Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications

The study of warped product manifolds has been a favourite topic in the field of geometry due to its applications in Physics and relativistic theories [1]. Many basic solutions to Einstein field equations are given by warped products [1]. The concept of modelling of space–time near black holes uses the idea of warped product manifolds [2]. Schwartzschild space–time is an example of warped product $P×r S2$ where the base $P=R×R+$ is a half plane $r>0$ and fibre $S2$ is the unit sphere. Under certain conditions, the Schwartzschild space–time becomes black hole. A cosmological model to model the universe as a space–time known as Robertson–Walker model is a warped product [3].

One of the important task in Physics and Engineering is to find the Dirichlet energy of smooth functions. Dirichlet energy is analogous to Kinetic energy. On a compact manifold $M$⁠, the Dirichlet energy of any smooth function $λ:M→R$ is given by

where $∇λ$ is the gradient of $λ$ and $dV$ is the volume element. It is obvious that $E(λ)≥0$ for any smooth function $λ$⁠. We know that the manifolds of non-zero (constant) curvature cannot be represented as a product manifold. So, considering the fact that a Riemannian product of manifolds cannot has negative curvature, the idea of warped product of manifolds came into existence. To construct the class of manifolds of negative or non-positive curvature, R. L. Bishop and B. O’Neill [4] introduced this idea of warped product manifolds. Warped product manifolds (see definition in Section 2) are a generalized setting of product manifolds. Since warping functions of the warped product manifolds are positive valued smooth functions, our interest is to find the Dirichlet energy of these functions.

Some intrinsic properties of warped product manifolds were studied in [4]. Initial extrinsic studies of warped product manifolds in the almost complex setting were performed by B. Y. Chen [5,6] while obtaining some existence results for CR-submanifolds to be CR-warped product submanifolds in Kaehler manifolds. On the other hand, in the almost contact settings contact CR-warped product submanifolds were explored by Hasegawa et al. [7]. Many other geometers have also explored warped product manifolds in contact settings and various existence results have been obtained [8–11].

Warped product pointwise semi-slant submanifold is another generalized class of warped product semi-slant submanifolds and contact CR-warped product submanifolds. In [12], Park studied the warped product pointwise semi-slant warped product submanifolds. After that, Ali and Ozel [13] extended this study in the setting of cosymplectic manifolds and they obtained some optimal inequalities related to the second fundamental form and warping function. Warped product pointwise semi-slant submanifolds for almost contact and almost complex manifolds were explored in (see [14–16]).

On the other hand the Gradient Ricci soliton are extensively investigated in the theory of relativity, physics as well as in the differential geometry. The classification results related to Ricci Soliton and Gradient Ricci solitons with the warped product structure have been established in [17–20]. Moreover, the Ricci curvature has a significant nature in Riemannian geometry, for example Ricci flat is a solution of Einstein field equation on a Riemannian manifold in which cosmological constant vanishes. More clearly, in the theory of general relativity the Ricci tensor is correlated with Einstein’s field equation to study the material contents of universe. So, in comparison with Riemannian curvature, the Ricci curvature is more significant in the theory of relativity and physics.

In the present article, we study warped product pointwise semi-slant submanifolds of cosymplectic space forms and obtain some interesting inequalities for warped product pointwise semi-slant submanifolds. We estimate the squared norm of the second fundamental form in terms of warping function and slant function. The equality case is discussed accordingly. We explored some applications of equality case of the derived inequality. More precisely, we calculate Dirichlet energy of the warping functions by using our derived inequality. Finally, we obtain the classification of the warped product pointwise semi-slant submanifolds admitting the gradient Ricci soliton, in terms of Ricci curvature and second fundamental form, some existence results are also established.

The paper is organized as follows: Section 2 is devoted to basic definitions, formulae and preliminary results which are required for the subsequent study of the paper. In Section 3, we explore the existence of warped product pointwise semi-slant submanifolds in cosymplectic space forms and prove our main results. A non-trivial example is given for the warped product pointwise semi-slant submanifolds in a cosymplectic manifold. Using the derived inequality, formulae for Dirichlet energy of warping function is obtained in Section 4. Conclusions are presented in Section 5. Bibliography is given at the end of the paper.

