Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

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Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

This version is not peer-reviewed.

Siraj Uddin,

Bang-Yen Chen^*,Rawan Bossly

Siraj Uddin,

Bang-Yen Chen^*,Rawan Bossly

This version is not peer-reviewed.

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Submitted:

13 May 2023

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15 May 2023

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A peer-reviewed article of this preprint also exists.

Abstract

Recently, we studied CR-slant warped products B1×fM⊥, where B1=MT×Mθ is the Riemannian product of holomorphic and proper slant submanifolds and M⊥ is a totally real submanifold in a nearly Kaehler manifold. In the continuation, in this paper, we study B2×fMθ, where B2=MT×M⊥ is a CR-product of a nearly Kaehler manifold and establish Chen’s inequality for the squared norm of the second fundamental form. Some special cases of Chen’s inequality are given.

Keywords:

Subject:

Computer Science and Mathematics - Geometry and Topology

MSC: 53B05; 53B20; 53C25; 53C40

1. Introduction

The study of CR-warped products was initiated by the second author in [6,7]. In [20], we studied CR-slant warped product submanifolds of the form

B_{1} \times_{f} M_{⊥}

, where

B_{1} = M_{T} \times M_{θ}

is the Riemannian product of holomorphic and proper slant submanifolds and

M_{⊥}

is a totally real submanifold of a nearly Kaehler manifold

\tilde{M}

. In fact, we established the following Chen’s inequality:

Theorem 1.

[20] Let

M = B_{1} \times_{f} M_{⊥}

be a CR-slant warped product submanifold of a nearly Kaehler manifold

\tilde{M}

such that M is

D^{⊥} \oplus D^{θ}

-mixed totally geodesic in

\tilde{M}

, where

B_{1} = M_{T} \times M_{θ}

is the Riemannian product of complex and proper slant submanifolds of

\tilde{M}

. Then:

(i): The second fundamental form h satisfies

$\begin{matrix} {∥ h ∥}^{2} \geq 2 s ∥ {\vec{\nabla}}^{T} {(ln f) ∥}^{2} + s {cot}^{2} θ {∥ {\vec{\nabla}}^{θ} (ln f) ∥}^{2} \end{matrix}$

(1)

where $s = dim M_{⊥}$ and ${\vec{\nabla}}^{T} (ln f)$ and ${\vec{\nabla}}^{θ} (ln f)$ denote the gradient components of $ln f$ along $M_{T}$ and $M_{θ}$ , respectively.
(ii): If the equality sign in (1) holds identically, then $M_{T}$ and $M_{θ}$ are totally geodesic, $B_{1}$ is mixed totally geodesic in $\tilde{M}$ and $M_{⊥}$ is totally umbilical in $\tilde{M}$ .

In the sequel, in this paper, we study CR-slant warped product submanifold

M = B_{2} \times_{f} M_{θ}^{n_{3}}

, where

B_{2} = M_{T}^{n_{1}} \times M_{⊥}^{n_{2}}

is the CR-product and

M_{θ}^{n_{3}}

is an

n_{3}

-dimensional proper

θ

-slant submanifold in a nearly Kaehler manifold

{\tilde{M}}^{2 m}

. We prove that the second fundamental form h of M satisfies the following inequality:

\begin{matrix} {∥ h ∥}^{2} \geq \frac{1}{9} n_{3} {cos}^{2} θ ∥ {\vec{\nabla}}^{⊥} {(ln f) ∥}^{2} + 2 n_{3} (1 + \frac{10}{9} {cot}^{2} θ) {∥ {\vec{\nabla}}^{T} (ln f) ∥}^{2} \end{matrix}

where

{\vec{\nabla}}^{⊥} (ln f)

and

{\vec{\nabla}}^{T} (ln f)

are the gradients of

ln f

along

M_{⊥}

and

M_{T}

, respectively. The equality case is discussed and some special cases of the inequality are given.

2. Basic definitions and formulas

Let

{\tilde{M}}^{2 m}

be an almost Hermitian manifold endowed with an almost complex structure J and a Riemannian metric

\tilde{g}

such that

\begin{matrix} J^{2} (X) = - X, \tilde{g} (J X, J Y) = \tilde{g} (X, Y) \end{matrix}

(2)

for any

X, Y \in Γ (T {\tilde{M}}^{2 m})

, where

Γ (T {\tilde{M}}^{2 m})

denotes the Lie algebra of vector fields on

{\tilde{M}}^{2 m}

. In addition, an almost Hermitian manifold is called Kaehler manifold if

\begin{matrix} ({\tilde{\nabla}}_{X}) Y = 0, \forall X, Y \in Γ (T {\tilde{M}}^{2 m}), \end{matrix}

where

\tilde{\nabla}

is the Levi-Civita connection on

{\tilde{M}}^{2 m}

. Furthermore, an almost Hermitian manifold

{\tilde{M}}^{2 m}

is nearly Kaehler if

({\tilde{\nabla}}_{X}) X = 0, \forall X \in Γ (T {\tilde{M}}^{2 m})

, equivalently

\begin{matrix} ({\tilde{\nabla}}_{X}) Y + ({\tilde{\nabla}}_{Y}) X = 0, \forall X, Y \in Γ (T {\tilde{M}}^{2 m}) . \end{matrix}

(3)

