta },\widehat{\eta },\widehat{g})$ and $({\widehat{M}}^{2n+1},\varphi ,\zeta ,\eta ,g)$, respectively; then, the relation between these curvature tensors is given as [20]

$$\begin{array}{ccc}\hfill R(X,Y,Z,W)& =& exp(-f)\left\{\widehat{R}(X,Y,Z,W)\right\}+{\displaystyle \frac{1}{2}}\{B(Y,Z)g(X,W)-B(X,Z)g(Y,W)\hfill \\ & +& B(X,W)g(Y,Z)-B(Y,W)g(X,Z)\}\hfill \\ & +& {\displaystyle \frac{1}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2\{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\},\hfill \end{array}$$

(1)

for all $X,Y,Z$ on $\widehat{M}$, where

$$\begin{array}{c}\hfill B=\nabla \omega -{\displaystyle \frac{1}{2}}\omega \otimes \omega ,\end{array}$$

$$\omega \left(X\right)=g\left(grad\right(f),X).$$

For the conformal Kenmotsu manifolds, we also have [20]

$$\begin{array}{c}\hfill {\nabla }_X\zeta ={\left(exp\left(f\right)\right)}^{{\displaystyle \frac{1}{2}}}\{X-\eta \left(X\right)\zeta \}-{\displaystyle \frac{1}{2}}\{\omega \left(\zeta \right)X-\eta \left(X\right)grad\left(f\right)\}.\end{array}$$

Example 1 

([20]). Consider a 3-dimensional manifold $M=\{(x,y,z)\in {\mathbb{R}}^3:x>0\}$ with the orthonormal frame

$$\begin{array}{cc}\hfill e_1& =x{\displaystyle \frac{\partial }{\partial z}},\hfill \\ \hfill e_2& =x{\displaystyle \frac{\partial }{\partial y}},\hfill \\ \hfill e_3& =-e^{x/2}x{\displaystyle \frac{\partial }{\partial x}}.\hfill \end{array}$$

Let g be the metric defined by

$$g(e_i,e_j)=\left\{\begin{array}{cc}{e}^{-x}\hfill & \mathrm{if}i=j=1,2\hfill \\ 1\hfill & \mathrm{if}i=j=3\hfill \\ 0\hfill & \mathrm{if}i\ne j.\hfill \end{array}\right.$$

Let η be the 1-form defined by

$$\eta \left(e_3\right)=1,\eta \left(e_2\right)=0,\eta \left(e_1\right)=0.$$

Define the almost contact structure as

$$\phi e_1=e_2,\phi e_2=-e_1,\phi e_3=0,\xi =e_3,\eta \left(e_3\right)=1.$$

By a contact transformation,

$$\tilde{g}=\mathit{exp}(\mathit{x})g,\tilde{\xi }=(exp{(-x)}^{\frac{1}{2}})\xi ,\tilde{\eta }=(exp{\left(x\right)}^{\frac{1}{2}})\eta ,\tilde{\phi }=\phi .$$

Then, $(M,\tilde{\phi },\tilde{\eta },\tilde{g})$ is a Kenmotsu manifold. So $(M,\phi ,\eta ,g)$ is a conformal Kenmotsu manifold, but not a Kenmotsu manifold, as follows:

$$\begin{array}{c}\hfill \left({\nabla }_X\phi \right)Y\ne -g(X,\phi Y)\xi -\eta \left(Y\right)\phi X.\end{array}$$

3. Main Results

Casorati Curvature Inequalities

Theorem 1. 

If M is an $(m+1)$-dimensional submanifold in a $(2n+1)$-dimensional conformal Kenmotsu space form $\widehat{M}\left(c\right)$, then

(i) 

The generalized normalized δ-Casorati curvature ${\delta }_C(r,m)$ satisfies

$$\begin{array}{ccc}\hfill {\delta }_C(r,m)& \ge & m(m+1)\left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]+exp(-f)\{{\displaystyle \frac{m(c+1)}{2}}\hfill \\ & -& {\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2\}-mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

(2)

(ii) 

The generalized normalized δ-Casorati curvature ${\widehat{\delta }}_C(r,m)$ satisfies

$$\begin{array}{ccc}\hfill {\widehat{\delta }}_C(r,m)& \ge & m(m+1)\left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]+exp(-f)\{{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}{\parallel P\parallel }^2\}\hfill \\ & -& mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}{sup[\parallel grad\left(f\right)\parallel }^2].\hfill \end{array}$$

(3)

In the inequality (2), the equality holds if, and only if, the submanifold M is invariantly quasi-umbilical with trivial normal connection in $\widehat{M}$, such that the shape operator $A_{e_i}$ with i ∈ $\{m+2,\dots ,2n+1\}$ with respect to suitable orthonormal tangent frame $\{e_1,\dots ,e_{m+1}\}$ and normal orthonormal frame $\{e_{m+2},\dots ,e_{2n+1}\}$ takes the following form:

$$A_{e_{m+2}}=\left(\begin{array}{cccccc}u& 0& 0& \cdots & 0& 0\\ 0& u& 0& \cdots & 0& 0\\ 0& 0& u& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & u& 0\\ 0& 0& 0& \cdots & 0& {\displaystyle \frac{m(m+1)}{r}}u\end{array}\right),A_{e_{m+3}}=\dots ,=A_{e_{2n+1}}=0.$$

(4)

Proof. 

