On warped product bi-slant submanifolds of Kenmotsu manifolds Open Access

In [18], K. Kenmotsu studied one class of almost contact metric manifolds known as Kenmotsu manifolds. He proved that:

• 1.Locally a Kenmotsu manifold is a warped product $I×fM$ of an interval $I$ and a Kaehler manifold $M$⁠, with warping function $f=cet$⁠, where $c$ is a nonzero constant.

• 2.A Kenmotsu manifold with constant sectional curvature is a space of constant curvature $−1$⁠, and so it is locally a hyperbolic space.

A $(2m+1)$-dimensional manifold $M˜$ is said to be an almost contact manifold if it admits an endomorphism $ϕ$ of its tangent bundle $TM˜$⁠, a vector field $ξ$ and a $1$-form $η$⁠, which satisfy:

There exists a compatible metric $g$⁠, which satisfies

for all vector fields $X,Y$ on $M˜$ [6]. In addition, an almost contact metric manifold $M˜$ is said to be a Kenmotsu manifold [18] if the relation

holds, where $∇˜$ is the Levi-Civita connection of $g$⁠. From (1.3), for a Kenmotsu manifold $M˜$⁠, we also have

As Kenmotsu manifolds are warped product manifolds, therefore it is interesting to investigate the geometry of its warped product submanifolds. The notion of warped submanifolds was first introduced by B.-Y. Chen as a CR-warped product submanifold of Kaehler manifolds in his series of articles [11,12]. He established a general sharp inequality between the main extrinsic invariant (the second fundamental form) and an intrinsic invariant (the warping function) of such submanifolds. Motivated by Chen’s work many geometers studied warped product submanifolds for different spaces (see, e.g., [4,17,19–25,29,30] among many others. For the most up-to-date overview of this subject, see [13–15]).

On the other hand, J.L. Cabrerizro et al. studied in [7] bi-slant submanifolds of almost contact metric manifolds. In [28], the first author and B.-Y. Chen investigated warped product bi-slant submanifolds in Kaehler manifolds. They proved that there do not exist any warped product bi-slant submanifolds of Kaehler manifolds other than hemi-slant warped products and CR-warped products. The non-existence of warped product bi-slant submanifolds is proved in [2] for cosymplectic manifolds.

In this paper, we study warped product bi-slant submanifolds of a Kenmotsu manifold. The geometry of such submanifolds in Kenmotsu manifolds is quite different from Sasakian and cosymplectic case because in case of Kenmotsu manifolds such submanifolds exist while there is no proper warped product bi-slant submanifolds in Sasakain and cosymplectic as well. On their existence, we establish a generalized Chen type sharp inequality for the squared norm of the second fundamental form in terms of the warping function and bi-slant angles. The equality case is considered. Some applications are given in the last section.

Let $ψ:Mn→M(n+d)$ be an isometric immersion of an $n$-dimensional Riemannian manifold $M$ into an $(n+d)$-dimensional Riemannian manifold $M˜$⁠. We denote by $∇$ and $∇˜$ the Levi-Civita connections on $M$ and $M˜$⁠, respectively. Then the Gauss and Weingarten formulas are respectively given by [31]

and

for any $X,Y∈TM$ and $V∈T⊥M$⁠, where $∇⊥$ is the normal connection in the normal bundle $T⊥M$ and $AV$ is the shape operator of $M$ with respect to $V$⁠. Moreover, $h:TM×TM→T⊥M$ is the second fundamental form of $M$ in $M˜$⁠. Furthermore, $AV$ and $h$ are related by

for any $X,Y∈TM$ and $V∈T⊥M$⁠.

A submanifold $M$ of a Riemannian manifold $M˜$ is said to be a totally umbilical submanifold if $h(X,Y)=g(X,Y)H$⁠, for any $X,Y∈TM$⁠, where $H=1n∑i=1nh(ei,ei)$ is the mean curvature vector of $M$⁠. A submanifold $M$ is said to be totally geodesic if $h(X,Y)=0$⁠. Also, one denotes

with $i,j=1,…,n;r=n+1,…,n+d$⁠, where ${e1,…,en}$ is an orthonormal basis of the tangent space $TpM$ and ${en+1,…,en+d}$ an orthonormal basis of the normal space $Tp⊥$⁠, for any $p∈M$⁠.