Clearly, every Kaehler manifold is nearly Kaehler but the converse is not true in general. The best known example of a nearly Kaehler non-Kaehlerian manifold is 6-dimensional sphere

S^{6}

Let

M^{n}

be a Riemannian manifold isometrically immersed in an almost Hermitian manifold

{\tilde{M}}^{2 m}, (n < 2 m)

and from now on we denote the metric

\tilde{g}

and the induced metric g on M by the same symbol g. Then the Gauss and Weingarten formulas are respectively given by (see, for instance, [6,10])

\begin{matrix} {\tilde{\nabla}}_{X} Y = \nabla_{X} Y + h (X, Y), \end{matrix}

(4)

\begin{matrix} {\tilde{\nabla}}_{X} ξ = - A_{ξ} X + \nabla_{X}^{⊥} ξ, \end{matrix}

(5)

for vector fields

X, Y \in Γ (T M)

and

ξ \in Γ (T^{⊥} M)

, where

Γ (T^{⊥} M)

is the set of all vector fields normal to M and ∇ and

\nabla^{⊥}

denote the induced connections on the tangent and normal bundles of M, respectively, and h is the second fundamental form, A is the shape operator of M and they are related by

\begin{matrix} g (A_{ξ} X, Y) = g (h (X, Y), ξ), \forall X, Y \in Γ (T M), ξ \in Γ (T^{⊥} M) . \end{matrix}

(6)

For each vector field X tangent to

M^{n}

, we write

\begin{matrix} J X = P X + F X . \end{matrix}

(7)

where

P X

and

F X

are the tangential and normal components of

J X

. Complex and totally real submanifolds are defined on the behaviour of almost complex structure J on

M^{n}

. A submanifold

M^{n}

of an almost Hermitian manifold

\tilde{M}

is complex if its tangent space remains the same under the action of almost complex structure J. On contrary, M is a totally real submanifold of

\tilde{M}

J X \in Γ (T^{⊥} M)

for any

X \in Γ (T M)

A submanifold M of an almost Hermitian manifold

\tilde{M}

is called CR-submanifold if there exists on M a differentiable holomorphic distribution

D : p \to D_{p} \subset T_{p} M

whose orthogonal complementary distribution

D^{⊥}

is totally real. A CR-submanifold M of an almost Hermitian manifold

\tilde{M}

is called aCR-product if it is a Riemannian product of a holomorphic submanifold

M_{T}

and a totally real submanifold

M_{⊥}

\tilde{M}

. The second author introduced the notion of CR-product of Kaehler manifolds in [3].

In [4], the second author introduced another important class of submanifolds and he called them slant submanifolds those are the generalization of complex and totally real submanifolds. He defined slant submanifolds as:

Definition 1.

A submanifold M of an almost Hermitian manifold

\tilde{M}

is called slant if for each

p \in M

, the Wirtinger angle

θ (X)

between

J X

and

T_{p} M

is constant on M, i.e., it does not depend on the choice of

X \in T_{p} M

and

p \in M

[4,5]. In this case, θ is called the slant angle of M.

Complex (holomorphic) and totally real submanifolds are slant submanifolds with slant angles 0 and

\frac{π}{2}

, respectively. A slant submanifold is called proper slant if it is neither holomorphic nor totally real.

More generally, a distribution

D

on M is called a slant distribution if the angle

θ (X)

between

J X

and

D_{p}

is independent of the choice of

p \in M

and of

0 \neq X \in D_{p}

He proved that a submanifold M of an almost Hermitian manifold

\tilde{M}

is slant if and only if [4]

\begin{matrix} P^{2} X = - ({cos}^{2} θ) X, X \in Γ (T M) . \end{matrix}

(8)

Clearly, from (7) and (8), we know that

\begin{matrix} g (P X, P Y) = ({cos}^{2} θ) g (X, Y), g (F X, F Y) = ({sin}^{2} θ) g (X, Y), \end{matrix}

(9)

for any vector fields

X, Y

tangent to M.

Definition 2.

A submanifold M of an almost Hermitian manifold

\tilde{M}

isCR-slant submanifold (skew CR-submanifold) if there exist orthogonal distributions

D, D^{⊥}

and

D^{θ}

such that the tangent bundle

T M

is spanned by

\begin{matrix} T M = D \oplus D^{⊥} \oplus D^{θ}, \end{matrix}

(10)

where

D, D^{⊥}

and

D^{θ}

are complex, totally real and proper slant distributions.

The normal bundle of a CR-slant submanifold M is decomposed by

\begin{matrix} T^{⊥} M = J D^{⊥} \oplus F D^{θ} \oplus ν, \end{matrix}

(11)

where

ν

is an invariant normal subbundle of

T^{⊥} M

A CR-slant product submanifold M is semi-slant mixed totally geodesic (resp., hemi-slant mixed totally geodesic) if its second fundamental satisfies

\begin{matrix} h (X_{1}, X_{2}) = 0 \forall X_{1} \in Γ (D), \forall X_{2} \in Γ (D^{θ}) \\ (r e s p ., h (X_{2}, X_{3}) = 0 \forall X_{2} \in Γ (D^{θ}), \forall X_{3} \in Γ (D^{⊥})) . \end{matrix}

3. CR-slant warped products $(M_{T} \times M_{⊥}) \times_{f} M_{θ}$

In this section, first we recall the definition of warped product manifolds which are the generalizations of Riemannian products. In 1969, Bishop and O’Neill [2] introduced the notion of warped product manifolds as follows:

Definition 3.