Gaussian equation for any $(m+1)$-dimensional submanifold M in $(2n+1)$ conformal Kenmotsu manifold $\widehat{M}$ is given as

$$\begin{array}{c}\hfill R(X,Y,Z,W)=R^{*}(X,Y,Z,W)+g(h(X,W),h(Y,Z))-g(h(X,Z),h(Y,W)),\end{array}$$

for all $X,Y,Z,W$ ∈ Γ(TM), where R and $R^{*}$ are curvature tensors with respect to $\widehat{\nabla }$ and ∇, respectively, while $\widehat{\nabla }$ and ∇ are Levi-Civita connections on $\widehat{M}$ and M, respectively.

The relation between curvature tensors R and $\widehat{R}$ of conformal Kenmotsu manifolds and Kenmotsu manifolds, respectively, is given in Equation (1), which is again established as

$$\begin{array}{ccc}\hfill R(X,Y,Z,W)& =& exp(-f)\left\{\widehat{R}(X,Y,Z,W)\right\}+{\displaystyle \frac{1}{2}}\{B(Y,Z)g(X,W)-B(X,Z)g(Y,W)\hfill \\ & +& B(X,W)g(Y,Z)-B(Y,W)g(X,Z)\}\hfill \\ & +& {\displaystyle \frac{1}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2\{g(Y,Z)g(X,W)-g(X,Z)g(Y,W)\}.\hfill \end{array}$$

(5)

Let {$e_1$,…$e_{m+1}$} be an orthonormal basis of the tangent space $T_{p}M$, and $\left\{e_{m+2},\dots ,e_{2n+1}\right\}$ be an orthonormal basis of the normal space $T_p^{\perp }M$, $p\in M$.

If a Kenmotsu manifold is assumed to be a Kenmotsu space form with ϕ-sectional curvature equal to constant c, then the curvature tensor $\widehat{R}$ is given as

$$\begin{array}{ccc}\hfill \widehat{R}(X,Y)Z& =& {\displaystyle \frac{c-3}{4}}\{\widehat{g}(Y,Z)X-\widehat{g}(X,Z)Y\}+{\displaystyle \frac{c+1}{4}}\{\eta \left(X\right)\eta \left(Z\right)Y\hfill \\ & -& \eta \left(Y\right)\eta \left(Z\right)X+\eta \left(Y\right)\widehat{g}(X,Z)\zeta -\eta \left(X\right)\widehat{g}(Y,Z)\zeta \hfill \\ & -& \widehat{g}(\varphi X,Z)\varphi Y+\widehat{g}(\varphi Y,Z)\varphi X+2\widehat{g}(X,\varphi Y)\varphi Z\}.\hfill \end{array}$$

(6)

Taking the inner product with vector field W in such a way that $X=W=e_i$ and $Y=Z=e_j$. By a Gaussian equation and Equation (6), Equation (5) can be simplified as

$$\begin{array}{ccc}\hfill 2\tau \left(p\right)& =& {(m+1)}^2{\left|\right|H\left|\right|}^2-{\left|\right|h\left|\right|}^2+exp(-f)\{{\displaystyle \frac{c-3}{4}}m(m+1)\hfill \\ & +& {\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2-{\displaystyle \frac{m(c+1)}{2}}\}\hfill \\ & +& mTrace\left(B\right)+{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2,\hfill \end{array}$$

(7)

where ${\left|\right|h\left|\right|}^2$ = ${\sum }_{i,j=1}^{m+1}g(h(e_1,e_j),h(e_i,h_j))$.

Now, consider the function Q which is associated with the following polynomial in the components $\left(h_{ij}^{\alpha }\right),i,j=1,\dots ,m+1;\alpha =m+2,\dots ,2n+1$ of second fundamental form h of M in $\widehat{M}$:

$$\begin{array}{ccc}\hfill Q& =& rC+{\displaystyle \frac{m(m+r+1)(m^2+m-r)}{r(m+1)}}C\left(L\right)-2\tau +exp(-f)\{{\displaystyle \frac{c-3}{4}}m(m+1)\hfill \\ & +& {\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2-{\displaystyle \frac{m(c+1)}{2}}\}+mTrace\left(B\right)+{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2,\hfill \end{array}$$

where L is the hyperplane of $T_{p}M$.