For a differentiable function $f$ on a $m$-dimensional manifold $M˜$⁠, the gradient $∇→f$ of $f$ is defined as $g(∇→f,X)=X(f)$⁠, for any $X$ tangent to $M˜$⁠.

For any vector field $X$ tangent to $M$⁠, we write

where $TX$ is the tangential component and $FX$ is the normal component of $ϕX$⁠. Thus, $T$ is an endomorphism on the tangent bundle $TM$ and $F$ is a normal bundle valued 1-form of $TM$⁠. A submanifold $M$ is called invariant if $F$ is identically zero, that is, $ϕX∈TM$ for any $X∈TM$⁠; while, $M$ is anti-invariant if $T$ is identically zero, that is, $ϕX∈T⊥M$⁠, for any $X∈TM$⁠.

Similarly, for any vector field $V$ normal to $M$⁠, we put

where $tV$ and $fV$ are the tangential and normal components of $ϕV$⁠, respectively.

Let $M$ be a submanifold of an almost contact metric manifold $(M˜,ϕ,ξ,η,g)$⁠. Hence, if we denote $D$ the orthogonal distribution to $ξ$ in $TM$⁠, then $TM=D⊕〈ξ〉$⁠, where $〈ξ〉$ is the $1$-dimensional distribution spanned by $ξ$⁠. For any nonzero vector $X$ tangent to $M$ at the point $p∈M$⁠, such that $X$ is not proportional to $ξp$⁠, we denote by $θ(X)$⁠, the angle between $ϕX$ and $TpM$⁠. In fact, since $ϕξ=0$⁠, $θ(X)$ agrees with the angle between $ϕX$ and $Dp$⁠. A submanifold $M$ of an almost contact metric manifold $M˜$ is said to be slant [8], if for any non-zero vector $X$ tangent to $M$ at $p$ such that $X$ is not proportional to $ξp$⁠, the angle $θ(X)$ between $ϕX$ and $TpM$ is constant, i.e., it does not depend on the choice of $p∈M$ and $X∈TpM−〈ξp〉$⁠. In this case $D$ is a slant distribution with slant angle $θ$⁠.

A slant submanifold is said to be proper slant, if neither $θ=0$ nor $θ=π2$ [9,10]. We note that on a slant submanifold if $θ=0$⁠, then it is an invariant submanifold and if $θ=π2$⁠, then it is an anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant.

A characterization of slant submanifolds was given in [8] as follows:

Furthermore, in such case, if $θ$ is slant angle, then $λ=cos2 θ$ .

The following relations are straightforward consequences of (2.7)

for any $X,Y$ tangent to $M$⁠.

The following useful relation is obtained as a consequence of (2.7) in [26].

for any$X∈TM$.

Another characterization of slant submanifolds was given in [7]:

J.L. Cabrerizo et al. [7] defined bi-slant submanifolds as follows:

• (i) The tangent bundle $TM$ admits the orthogonal direct decomposition: $TM=D1⊕D2⊕〈ξ〉$⁠.

• (ii) Each $Di∀i=1,2$ is a slant distribution with slant angle $θi$⁠.

Given a bi-slant submanifold $M$⁠, for any $X∈TM$⁠, we put

where $PiX$ denotes the component of $X$ in $Di$⁠, for any $i=1,2$⁠. In particular, if $X∈Di$⁠, then we obtain $X=PiX$⁠. If we define $Ti=Pi∘T$⁠, then we have

for any $X∈TM$⁠. Given $i=1,2$⁠, from Theorem 3, we get

for any $X∈Di$⁠.

Non-trivial examples of bi-slant submanifolds are given in [7].

Let $M1×fM2$ be a warped product manifold of two Riemannian manifolds $M1$ and $M2$⁠. Then from a result of [5], we have

for any vector fields $X,Z$ tangent to $M1,M2$⁠, respectively.