A warped product

B \times_{f} F

of two Riemannian manifolds

(B, g_{B})

and

(F, g_{F})

is the product manifold

M = B \times F

equipped with the product structure

\begin{matrix} g_{M} (X, Y) = g_{B} (π_{1 *} X, π_{1 *} Y) + {(f \circ π_{1})}^{2} g_{F} (π_{2 *} X, π_{2 *} Y) \end{matrix}

where

f : B \to (0, \infty)

and

π_{1} : M \to B, π_{2} : M \to F

are projection maps given by

π_{1} (p, q) = p

and

π_{2} (p, q) = q

for any

(p, q) \in B \times F

and ∗ denotes the symbol for tangent map.

The function f is called warping function, if f is constant, then M is simply a Riemannian product. It is known that, for any vector field X on B and a vector field Z on F, we have

\begin{matrix} \nabla_{X} Z = \nabla_{Z} X = X (ln f) Z \end{matrix}

(12)

where ∇ is the Levi-Civita connection on M. Further, it is well known that the base manifold B is totally geodesic and the fiber F is totally umbilical in M.

Now, we define CR-slant warped products

(M_{T} \times M_{⊥}) \times_{f} M_{θ}

Definition 4.

A submanifold M of an almost Hermitian manifold

\tilde{M}

is said to be CR-slant warped product submanifold if it is a warped product of CR-product

M_{T} \times M_{⊥}

and a proper θ-slant submanifold

M_{θ}

\tilde{M}

In [20], we discussed CR-slant warped product submanifolds of the form

B_{1} \times_{f} M_{⊥}

, where

B_{1} = M_{T} \times M_{θ}

. In this section we study CR-slant warped products of the form

B_{2} \times_{f} M_{θ}

, where

B_{2} = M_{T} \times M_{⊥}

. For this, we use the following conventions:

X_{1}, Y_{1}, \dots

are vector fields on

D

and

X_{2}, Y_{2} \dots

are vector fields on

D^{θ}

, while

X_{3}, Y_{3}, \dots

are vector fields on

D^{⊥}

First, we have the following preparatory lemmas.

Lemma 1.

On a CR-slant warped product submanifold

M = B_{2} \times_{f} M_{θ}

of a nearly Kaehler manifold

\tilde{M}

, we have

(i): $g (h (X_{1}, Y_{1}), F X_{2}) = 0,$
(ii): $2 g (h (X_{3}, Y_{3}), F X_{2}) = g (h (X_{2}, X_{3}), J Y_{3}) + g (h (X_{2}, Y_{3}), J X_{3}),$

for any

X_{1}, Y_{1} \in Γ (T M_{T}), X_{2} \in Γ (T M_{θ})

and

X_{3}, Y_{3} \in Γ (T M_{⊥})

, where

B_{2} = M_{T} \times M_{⊥}

is the CR-product submanifold in

\tilde{M}

Proof.

The first part is easy to prove by using (4), (3) and (12). For the second part, we have

\begin{matrix} g (h (X_{3}, Y_{3}), F X_{2}) = g ({\tilde{\nabla}}_{X_{3}} Y_{3}, J X_{2}) + g ({\tilde{\nabla}}_{X_{3}} P X_{2}, Y_{3}) - g (J \nabla_{X_{3}} Y_{3}, X_{2}) + g (\nabla_{X_{3}} Y_{3}, P X_{2}) \end{matrix}

for any

X_{2} \in Γ (T M_{θ})

and

X_{3}, Y_{3} \in Γ (T M_{⊥})

. Since

\nabla_{X_{3}} Y_{3} \in Γ (T M_{⊥})

, then using orthogonality of vector fields and covariant derivative property of J with (12), we find

\begin{matrix} g (h (X_{3}, Y_{3}), F X_{2}) & = g (({\tilde{\nabla}}_{X_{3}} J) Y_{3}, X_{2}) - g ({\tilde{\nabla}}_{X_{3}} J Y_{3}, X_{2}) + X_{3} (ln f) g (P X_{2}, Y_{3}) \\ = g (({\tilde{\nabla}}_{X_{3}} J) Y_{3}, X_{2}) + g (h (X_{2}, X_{3}), J Y_{3}) \end{matrix}

(13)

Similarly, by interchanging

X_{3}

with

Y_{3}

in (13), we brain

\begin{matrix} g (h (X_{3}, Y_{3}), F X_{2}) = g (({\tilde{\nabla}}_{Y_{3}} J) X_{3}, X_{2}) + g (h (X_{2}, Y_{3}), J X_{3}) . \end{matrix}

(14)

Hence, the second part immediately follows from (13) and (14). □

Lemma 2.

Let

M = B_{2} \times_{f} M_{θ}

be a CR-slant warped product submanifold of a nearly Kaehler manifold

\tilde{M}

such that

B_{2} = M_{T} \times M_{⊥}

is the CR-product submanifold in

\tilde{M}

. Then, we have

\begin{matrix} g (h (X_{1}, X_{3}), F X_{2}) = \frac{1}{2} g (h (X_{1}, X_{2}), J X_{3}) \end{matrix}

(15)

for any

X_{1} \in Γ (T M_{T}), X_{2} \in Γ (T M_{θ})

and

X_{3} \in Γ (T M_{⊥})

Proof.