If L is assumed to be spanned by $e_1,\dots ,e_m$, then we have

$$\begin{array}{ccc}\hfill Q& =& {\displaystyle \frac{r}{m+1}}\sum _{\alpha =m+2}^{2n+1}\sum _{i,j=1}^{m+1}{\left(h_{ij}\right)}^2+{\displaystyle \frac{(m+r+1)(m^2+m-r)}{r(m+1)}}\sum _{\alpha =m+2}^{2n+1}\sum _{i,j=1}^{m+1}{\left(h_{ij}\right)}^2\hfill \\ & -& 2\tau +exp(-f)\{{\displaystyle \frac{c-3}{4}}m(m+1)+{\displaystyle \frac{3(c+1)}{4}}||P|{|}^2-{\displaystyle \frac{m(c+1)}{2}}\}\hfill \\ & +& mTrace\left(B\right)+{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

Using Equation (7), we obtain

$$\begin{array}{ccc}\hfill Q& =& {\displaystyle \frac{m+r+1}{m+1}}\sum _{\alpha =m+2}^{2n+1}\sum _{i,j=1}^{m+1}{\left(h_{ij}\right)}^2\hfill \\ & +& {\displaystyle \frac{(m+r+1)(m^2+m-r)}{r(m+1)}}\sum _{\alpha =m+2}^{2n+1}\sum _{i,j=1}^{m+1}{\left(h_{ij}\right)}^2-\sum _{\alpha =n+2}^{2n+1}{\left(\sum _{i+1}^{m+1}{h}_{ii}^{\alpha }\right)}^2.\hfill \end{array}$$

The foregoing equation can be written as

$$\begin{array}{ccc}\hfill Q& =& \sum _{\alpha =m+2}^{2n+1}\sum _{i=1}^m\left[{\displaystyle \frac{(m^2+m(r+1)-r)}{r}}{\left(h_{ii}^{\alpha }\right)}^2+{\displaystyle \frac{2(m+r+1)}{m+1}}{\left(h_{im+1}^{\alpha }\right)}^2\right]\hfill \\ & +& \sum _{\alpha =m+2}^{2n+1}\left[{\displaystyle \frac{2m(m+r+1)}{r}}\sum _{1\le i<j\le m}{\left(h_{ij}^{\alpha }\right)}^2-2\sum _{1\le i<j\le m+1}{h}_{ii}^{\alpha }{h}_{jj}^{\alpha }\right]+{\displaystyle \frac{r}{m+1}}\sum _{\alpha =m+2}^{2n+1}{\left(h_{m+1m+1}^{\alpha }\right)}^2.\hfill \end{array}$$

(8)

Hence, Q is a quadratic polynomial in the components of the second fundamental form, and from Equation (8), we deduce its critical points:

$$\begin{array}{ccc}\hfill h^c& =& \left[h_{11}^{m+1},h_{12}^{m+2},\dots ,h_{m+1m+1}^{m+2},\dots ,h_{11}^{2n+1},h_{12}^{2n+1},\dots ,h_{m+1m+1}^{2n+1}\right]\hfill \end{array}$$

are the solutions of the following system of linear homogeneous equations:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial Q}{\partial h_{ii}^{\alpha }}}& =& {\displaystyle \frac{2m(m+r+1)}{r}}{h}_{ii}^{\alpha }-2\sum _{k=1}^{m+1}{h}_{kk}^{\alpha }=0,\hfill \\ \hfill {\displaystyle \frac{\partial Q}{\partial h_{m+1m+1}^{\alpha }}}& =& {\displaystyle \frac{2r}{m+1}}{h}_{m+1}^{\alpha }-2\sum _{k=1}^{m+1}{h}_{kk}^{\alpha }=0,\hfill \\ \hfill {\displaystyle \frac{\partial Q}{\partial h_{ij}^{\alpha }}}& =& {\displaystyle \frac{4m(m+r+1)}{r}}{h}_{ij}^{\alpha }=0,\hfill \\ \hfill {\displaystyle \frac{\partial Q}{\partial h_{im+1}^{\alpha }}}& =& {\displaystyle \frac{4(m+r+1)}{m+1}}{h}_{im+1}^{\alpha }=0,\hfill \end{array}$$

with $i,j\in \{1,\dots ,m\}$, $i\ne j$ and $\alpha \in \{m+2,\dots ,2n+1\}$.