Recently, B.-Y. Chen and the first author introduced the notion of warped product bi-slant submanifolds of Kaehler manifolds [28]. They proved the non-existence of proper warped product bi-slant submanifolds. Then, they introduced the notion of warped product pointwise bi-slant submanifolds and obtained several fundamental results [16]. In this section, we give some useful lemmas for warped product bi-slant submanifolds of Kenmotsu manifolds. First we define these submanifolds as follows:

A warped product $M1×fM2$ of two slant submanifolds $M1$ and $M2$ with slant angles $θ1$ and $θ2$ of a Kenmotsu manifold $M˜$ is called a warped product bi-slant submanifold.

A warped product bi-slant submanifold $M=M1×fM2$ is called proper if both $M1$ and $M2$ are proper slant submanifolds with slant angles $θ1,θ2≠0,π2$ of $M˜$⁠. A warped product bi-slant submanifold $M1×fM2$ is a contact CR-warped product if $θ1=0,θ2=π2$ or $θ2=0,θ1=π2$⁠; such submanifolds were discussed in [3,27]. Also, it is a warped product pseudo-slant submanifold if $θ1=θ$ and $θ2=π2$ [1] or $θ2=θ$ and $θ1=π2$ [23]. The warped product bi-slant submanifolds with slant angles $θ1=0,θ2=θ$ or $θ2=0,θ1=θ$ were discussed in [25,30].

Let $M=M1×fM2$ be a warped product bi-slant submanifold of a Kenmotsu manifold $M˜$ such that the structure vector field $ξ$ is tangent to $M$⁠, where $M1$ and $M2$ are proper slant submanifolds of $M˜$⁠. Then, we distinguish 2 cases:

• $ξ$ is tangent to $M1$⁠;

• $ξ$ is tangent to $M2$⁠.

From (1.4), (2.1) and (3.1), the second case is trivial i.e., there does not exist any proper warped product bi-slant submanifold of a Kenmotsu manifold when the structure vector field is tangent to the fiber.

Now, we start with the case (i). Throughout this paper, we assume that the tangent spaces of $M1$ and $M2$⁠, respectively are $D1$ and $D2$⁠. From now on, we use the following conventions: $X1,Y1$ are vector fields in $D1$ and $X2,Y2$ are vector fields in $D2$⁠.

• (i) $ξ(ln f)=1;$

• (ii) $g(h(X1,Y1),FX2)=g(h(X1,X2),FY1),$

for any $X1,Y1∈D1$ and $X2∈D2$ .

Proof. First part is trivial and can be obtained by using (1.4), (2.1) and (3.1). For the second part, we have

for any $X1,Y1∈D1$ and $X2∈D2$⁠. Also, from (1.2) and the fact that $ξ$ is tangent to $M1$⁠, we have

Form (3.2) and (3.3), we obtain

Then using the covariant derivative property of Riemannian connection and (1.2), (1.3), (2.1) and (3.1), we derive

By the orthogonality of vector fields, the left hand side and the first term in the right hand side vanish identically. Then using (1.3), (2.6) and (2.10), we find

Using (1.4), (2.2), (2.3), (3.1) and the orthogonality of vector fields, we derive

Interchanging $Y1$ by $T1Y1$ in (3.4), we obtain

Since for a submanifold $M$ of a Kenmotsu manifold $M˜$⁠, $h(X,ξ)=0,∀X∈TM$⁠, then the second part of the lemma follows from above relation. Hence, the proof is complete. ■

for any$X1∈D1$and$X2,Y2∈D2$.