For any

X_{1} \in Γ (T M_{T}), X_{2} \in Γ (T M_{θ})

and

X_{3} \in Γ (T M_{⊥})

, we have

\begin{matrix} g (h (X_{1}, X_{3}), F X_{2}) = g (({\tilde{\nabla}}_{X_{3}} J) X_{1}, X_{2}) - g ({\tilde{\nabla}}_{X_{3}} J X_{1}, X_{2}) = g (({\tilde{\nabla}}_{X_{3}} J) X_{1}, X_{2}) . \end{matrix}

(16)

On the other hand, we know that

\begin{matrix} g (h (X_{1}, X_{3}), F X_{2}) = g (({\tilde{\nabla}}_{X_{1}} J) X_{3}, X_{2}) - g ({\tilde{\nabla}}_{X_{1}} J X_{3}, X_{2}) + g (X_{3}, {\tilde{\nabla}}_{X_{1}} P X_{2}) . \end{matrix}

(17)

Then, the lemma follows from (16) and (17) with the help of (3) and (12). □

Lemma 3.

For a proper CR-slant warped product

M = B_{2} \times_{f} M_{θ}

such that

B_{2} = M_{T} \times M_{⊥}

in a nearly Kaehler manifold

\tilde{M}

, we have

\begin{matrix} g (h (J X_{1}, X_{2}), F Y_{2}) = X_{1} (ln f) g (X_{2}, Y_{2}) + \frac{1}{3} J X_{1} (ln f) g (X_{2}, P Y_{2}) \end{matrix}

(18)

for any

X_{1} \in Γ (T M_{T}), X_{2}, Y_{2} \in Γ (T M_{θ})

Proof.

From (4) and (12), we have

\begin{matrix} g (h (X_{1}, X_{2}), F Y_{2}) = g (({\tilde{\nabla}}_{X_{2}} J) X_{1}, Y_{2}) - J X_{1} (ln f) g (X_{2}, Y_{2}), \end{matrix}

(19)

for any orthogonal vector fields

X_{1} \in Γ (T M_{T}), X_{2}, Y_{2} \in Γ (T M_{θ})

. On the other hand, we derive

\begin{matrix} g (h (X_{1}, X_{2}), F Y_{2}) = g (({\tilde{\nabla}}_{X_{1}} J) X_{2}, Y_{2}) - X_{1} (ln f) g (P X_{2}, Y_{2}) + g (h (X_{1}, Y_{2}), F X_{2}) . \end{matrix}

(20)

Then, from (19) and (20), we find

\begin{matrix} 2 g (h (X_{1}, X_{2}), F Y_{2}) = X_{1} (ln f) g (X_{2}, P Y_{2}) - J X_{1} (ln f) g (X_{2}, Y_{2}) + g (h (X_{1}, Y_{2}), F X_{2}) . \end{matrix}

(21)

Interchanging

X_{2}

Y_{2}

, we obtain

\begin{matrix} 2 g (h (X_{1}, Y_{2}), F X_{2}) = X_{1} (ln f) g (P X_{2}, Y_{2}) - J X_{1} (ln f) g (X_{2}, Y_{2}) + g (h (X_{1}, X_{2}), F Y_{2}) . \end{matrix}

(22)

Then, from (21) and (22), we derive

\begin{matrix} g (h (X_{1}, X_{2}), F Y_{2}) = - J X_{1} (ln f) g (X_{2}, Y_{2}) + \frac{1}{3} X_{1} (ln f) g (X_{2}, P Y_{2}) . \end{matrix}

(23)

Hence, (18) follows immediately by interchanging

X_{1}

with

J X_{1}

in (23), which proves the lemma completely. □

The following relations are immediate consequences of (18).

\begin{matrix} g (h (J X_{1}, P X_{2}), F Y_{2}) = X_{1} (ln f) g (P X_{2}, Y_{2}) + \frac{1}{3} {cos}^{2} θ J X_{1} (ln f) g (X_{2}, Y_{2}), \end{matrix}

(24)

\begin{matrix} g (h (J X_{1}, P X_{2}), F P Y_{2}) = {cos}^{2} θ X_{1} (ln f) g (X_{2}, Y_{2}) + \frac{1}{3} {cos}^{2} θ J X_{1} (ln f) g (X_{2}, P Y_{2}), \end{matrix}

(25)

\begin{matrix} g (h (J X_{1}, X_{2}), F P Y_{2}) = X_{1} (ln f) g (X_{2}, P Y_{2}) - \frac{1}{3} {cos}^{2} θ J X_{1} (ln f) g (X_{2}, Y_{2}) . \end{matrix}

(26)

Lemma 4.

Let

M = B_{2} \times_{f} M_{θ}

be a CR-slant warped product submanifold of a nearly Kaehler manifold

\tilde{M}

such that

B_{2} = M_{T} \times M_{⊥}

is the CR-product submanifold in

\tilde{M}

. Then, we have

\begin{matrix} g (h (X_{2}, Y_{2}), J X_{3}) = g (h (X_{2}, X_{3}), F Y_{2}) + \frac{1}{3} X_{3} (ln f) g (X_{2}, P Y_{2}) \end{matrix}

(27)

for any

X_{2}, Y_{2} \in Γ (T M_{θ})

and

X_{3} \in Γ (T M_{⊥})

Proof.