The above system of equations shows that every solution $h^c$ has $h_{ij}^{\alpha }=0$ for $i\ne j$. The Hessian matrix can be computed as

$$H\left(p\right)=\left(\begin{array}{ccc}{H}_1& 0& 0\\ 0& H_2& 0\\ 0& 0& H_3\end{array}\right),$$

Here, $H_1$ is given as

$$H_1=\left(\begin{array}{cccccc}{\displaystyle \frac{2m(m+r+1)}{r}}-2& -2& -2& \cdots & -2& -2\\ -2& {\displaystyle \frac{2m(m+r+1)}{r}}-2& -2& \cdots & -2& -2\\ -2& -2& {\displaystyle \frac{2m(m+r+1)}{r}}-2& \cdots & -2& -2\\ ⋮& ⋮& \ddots & ⋮\\ -2& -2& -2& \cdots & {\displaystyle \frac{2m(m+r+1)}{r}}-2& -2\\ -2& -2& -2& \cdots & -2& {\displaystyle \frac{2r}{m+1}}\end{array}\right),$$

$$\begin{array}{ccc}\hfill H_2& =& diag\left({\displaystyle \frac{4m(m+r+1)}{r}},{\displaystyle \frac{4m(m+r+1)}{r}},\dots ,{\displaystyle \frac{4m(m+r+1)}{r}}\right),\hfill \\ \hfill H_3& =& diag\left({\displaystyle \frac{4(m+r+1)}{m+1}},{\displaystyle \frac{4(m+r+1)}{m+1}},\dots ,{\displaystyle \frac{4(m+r+1)}{m+1}}\right).\hfill \end{array}$$

It can be found by direct computation that the Hessian matrix $H\left(Q\right)$ of Q has the following eigenvalues:

$$\begin{array}{ccc}\hfill {\lambda }_{11}& =& 0,{\lambda }_{22}={\displaystyle \frac{2(m{(m+1)}^2+r^2)}{r(m+1)}},{\lambda }_{33}=\dots ,={\lambda }_{m+1m+1}={\displaystyle \frac{2m(m+r+1)}{r}},\hfill \\ \hfill {\lambda }_{ij}& =& {\displaystyle \frac{4m(m+r+1)}{r}},{\lambda }_{im+1}={\displaystyle \frac{4(m+r+1)}{m+1}};,\forall i,j\in \{1,\dots ,m\},i\ne j.\hfill \end{array}$$

It follows that the Hessian matrix is positive, semi-definite, and admits one eigenvalue equal to zero. Hence, we can see that Q is parabolic and reaches a minimum at $h^c$. In fact, at the critical point $h^c$, the function Q reaches the global minimum. So, we have $Q\left(h^c\right)=0$. Therefore, we deduce $Q\ge 0$, and this implies that

$$\begin{array}{ccc}\hfill 2\tau & \le & rC+{\displaystyle \frac{m(m+r+1)(m^2+m-r)}{r(m+1)}}C\left(L\right)+exp(-f)\{{\displaystyle \frac{c-3}{4}}m(m+1)\hfill \\ & +& {\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2-{\displaystyle \frac{m(c+1)}{2}}\}+mTrace\left(B\right)+{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

(9)

We know that the normalized scalar curvature ρ is given as

$$\rho ={\displaystyle \frac{2\tau }{m(m+1)}}.$$

Using the foregoing equation in Equation (9), we obtain

$$\begin{array}{ccc}\hfill \rho & \le & {\displaystyle \frac{r}{m(m+1)}}C+{\displaystyle \frac{(m+r+1)(m^2+m-r)}{r{(m+1)}^2}}C\left(L\right)+exp(-f)\{{\displaystyle \frac{c-3}{4}}\hfill \\ & +& {\displaystyle \frac{3(c+1)}{4m(m+1)}}{\left|\right|P\left|\right|}^2-{\displaystyle \frac{(c+1)}{2(m+1)}}\}+{\displaystyle \frac{1}{m+1}}Trace\left(B\right)+{\displaystyle \frac{1}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

The foregoing equation is equivalent to

$$\begin{array}{ccc}\hfill rC+{\displaystyle \frac{m(m+r+1)(m^2+m-r)}{r(m+1)}}C\left(L\right)& \ge & m(m+1)\left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f)\{{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}||P|{|}^2\}\hfill \\ & -& mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

The required inequalities follow by taking the infimum and supremum, respectively, over all the tangent hyperplanes $T_{p}M$.

On the other hand, it is clear that the equality sign follows if, and only if, $h_{ij}^{\alpha }=0$, $\forall i,j$ ∈ $\{1,\dots ,m+1\}$, $i\ne j$,

and

$$h_{m+1m+1}^{\alpha }={\displaystyle \frac{m(m+1)}{r}}{h}_{11}^{\alpha }=,\dots ,={\displaystyle \frac{m(m+1)}{r}}{h}_{mm}^{\alpha },$$

for all α ∈ $\{m+2,\dots ,2n+1\}$.

If ζ is tangent to M, then by the Gaussian formula, we have

$${\nabla }_{e_i}^{*}{\zeta }_p={\nabla }_{e_i}\zeta +h(e_i,{\zeta }_p),$$

for $i=1,\dots ,m+1$.