Proof. For any $X1∈D1$ and $X2,Y2∈D2$⁠, we have

On the other hand, we also have

Using (1.3) and (2.5), we arrive at

From (1.2), (2.1) and (3.1), the above equation takes the form

Again, using (1.3) and (2.6), we derive

Then, from (2.10) and (3.1), we obtain

Thus, the result follows from (3.7) and (3.8), which proves the lemma completely. ■

The following useful relations are easily derived by interchanging $X1$ by $T1X1$⁠, $X2$ by $T2X2$ and $Y2$ by $T2Y2$ in Lemma 2.

and

Let $M=M1×fM2$ be a warped product bi-slant submanifolds of a Kenmotsu manifold $M˜$⁠; we decompose the normal bundle of $M$ as follows

where $μ$ is a $ϕ$-invariant normal subbundle of $T⊥M$⁠.

A warped product bi-slant submanifold $M=M1×fM2$ of a Kenmotsu manifold $M˜$ is said to be mixed totally geodesic, if $h(X,Z)=0$⁠, for any $X∈D1$ and $Z∈D2$⁠, where $D1$ and $D2$ are the tangent bundles of $M1$ and $M2$⁠, respectively.

Now, we set the following frame fields for an $n$-dimensional warped product bi-slant submanifold $M=M1×fM2$ of a $(2m+1)$-dimensional Kenmotsu manifold $M˜$ such that $ξ$ is tangent to $M1$⁠, where $M1$ and $M2$ are proper slant submanifolds of $M˜$ with slant angles $θ1$ and $θ2$ respectively. Let us consider the dimensions $dim(M1)=2p+1$ and $dimM2=2q$⁠, i.e., $n=2p+1+2q$⁠. Then the orthonormal frames of the corresponding tangent spaces $D1$ and $D2$⁠, respectively are given by ${e1,…,ep,ep+1=secθ1T1e1,…,e2p=secθ1T1ep,e2p+1=ξ}$ and ${e2p+2=e¯1,…,e2p+1+q=e¯q,e2p+q+2=e¯q+1=secθ2T2e¯1,…,en=e¯2q=secθ2T2e¯q}$⁠. Thus, the orthonormal frame fields of the normal subbundles of $FD1,FD2$ and $μ$⁠, respectively are ${en+1=e˜1=cscθ1Fe1,…,en+p=e˜p=cscθ1Fep,en+p+1=e˜p+1=cscθ1secθ1FT1e1,…,en+2p=e˜2p=cscθ1 secθ1FT1ep},{en+2p+1=e˜2p+1=e^1=cscθ2Fe¯1,…,en+2p+q=e˜2p+q=e^q=cscθ2Fe¯q,en+2p+q+1=e˜2p+q+1=e^q+1=cscθ2 secθ2FT2e¯1,…,en+2p+2q=e˜2p+2q=e^2q=cscθ2 secθ2FT2e¯q}$ and ${e2n=e˜n,…,e2m+1=e˜2(m−n+1)}$⁠.

Now, using the results of Section 3 and the above frame fields, we give following main result of this paper.

• (i) The second fundamental form $h$ of $M$ satisfies the following inequality

  $‖h‖2≥2q csc2θ1(cos2θ1+cos2θ2)(‖∇→(ln f)‖2−1),$

  (4.2)

  where$2q=dimM2$and$∇→(ln f)$is the gradient of$ln f$along$M1$.

• (ii) If equality sign in (i) holds identically, then:

  • (a) $M1$ is a totally geodesic submanifold of $M˜$;

  • (b) $M2$ is a totally umbilical submanifold of $M˜$⁠.

Proof. From (2.4), we have

Splitting the above expression for the tangent bundles of $M1$ and $M2$⁠, we derive

Since $M$ is mixed totally geodesic, then the second term in the right hand side of (4.3) is identically zero. Thus, we find

The third and sixth terms have $μ$-components and we could not find any relation for warped products in terms of $μ$-components, therefore we shall leave these two terms. Also, we could not find any relations for $g(h(ei,ej),e˜r),i,j=1,…,2p+1,r=1,…,2p$ and $g(h(e¯i,e¯j),e^r),i,j,r=1,…,2q$⁠. Therefore, we also leave the first and fifth terms. By using Lemma 1(ii) for a mixed totally geodesic warped product the second term in the right hand side is also zero. Thus, the evaluated term is only fourth term which can be expressed by using the constructed frame fields as follows