From the definition of covariant derivative with (4) and (7), we have

\begin{matrix} g (h (X_{2}, X_{3}), F Y_{2}) = g (({\tilde{\nabla}}_{X_{3}} J) X_{2}, Y_{2}) - g ({\tilde{\nabla}}_{X_{3}} P X_{2}, Y_{2}) - g ({\tilde{\nabla}}_{X_{3}} F X_{2}, Y_{2}) - g ({\tilde{\nabla}}_{X_{3}} X_{2}, P Y_{2}) . \end{matrix}

Again using (4), (5) and (12), we find

\begin{matrix} g (h (X_{2}, X_{3}), F Y_{2}) = g (({\tilde{\nabla}}_{X_{3}} J) X_{2}, Y_{2}) + g (h (Y_{2}, X_{3}), F X_{2}) . \end{matrix}

(28)

On the other hand, we derive

\begin{matrix} g (h (X_{2}, X_{3}), F Y_{2}) & = g (({\tilde{\nabla}}_{X_{2}} J) X_{3}, Y_{2}) - g ({\tilde{\nabla}}_{X_{2}} J X_{3}, Y_{2}) - g ({\tilde{\nabla}}_{X_{2}} X_{3}, P Y_{2}) \\ = g (({\tilde{\nabla}}_{X_{2}} J) X_{3}, Y_{2}) + g (h (X_{2}, Y_{2}), J X_{3}) - X_{3} (ln f) g (X_{2}, P Y_{2}) . \end{matrix}

(29)

Then, from (28) and (29), we get

\begin{matrix} 2 g (h (X_{2}, X_{3}), F Y_{2}) = g (h (X_{2}, Y_{2}), J X_{3}) + g (h (Y_{2}, X_{3}), F X_{2}) - X_{3} (ln f) g (X_{2}, P Y_{2}) . \end{matrix}

(30)

Interchanging

X_{2}

Y_{2}

, we obtain

\begin{matrix} 2 g (h (Y_{2}, X_{3}), F X_{2}) = g (h (X_{2}, Y_{2}), J X_{3}) + g (h (X_{2}, X_{3}), F Y_{2}) + X_{3} (ln f) g (X_{2}, P Y_{2}) . \end{matrix}

(31)

Then, from (30) and (31), we get (27); which proves the Lemma completely. □

4. Chen’s inequality and its consequences

In this section first we prove the following main result by using Lemma 3.

Theorem 2.

Let

M = B_{2} \times_{f} M_{θ}

be a proper CR-slant warped product submanifold of a nearly Kaehler manifold

\tilde{M}

. Then, M is simply Riemannian product if and only if either M is semi-slant mixed totally geodesic i.e.,

h (X_{1}, X_{2}) = 0, \forall X_{1} \in Γ (D), X_{2} \in Γ (D^{θ})

h (D, D^{θ})

is orthogonal to

F D^{θ}

Proof.

From Lemma 3, we find

\begin{matrix} g (h (J X_{1}, X_{2}), F Y_{2}) = \frac{1}{3} J X_{1} (ln f) g (X_{2}, P Y_{2}) + X_{1} (ln f) g (X_{2}, Y_{2}), \end{matrix}

(32)

for any

X_{1} \in Γ (D), X_{2}, Y_{2} \in Γ (D^{θ})

. Then, from () and (32), we derive

\begin{matrix} g (h (J X_{1}, X_{2}), F Y_{2}) + \frac{1}{3} g (h (X_{1}, X_{2}), F P Y_{2}) = (1 - \frac{1}{9} {cos}^{2} θ) X_{1} (ln f) g (X_{2}, Y_{2}) . \end{matrix}

(33)

If M is semi-slant mixed totally geodesic or

h (D, D^{θ})

is orthogonal to

F D^{θ}

then from (33), we find

\begin{matrix} (1 - \frac{1}{9} {cos}^{2} θ) X_{1} (ln f) g (X_{2}, Y_{2}) = 0 \end{matrix}

Since g is a Riemannian metric and

- 1 \leq cos θ \leq 1

, then from above equation we get

X_{1} (ln f) = 0

, i.e., f is constant along

M_{T}

Conversely, if f is constant then again from (33), we get

\begin{matrix} g (h (J X_{1}, X_{2}), F Y_{2}) + \frac{1}{3} g (h (X_{1}, X_{2}), F P Y_{2}) = 0 . \end{matrix}

(34)

Interchanging

X_{1}

J X_{1}

and

Y_{2}

P Y_{2}

in (34), we derive

\begin{matrix} g (h (X_{1}, X_{2}), F P Y_{2}) + \frac{1}{3} {cos}^{2} θ g (h (J X_{1}, X_{2}), F Y_{2}) = 0 . \end{matrix}

(35)

Then, from (34) and (35), we obtain

\begin{matrix} (1 - \frac{1}{9} {cos}^{2} θ) g (h (J X_{1}, X_{2}), F Y_{2}) = 0 . \end{matrix}

(36)

Since

- 1 \leq cos θ \leq 1

for any value of

θ \in R

, thus we find either

h (D, D^{θ}) = {0}

h (D, D^{θ})

is orthogonal to

F D^{θ}

, which completes the proof. □

Next, we derive the Chen’s inequality for CR-slant wanted products

M = B_{2} \times_{f} M_{θ}

, where

B_{2} = M_{T} \times M_{⊥}

is a CR-product in a nearly Kaehler manifold.