Furthermore, from [20], we have

$${\nabla }_{e_i}\zeta +h(e_i,{\zeta }_p)={\left(exp\left(f\right)\right)}^{{\displaystyle \frac{1}{2}}}\{e_i-\eta \left(e_i\right){\zeta }_p\}-{\displaystyle \frac{1}{2}}\{\omega \left({\zeta }_p\right)e_i-\eta \left(e_i\right)grad\left(f\right)\}.$$

By equating the normal components, we obtain

$$h(e_i,{\zeta }_p)={\displaystyle \frac{1}{2}}\eta \left(e_i\right)grad\left(f\right).$$

Since $h(e_i,{\zeta }_p)$≠ 0, we have that the equality holds only when the submanifold is invariantly quasi-umbilical and the shape operator takes the form given in Equation (4). Similarly, inequality (3) can be proved. □

Remark 1. 

The results established in inequalities (2) and (3) are generalized ones. These inequalities can be further modified, but those results depend on the sign of the term

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{{3(c+1)\left|\right|P\left|\right|}^2}{4}}-{\displaystyle \frac{m(m+1)(c-3)}{4}}\right].\end{array}$$

If the above term is positive, the inequalities will take the following form:

$$\begin{array}{ccc}\hfill {\delta }_C(r,m)& \ge & m(m+1)\left[\rho -inf\left(exp(-f)\right){\displaystyle \frac{c-3}{4}}\right]+inf\left(exp(-f)\right)\{{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2\}\hfill \\ & -& mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \\ \hfill {\widehat{\delta }}_C(r,m)& \ge & m(m+1)\left[\rho -inf\left(exp(-f)\right){\displaystyle \frac{c-3}{4}}\right]+inf\left(exp(-f)\right)\{{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2\}\hfill \\ & -& mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

Similarly, inequalities for the negative sign of the established term can be also derived.

Corollary 1. 

If M is an $(m+1)$-dimensional submanifold in $(2n+1)$-dimensional conformal Kenmotsu space form, then

(i) 

The normalized δ-Casorati curvature ${\delta }_C\left(m\right)$ satisfies the following:

$$\begin{array}{ccc}\hfill {\delta }_C\left(m\right)& \ge & \left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f){\displaystyle \frac{1}{m(m+1)}}\{{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2\}\hfill \\ & -& {\displaystyle \frac{1}{m+1}}Trace\left(B\right)-{\displaystyle \frac{1}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

(10)

(ii) 

The normalized δ-Casorati curvature $\widehat{{\delta }_C\left(m\right)}$ satisfies the following:

$$\begin{array}{ccc}\hfill {\widehat{\delta }}_C\left(m\right)& \ge & \left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f){\displaystyle \frac{1}{m(m+1)}}\{{\displaystyle \frac{m(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2\}\hfill \\ & -& {\displaystyle \frac{1}{m+1}}Trace\left(B\right)-{\displaystyle \frac{1}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

(11)

In the inequalities (10) and (11), the equality holds if, and only if, the submanifold M is invariantly quasi-umbilical with trivial normal connection in $\widehat{M}$, such that the shape operator $A_{e_i}$ with i ∈ $\{m+2,\dots ,2n+1\}$ with respect to suitable orthonormal tangent frame $\{e_1,\dots ,e_{m+1}\}$ and normal orthonormal frame $\{e_{m+2},\dots ,e_{2n+1}\}$, takes the following forms:

$$A_{e_{m+2}}=\left(\begin{array}{cccccc}u& 0& 0& \cdots & 0& 0\\ 0& u& 0& \cdots & 0& 0\\ 0& 0& u& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & u& 0\\ 0& 0& 0& \cdots & 0& 2u\end{array}\right),$$

$$A_{e_{m+3}}=\dots ,=A_{e_{2n+1}}=0.$$

and

$$A_{e_{m+2}}=\left(\begin{array}{cccccc}2u& 0& 0& \cdots & 0& 0\\ 0& 2u& 0& \cdots & 0& 0\\ 0& 0& 2u& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 2u& 0\\ 0& 0& 0& \cdots & 0& u\end{array}\right),$$

$$A_{e_{m+3}}=\dots ,=A_{e_{2n+1}}=0.$$

Proof. 

It can be easily seen that

$${[{\delta }_C({\displaystyle \frac{m(m+1)}{2}},m)]}_p=m(m+1){\left[{\delta }_C\left(m\right)\right]}_p,$$

and

$${[{\widehat{\delta }}_C(2m(m+1),m)]}_p=m(m+1){[{\widehat{\delta }}_C\left(m\right)]}_p.$$

The proof follows by putting $r={\displaystyle \frac{m(m+1)}{2}}$ in (2) and $r=2m(m+1)$ in (3) and using above two equations. □

Corollary 2. 

If M is an $(m+1)$-dimensional θ-slant submanifold in $(2n+1)$-dimensional conformal Kenmotsu space form, then

(i) 

The generalized normalized δ-Casorati curvature ${\delta }_C(r,m)$ satisfies the following:

$$\begin{array}{ccc}\hfill {\delta }_C(r,m)& \ge & m(m+1)\left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f)m\{{\displaystyle \frac{(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}co{s}^2\left(\theta \right)\}\hfill \\ & -& mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

(ii) 

The generalized normalized δ-Casorati curvature ${\widehat{\delta }}_C(r,m)$ satisfies the following:

$$\begin{array}{ccc}\hfill {\widehat{\delta }}_C(r,m)& \ge & m(m+1)\left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f)m\{{\displaystyle \frac{(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}co{s}^2\left(\theta \right)\}\hfill \\ & -& mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

The equality in the foregoing inequalities holds if, and only if, the submanifold M is invariantly quasi-umbilical with trivial normal connection in $\widehat{M}$.

Proof. 

The proof follows immediately from Theorem 1. By [21], for $(m+1)$-dimensional slant submanifold with θ slant angle, tangent to the structure vector field ζ, we have:

$${\left|\right|P\left|\right|}^2=mco{s}^2\left(\theta \right)$$

(12)

□

Corollary 3. 

If M is an $(m+1)$-dimensional θ-slant submanifold in $(2n+1)$-dimensional conformal Kenmotsu manifold, then

(i) 

The normalized δ-Casorati curvature ${\delta }_C\left(m\right)$ satisfies the following:

$$\begin{array}{ccc}\hfill {\delta }_C\left(m\right)& \ge & \left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f){\displaystyle \frac{1}{(m+1)}}\{{\displaystyle \frac{(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}co{s}^2\left(\theta \right)\}\hfill \\ & -& {\displaystyle \frac{1}{m+1}}Trace\left(B\right)-{\displaystyle \frac{1}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

(ii) 

The normalized δ-Casorati curvature ${\widehat{\delta }}_C\left(m\right)$ satisfies the following:

$$\begin{array}{ccc}\hfill {\widehat{\delta }}_C\left(m\right)& \ge & \left[\rho -exp(-f){\displaystyle \frac{c-3}{4}}\right]\hfill \\ & +& exp(-f){\displaystyle \frac{1}{(m+1)}}\{{\displaystyle \frac{(c+1)}{2}}-{\displaystyle \frac{3(c+1)}{4}}co{s}^2\left(\theta \right)\}\hfill \\ & -& {\displaystyle \frac{1}{m+1}}Trace\left(B\right)-{\displaystyle \frac{1}{4}}sup\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

The equality in the foregoing inequalities holds if, and only if, the submanifold M is invariantly quasi-umbilical with trivial normal connection in $\widehat{M}$.

Proof. 

The proof follows from Corollaries 1 and 2. □

4. Chen Invariant Inequalities

Before introducing chen invariants we recall the following lemma from [1]:

Lemma 1. 

If $k\ge 2$ and $a_1,a_2,\dots a_k,a$ are real numbers such that

$${\left(\sum _i^k{a}_i\right)}^2=(k-1)\left(\sum _{i=1}^k{\left(a_i\right)}^2+a\right),$$

then $2a_1{a}_2\ge a$ with equality holding if, and only if,

$$a_1+a_2=a_3=\dots =a_k.$$

Theorem 2. 

Let M be any $(m+1)$-dimensional submanifold of the conformal Kenmotsu space form ${\widehat{M}}^{2n+1}\left(c\right)$, which has the constant sectional curvature c. Then, for any point p and any plane π ⊂ $T_{p}M$, we have

$$\begin{array}{ccc}\hfill \tau -K\left(\pi \right)& \le & {\displaystyle \frac{{(m+1)}^2(m+2)}{2m}}{\left|\right|H\left|\right|}^2+exp(-f){\displaystyle \frac{(c-3)(m+1)(m-2)}{8}}\hfill \\ & +& exp(-f)\{{\displaystyle \frac{3(c+1)}{8}}|\left|P\right|{|}^2-{\displaystyle \frac{m(c+1)}{4}}-{\displaystyle \frac{3(c+1)}{4}}{g}^2(e_1,\varphi e_2)\}\hfill \\ & +& {\displaystyle \frac{1}{2}}\{mTraceB-(B(e_1,e_1)+B(e_2,e_2))\}+{\displaystyle \frac{m^2+m-2}{8}}inf\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

(13)

Equality holds if the shape operator takes the following form:

$$A_{m+2}=\left(\begin{array}{ccc}t& 0& 0\\ 0& u& 0\\ 0& 0& \lambda I_{m-1}\end{array}\right)$$

such that $t+u=\lambda$,

$$A_{e_r}=\left(\begin{array}{ccc}{h}_{11}^r& h_{12}^r& 0\\ h_{12}^r& -h_{11}^r& 0\\ 0& 0& 0_{m-1}\end{array}\right)$$

where $r\in \{m+3,\dots ,2n+1\}$.