Theorem 3.

Let

M = (M_{T}^{n_{1}} \times M_{⊥}^{n_{2}}) \times_{f} M_{θ}^{n_{3}}

be a CR-slant warped product submanifold of a nearly Kaehler manifold

\tilde{M}

such that M is hemi-slant mixed totally geodesic. Then, the squared norm of the second fundamental form satisfies

\begin{matrix} {∥ h ∥}^{2} \geq \frac{1}{9} n_{3} {cos}^{2} θ ∥ {\vec{\nabla}}^{⊥} {(ln f) ∥}^{2} + 2 n_{3} (1 + \frac{10}{9} {cot}^{2} θ) {∥ {\vec{\nabla}}^{T} (ln f) ∥}^{2} \end{matrix}

(37)

where

{\vec{\nabla}}^{T} (ln f)

and

{\vec{\nabla}}^{⊥} (ln f)

denote the gradient components of

ln f

along

M_{T}

and

M_{⊥}

, respectively.

Furthermore, if the equality holds in (37), then

M_{T} \times M_{⊥}

is totally geodesic and

M_{θ}

is totally umbilical in

\tilde{M}

. Moreover, M is not a semi-slant mixed totally geodesic submanifold of

\tilde{M}

Proof.

If we denote the tangent bundles of

M_{T}, M_{⊥}

and

M_{θ}

D, D^{⊥}

and

D^{θ}

, respectively; then we use the following frame fields for the CR-slant warped product

\begin{matrix} D = Span {e_{1}, \dots, e_{p}, e_{p + 1} = J e_{1}, \dots, e_{n_{1}} = e_{2 p} = J e_{p}}, \\ D^{⊥} = Span {e_{n_{1} + 1} = {\hat{e}}_{1}, \dots, e_{n_{1} + n_{2}} = {\hat{e}}_{n_{2}}}, \\ D^{θ} = Span {e_{n_{1} + n_{2} + 1} = e_{1}^{*}, \dots, e_{n_{1} + n_{2} + q} = e_{q}^{*}, e_{n_{1} + n_{2} + q + 1} = sec θ P e_{1}^{*}, \dots, \\ e_{n} = e_{2 q}^{*} = sec θ P e_{q}^{*}} . \end{matrix}

And the normal bundle frame will be

\begin{matrix} J D^{⊥} = Span {e_{n + 1} = {\tilde{e}}_{1} = J {\hat{e}}_{1}, \dots, e_{n + n_{2}} = {\tilde{e}}_{n_{2}} = J {\hat{e}}_{n_{2}}}, \\ F D^{θ} = Span {e_{n + n_{2} + 1} = {\tilde{e}}_{n_{2} + 1} = E_{1}^{*} = csc θ F e_{1}^{*}, \dots, e_{n + n_{2} + q} = {\tilde{e}}_{n_{2} + q} = E_{q}^{*} = csc θ F e_{q}^{*}, \\ e_{n + n_{2} + q + 1} = {\tilde{e}}_{n_{2} + q + 1} = E_{q + 1}^{*} = csc θ sec θ F P e_{1}^{*}, \dots, \\ e_{n + n_{2} + n_{3}} = {\tilde{e}}_{n_{2} + n_{3}} = E_{n_{3}}^{*} = csc θ sec θ F P e_{q}^{*}}, \\ ν = Span {e_{n + n_{2} + n_{3} + 1} = {\tilde{e}}_{n_{2} + n_{3} + 1}, \dots, e_{2 m} = {\tilde{e}}_{2 m - n - n_{2} - n_{3}}} . \end{matrix}

From the definition of h, we find

\begin{matrix} {∥ h ∥}^{2} & = {∥ h (D, D) ∥}^{2} + ∥ h (D^{⊥}, D^{⊥}) ∥^{2} + {∥ h (D^{θ}, D^{θ}) ∥}^{2} \\ + 2 (∥ h (D, D^{⊥}) ∥^{2} + ∥ h (D, D^{θ}) ∥^{2} + {∥ h (D^{⊥}, D^{θ}) ∥}^{2}) . \end{matrix}

(38)

Using the frame fields and preparatory lemmas, we solve each term of (38) as follows:

\begin{matrix} {∥ h (D, D) ∥}^{2} = & \sum_{k = 1}^{n_{2}} \sum_{i, j = 1}^{n_{1}} {(g (h (e_{i}, e_{j}), J {\hat{e}}_{k}))}^{2} + \sum_{k = 1}^{n_{3}} \sum_{i, j = 1}^{n_{1}} {(g (h (e_{i}, e_{j}), E_{k}^{*}))}^{2} \\ + \sum_{k = 1}^{2 m - n - n_{2} - n_{3}} \sum_{i, j = 1}^{n_{1}} {(g (h (e_{i}, e_{j}), {\tilde{e}}_{k}))}^{2} . \end{matrix}

Leaving the

ν

-components terms and the is no warped product relation for the first term, then from Lemma 1 (i), we get

\begin{matrix} {∥ h (D, D) ∥}^{2} \geq 0 . \end{matrix}

(39)