Proof. 

Since M is submanifold of $\widehat{M}\left(c\right)$, from (7), we have

$$\begin{array}{ccc}\hfill 2\tau \left(p\right)& =& {(m+1)}^2{\left|\right|H\left|\right|}^2-{\left|\right|h\left|\right|}^2+exp(-f)\{{\displaystyle \frac{c-3}{4}}m(m+1)\hfill \\ & +& {\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2-{\displaystyle \frac{m(c+1)}{2}}\}\hfill \\ & +& mTrace\left(B\right)+{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

Let

$$\begin{array}{ccc}\hfill ϵ& =& 2\tau -{\displaystyle \frac{{(m+1)}^2(m-1)}{m}}{\left|\right|H\left|\right|}^2\hfill \\ & -& exp(-f){\displaystyle \frac{c-3}{4}}(m+1)(m-2)-exp(-f)\{{\displaystyle \frac{3(c+1)}{4}}{\left|\right|P\left|\right|}^2\hfill \\ & -& {\displaystyle \frac{m(c+1)}{2}}\}-mTrace\left(B\right)-{\displaystyle \frac{m(m+1)}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

Then, the foregoing equations imply that

$$\begin{array}{c}\hfill ϵ={\displaystyle \frac{{(m+1)}^2}{m}}{\left|\right|H\left|\right|}^2-{\left|\right|h\left|\right|}^2+exp(-f){\displaystyle \frac{c-3}{2}},\end{array}$$

or

$$\begin{array}{c}\hfill {(m+1)}^2{\left|\right|H\left|\right|}^2={\left|\right|h\left|\right|}^2+m\left(ϵ-exp(-f){\displaystyle \frac{c-3}{2}}\right).\end{array}$$

(14)

Let p ∈ M, π ⊂ $T_{p}M$. We choose an orthonormal basis $\{e_1,\dots e_{m+1}\}$ of $T_{p}M$ and $\{e_{m+2},\dots ,e_{2n+1}\}$ of $T_p^{\perp }M$, such that ${\zeta }_p=e_{m+1}$. Further assume that π is spanned by $e_1$ and $e_2$ and $e_{m+2}$ is in the direction of mean curvature vector H. Then, by Equation (14), we have

$$\begin{array}{ccc}\hfill {\left(\sum _{i=1}^{m+1}{h}_{ii}^{m+2}\right)}^2& =& m\left(\sum _{i=1}^{m+1}{\left(h_{ii}^{m+2}\right)}^2+\sum _{i\ne j}{\left(h_{ii}^{m+2}\right)}^2+\sum _{r=m+3}^{2m+1}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2+ϵ-exp(-f){\displaystyle \frac{c-3}{2}}\right).\hfill \end{array}$$

Now, by using Lemma 1, we have

$$\begin{array}{c}\hfill 2h_{11}^{m+2}{h}_{22}^{m+2}\ge \sum _{i\ne j}{\left(h_{ii}^{m+2}\right)}^2+\sum _{r=m+3}^{2m+1}\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2+ϵ-exp(-f){\displaystyle \frac{c-3}{2}}.\end{array}$$

(15)

Now, by the definition of sectional curvature of the plane π and using the Gaussian equations, Equations (6) and (7), we have

$$\begin{array}{ccc}\hfill K\left(\pi \right)& =& g(h(e_1,e_1),h(e_2,e_2))-g(h(e_1,e_2),h(e_1,e_2))\hfill \\ & +& exp(-f)\left({\displaystyle \frac{c-3}{4}}+{\displaystyle \frac{3(c+1)}{4}}{g}^2(e_1,\varphi e_2)\right)+{\displaystyle \frac{1}{2}}\left(B(e_1,e_1)+B(e_2,e_2)\right)+{\displaystyle \frac{1}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2,\hfill \end{array}$$

or

$$\begin{array}{ccc}\hfill K\left(\pi \right)& =& \sum _{r=m+2}^{2n+1}g(h(e_1,e_1),e_r)g(h(e_2,e_2),e_r)-\sum _{r=m+2}^{2n+1}g(h(e_1,e_2),e_r)g(h(e_1,e_2),e_r)\hfill \\ & +& exp(-f)\left({\displaystyle \frac{c-3}{4}}+{\displaystyle \frac{3(c+1)}{4}}{g}^2(e_1,\varphi e_2)\right)+{\displaystyle \frac{1}{2}}\left(B(e_1,e_1)+B(e_2,e_2)\right)+{\displaystyle \frac{1}{4}}{\left|\right|grad\left(f\right)\left|\right|}^2.\hfill \end{array}$$