Similarly, for the second term of (38), we derive

\begin{matrix} ∥ h (D^{⊥}, D^{⊥}) ∥^{2} = & \sum_{k = 1}^{n_{2}} \sum_{i, j = 1}^{n_{2}} {(g (h ({\hat{e}}_{i}, {\hat{e}}_{j}), J {\hat{e}}_{k}))}^{2} + \sum_{k = 1}^{n_{3}} \sum_{i, j = 1}^{n_{2}} {(g (h ({\hat{e}}_{i}, {\hat{e}}_{j}), E_{k}^{*}))}^{2} \\ + \sum_{k = 1}^{2 m - n - n_{2} - n_{3}} \sum_{i, j = 1}^{n_{2}} {(g (h ({\hat{e}}_{i}, {\hat{e}}_{j}), {\tilde{e}}_{k}))}^{2} . \end{matrix}

Using Lemma 1 (ii) with the given hemi-slant totally geodesic condition and leaving the first and last positive terms, we find

\begin{matrix} ∥ h (D^{⊥}, D^{⊥}) ∥^{2} \geq 0 . \end{matrix}

(40)

For the third term of (38), we find

\begin{matrix} ∥ h (D^{θ}, D^{θ}) ∥^{2} = & \sum_{k = 1}^{n_{2}} \sum_{i, j = 1}^{n_{3}} {(g (h (e_{i}^{*}, e_{j}^{*}), J {\hat{e}}_{k}))}^{2} + \sum_{k = 1}^{n_{3}} \sum_{i, j = 1}^{n_{3}} {(g (h (e_{i}^{*}, e_{j}^{*}), E_{k}^{*}))}^{2} \\ + \sum_{k = 1}^{2 n - n_{2} - n_{3}} \sum_{i, j = 1}^{n_{3}} {(g (h (e_{i}^{*}, e_{j}^{*}), {\tilde{e}}_{k}))}^{2} \end{matrix}

Leaving the last two positive terms and using Lemma 4 with mixed totally geodesic condition, we get

\begin{matrix} ∥ h (D^{θ}, D^{θ}) ∥^{2} \geq \frac{2 q}{9} {cos}^{2} θ \sum_{k = 1}^{n_{2}} {(e_{k} (ln f))}^{2} = \frac{1}{9} n_{3} {cos}^{2} θ {∥ {\vec{\nabla}}^{⊥} (ln f) ∥}^{2} . \end{matrix}

(41)

Similarly, we derive the other terms of (38) as follows

\begin{matrix} ∥ h (D, D^{⊥}) ∥^{2} = & \sum_{k, j = 1}^{n_{2}} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, {\hat{e}}_{j}), J {\hat{e}}_{k}))}^{2} + \sum_{k = 1}^{n_{3}} \sum_{i = 1}^{n_{1}} \sum_{j = 1}^{n_{2}} {(g (h (e_{i}, {\hat{e}}_{j}), E_{k}^{*}))}^{2} \\ + \sum_{k = 1}^{2 m - n - n_{2} - n_{3}} \sum_{i = 1}^{n_{1}} \sum_{j = 1}^{n_{2}} {(g (h (e_{i}, {\hat{e}}_{j}), {\tilde{e}}_{k}))}^{2} \end{matrix}

There is no relation for the first positive term in terms of warped products and leaving the last

ν

-components term. Then, using Lemma 2, we derive

\begin{matrix} ∥ h (D, D^{⊥}) ∥^{2} & \geq \frac{1}{4} \sum_{j = 1}^{n_{2}} \sum_{i = 1}^{n_{1}} \sum_{k = 1}^{n_{3}} {(g (h (e_{i}, e_{k}^{*}), J {\hat{e}}_{j}))}^{2} \geq 0 . \end{matrix}

(42)

On the other hand, we also have

\begin{matrix} ∥ h (D, D^{θ}) ∥^{2} = & \sum_{k = 1}^{n_{2}} \sum_{j = 1}^{n_{3}} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, e_{j}^{*}), J {\hat{e}}_{k}))}^{2} + \sum_{k, j = 1}^{n_{3}} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, e_{j}^{*}), E_{k}^{*}))}^{2} \\ + \sum_{k = 1}^{2 m - n - n_{2} - n_{3}} \sum_{i = 1}^{n_{1}} \sum_{j = 1}^{n_{3}} {(g (h (e_{i}, e_{j}^{*}), {\tilde{e}}_{k}))}^{2} \end{matrix}

For the first term we use (42) and omit the

ν

-components terms and using frame fields of

D^{θ}

and

F D^{θ}

, we derive

\begin{matrix} ∥ h (D, D^{θ}) ∥^{2} & \geq {csc}^{2} θ \sum_{k, j = 1}^{q} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, e_{j}^{*}), F e_{k}^{*}))}^{2} + {csc}^{2} θ {sec}^{2} θ \sum_{k, j = 1}^{q} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, T e_{j}^{*}), F e_{k}^{*}))}^{2} \\ + {csc}^{2} θ {sec}^{2} θ \sum_{k, j = 1}^{q} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, e_{j}^{*}), F T e_{k}^{*}))}^{2} \\ + {csc}^{2} θ {sec}^{4} θ \sum_{k, j = 1}^{q} \sum_{i = 1}^{n_{1}} {(g (h (e_{i}, T e_{j}^{*}), F T e_{k}^{*}))}^{2} . \end{matrix}

Using Lemma 3 with (23)-(), we obtain

\begin{matrix} ∥ h (D, D^{θ}) ∥^{2} & \geq 2 q {csc}^{2} θ \sum_{i = 1}^{n_{1}} {(e_{i} (ln f))}^{2} + \frac{2 q}{9} {cot}^{2} θ \sum_{i = 1}^{n_{1}} {(e_{i} (ln f))}^{2} \\ = n_{3} ({csc}^{2} θ + \frac{1}{9} n_{3} {cot}^{2} θ) {∥ {\vec{\nabla}}^{T} (ln f) ∥}^{2} . \end{matrix}

(43)

Last term of (38) is identically zero by the hemi-slant mixed totally geodesic condition. Then, for all values of h from (39)-(43), finally we get the required inequality (37).

For the equality case, since M is

D^{⊥} \oplus D^{θ}

-mixed totally geodesic, i.e.,

\begin{matrix} h (D^{⊥}, D^{θ}) = {0} . \end{matrix}

(44)

Form the leaving and vanishing terms, we also find

\begin{matrix} h (D, D) = {0}, h (D^{⊥}, D^{⊥}) = {0}, h (D, D^{⊥}) = {0}, \\ h (D^{θ}, D^{θ}) \subseteq J D^{⊥}, h (D, D^{θ}) \subseteq F D^{θ} . \end{matrix}

(45)

Then,

M_{T} \times M_{⊥}

is totally geodesic and

M_{θ}

is totally umbilical in

\tilde{M}

due to the fact that

M_{T} \times M_{⊥}

is totally geodesic and

M_{θ}

is totally umbilical in M[2,6] with equality holding case of (45). Furthermore, due to Theorem 2 and Lemma 2, we observe that M is not a

D \oplus D^{θ}

-mixed totally submanifold of

\tilde{M}

. Hence, the proof is complete. □

Now, we give the following consequences of Theorem 3.

A warped submanifold of the form

M = M_{θ} \times_{f} M_{⊥}

in a nearly Kaehler manifold

\tilde{M}

is called hemi-slant if

M_{⊥}

is a totally real submanifold and

M_{θ}

is a proper slant submanifold.

dim M_{T} = 0

in Theorem 3, then we have

Theorem 4.

Let

M = M_{⊥}^{n_{1}} \times_{f} M_{θ}^{n_{2}}

be a mixed totally geodesic hemi-slant warped product submanifold in a nearly Kaehler manifold

\tilde{M}

. Then

(i): The second fundamental form h of M satisfies

$\begin{matrix} {| | h | |}^{2} \geq \frac{1}{9} n_{2} {cos}^{2} θ | | {\vec{\nabla}}^{⊥} (ln f) {| |}^{2}, \end{matrix}$

(46)

where ${\vec{\nabla}}^{⊥} (ln f)$ is the gradient of $ln f$ along $M_{⊥}$ .
(ii): if the equality sign of (46) holds identically, then $M_{⊥}$ and $M_{θ}$ are totally geodesic and totally umbilical submanifolds of $\tilde{M}$ , respectively.

On the other hand, if

M_{⊥} = {0}

, we have the following special case of Theorem 3.

Theorem 5.

[1]Let

M = M_{T}^{n_{1}} \times_{f} M_{θ}^{n_{2}}

be a semi-slant warped product submanifold in a nearly Kaehler manifold

\tilde{M}

. Then, we have

(i): The second fundamental form h and the warping function f satisfy

$\begin{matrix} {∥ h ∥}^{2} \geq 2 n_{2} (1 + \frac{10}{9} {cot}^{2} θ) {∥ {\vec{\nabla}}^{T} (ln f) ∥}^{2} . \end{matrix}$

(47)

where ${\vec{\nabla}}^{T} ln f$ is gradient of $ln f$ along $M_{T}$ .
(ii): If the equality sign in (47) holds identically, then $M_{T}$ is totally geodesic and $M_{θ}$ is totally umbilical in $\tilde{M}$ . Moreover, M is a minimal submanifold in $\tilde{M}$ .

Also, if

dim M_{⊥} = 0

and

θ = \frac{π}{2}

in Theorem 3, then

M = M_{T}^{n_{1}} \times_{f} M_{⊥}^{n_{2}}

is a CR-warped product submanifold of a nearly Kaehler manifold

\tilde{M}

and they were studied in [17] and hence the main Theorem 4.2 of [17] is a special case of Theorem 3.

Author Contributions

Writing—original draft, S.U. and B.-Y. C.; Writing—review & editing, R. M. B. All authors have read and agreed to the published version of the manuscript.

Funding

This research work was funded by Institutional Fund Projects under grant no. (IFPIP: 1156-130-1443). The authors gratefully acknowledge the technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

Data Availability Statement

Not applicable

Conflicts of Interest

The authors declare no conflict of interest.

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Geometry of CR-Slant Warped Products in Nearly Kaehler Manifolds

Abstract

Keywords:

Subject:

1. Introduction

2. Basic definitions and formulas

3. CR-slant warped products M T × M ⊥ × f M θ

4. Chen’s inequality and its consequences

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

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3. CR-slant warped products $(M_{T} \times M_{⊥}) \times_{f} M_{θ}$