From the foregoing equations, we obtain

$$\begin{array}{ccc}\hfill K\left(\pi \right)& \ge & \sum _{m+2}^{2n+1}\sum _{j>2}\{{(h_{1j}^r)}^2+{(h_{2j}^r)}^2\}+{\displaystyle \frac{1}{2}}\sum _{1\ne j>2}{(h_{ij}^{m+2})}^2\hfill \\ & +& {\displaystyle \frac{1}{2}}\sum _{m+3}^{2n+1}\sum _{j>2}{(h_{1j}^r)}^2+{\displaystyle \frac{1}{2}}\sum _{m+3}^{2n+1}{\left(h_{11}^r+h_{22}^r\right)}^2+{\displaystyle \frac{ϵ}{2}}\hfill \\ & +& exp(-f){\displaystyle \frac{3(c+1)}{4}}{g}^2(e_1,\varphi e_2)+{\displaystyle \frac{1}{2}}\left(B(e_1,e_1)+B(e_2,e_2)\right)+{\displaystyle \frac{1}{4}}\left|\right|grad\left(f\right){\left|\right|}^2,\hfill \end{array}$$

or,

$$\begin{array}{c}\hfill K\left(\pi \right)\ge {\displaystyle \frac{ϵ}{2}}+exp(-f){\displaystyle \frac{3(c+1)}{4}}{g}^2(e_1,\varphi e_2)+{\displaystyle \frac{1}{2}}\left(B(e_1,e_1)+B(e_2,e_2)\right)+{\displaystyle \frac{1}{4}}\left|\right|grad\left(f\right){\left|\right|}^2.\end{array}$$

(16)

By substituting the value of ϵ and taking the infimum value of ${\left|\right|gradf\left|\right|}^2$, we obtain the required inequality.

In (13), the equality holds if, and only if, Equations (15) and (16) become equalities. In such cases, we have

$$\begin{array}{ccc}\hfill h_{1j}^{m+2}& =& h_{2j}^{m+2}=h_{ij}^{m+2}=0i\ne j\ge 2,\hfill \\ \hfill h_{ij}^r& =& 0\forall i\ne j,r=m+3,\dots ,2n+1,i,j=3,\dots ,m+3,\hfill \\ \hfill h_{11}^r& +& h_{22}^r=0\forall r=m+3,\dots ,2n+1,\hfill \\ \hfill h_{11}^{m+2}& +& h_{22}^{m+2}=h_{33}^{m+2}=\dots =h_{m+1m+1}^{m+2}.\hfill \end{array}$$

Now, we can choose $e_1$ and $e_2$ so that $h_{12}^{m+2}=0$, and let $t=h_{11}^r$, $u=h_{22}^r$ and $\lambda =h_{33}^{m+2}=\dots =h_{m+1m+1}^{m+2}$.

This shows that the shape operator takes the desired form. □

Corollary 4. 

Let M be any $(m+1)$-dimensional θ-slant submanifold of the conformal Kenmotsu space form ${\widehat{M}}^{2n+1}\left(c\right)$, which has the constant sectional curvature c. Then for any point p and any plane π ⊂ $T_{p}M$, we have:

$$\begin{array}{ccc}\hfill \tau -K\left(\pi \right)& \le & {\displaystyle \frac{{(m+1)}^2(m+2)}{2m}}{\left|\right|H\left|\right|}^2+exp(-f){\displaystyle \frac{(c-3)(m+1)(m-2)}{8}}\hfill \\ & +& exp(-f)\{{\displaystyle \frac{3(c+1)}{8}}co{s}^2\left(\theta \right)-{\displaystyle \frac{m(c+1)}{4}}-{\displaystyle \frac{3(c+1)}{4}}{g}^2(e_1,\varphi e_2)\}\hfill \\ & +& {\displaystyle \frac{1}{2}}\{mTraceB-(B(e_1,e_1)+B(e_2,e_2))\}+{\displaystyle \frac{m^2+m-2}{8}}inf\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

Proof. 

Using (12) in Equation (13), we get the desired inequality. □

Corollary 5. 

Let M be any $(m+1)$-dimensional submanifold of the conformal Kenmotsu space form ${\widehat{M}}^{2n+1}\left(c\right)$, which has the constant sectional curvature $c=-1$. Then, for any point p and any plane $\pi \subset T_{p}M$, we have

$$\begin{array}{ccc}\hfill \tau -K\left(\pi \right)& \le & {\displaystyle \frac{{(m+1)}^2(m+2)}{2m}}{\left|\right|H\left|\right|}^2-exp(-f){\displaystyle \frac{(m+1)(m-2)}{2}}\hfill \\ & +& {\displaystyle \frac{1}{2}}\{mTraceB-(B(e_1,e_1)+B(e_2,e_2))\}+{\displaystyle \frac{m^2+m-2}{8}}inf\left[{\left|\right|grad\left(f\right)\left|\right|}^2\right].\hfill \end{array}$$

Proof. 

The proof follows by putting the value of $c=-1$ in Equation (13). □