A $(2n+1)$-dimensional $C∞$-manifold $M¯$ is said to have an almost contact structure if on $M¯$ there exist a tensor field $Ψ$ of type $(1,1)$⁠, a vector field $η$ and a 1-form $ζ$ satisfying the following properties [21]

The manifold $M¯$ with the structure $(Ψ,η,ζ)$ is called almost contact metric manifold. There exists a Riemannian metric $g$ on an almost contact metric manifold $M¯$⁠, satisfying the following

for all $Y,V∈TM¯$ where $TM¯$ is the tangent bundle of $M¯$⁠.

An almost contact metric structure $(Ψ,η,ζ,g)$ is said to be cosymplectic manifold if it satisfies the following tensorial equation [21]

for any $Y,V∈TM¯$⁠, where $∇¯$ denotes the Riemannian connection of the metric $g$⁠. Moreover, for a cosymplectic manifold

A cosymplectic manifold $M¯$ is said to be a cosymplectic space form [21] if it has constant $Ψ$-holomorphic sectional curvature $c$ and is denoted by $M¯(c)$⁠. The curvature tensor $R¯$ of cosymplectic space form $M¯(c)$ is given by

for any vector fields $Y1,Y2,V$ on $M¯$⁠.

Let $M$ be a submanifold of an almost contact metric manifold $M¯$ with induced metric $g$⁠. The Riemannian connection $∇¯$ of $M¯$ induces canonically the connections $∇$ and $∇⊥$ on the tangent bundle $TM$ and the normal bundle $T⊥M$ of $M$ respectively, then the Gauss and Weingarten formulae are given by

for each $Y1,Y2∈TM$ and $ξ∈T⊥M$⁠, where $σ$ and $Aξ$ are the second fundamental form and the shape operator respectively for the immersion of $M$ into $M¯$⁠, they verify the relation

where $g$ denotes the Riemannian metric on $M¯$ as well as the induced metric on $M$⁠.

For a submanifold $M$ of a Riemannian manifold $M¯$⁠, the equation of Codazzi is given by

where $(R¯(Y1,Y2)V)⊥$ denotes the normal component of the curvature tensor $R¯(Y1,Y2)V$⁠.

If $PY$ and $FY$ denote the tangential and normal component of $ΨY$ respectively for any $Y∈TM$⁠, we can write

Similarly, for any $ξ∈T⊥M$⁠, we write

where $tξ$ and $fξ$ are the tangential and normal components of $Ψξ$ respectively. Thus $P$ (resp. $f$⁠) is 1–1 tensor field on $TM$ (resp. $T⊥M$⁠) and $t$ (resp. $F$⁠) is a tangential (resp. normal) valued 1-form on $T⊥M$ (resp. $TM$⁠). The covariant derivatives of the tensor fields $Ψ$⁠, $P$ and $F$ are defined as

From Eqs. (2.3), (2.6), (2.7), (2.9) and (2.10), we have

The mean curvature vector $H$ of $M$ is given by

where $m$ is the dimension of $M$ and ${e1,e2,…,em}$ is a local orthonormal frame of vector fields on $M$⁠. The squared norm of the second fundamental form $σ$ is defined as

A submanifold $M$ of $M¯$ is said to be a totally geodesic submanifold if $σ(Y1,Y2)=0$ and totally umbilical submanifold if $σ(Y1,Y2)=g(Y1,Y2)H$⁠, for each $Y1,Y2∈TM.$

For slant submanifolds of the contact metric manifolds J. L. Cabrerizo et al. [22] proved the following lemma.

where$λ=−cos2 θ$.

The notion of pointwise slant submanifolds was introduced by F. Etayo [23] as a natural generalization of the slant submanifolds in the setting of almost Hermitian manifolds. Later, B. Y. Chen and O. J. Garay [24] investigated pointwise slant submanifolds for almost Hermitian manifolds and obtained some fundamental results. A step forward, K. S. Park [12] extended the concept of pointwise slant submanifolds in almost contact metric manifolds. Recently, Uddin and Al-Khalidi [25] modified the definition of pointwise slant submanifolds for almost contact metric manifolds. More precisely, a submanifold $M$ of an almost contact metric manifold $M¯$ is said to be pointwise slant submanifold if for any $X∈TxM$ such that $η$ is tangential to $M$⁠, the angle $θ(X)$ between $ΨX$ and $TxM−{0}$ is independent of the choice of non zero vector field $X∈TpM−{0}$⁠. In this case $θ$ is treated as the function on $M$⁠, which is called the slant function of the point wise slant function. Now, we have the following characterizing theorem

for some real valued function $θ$ on $TM$

Thus, one has the following consequences of the above formula.

for all $Y1,Y2∈TM$⁠.

Pointwise semi-slant submanifolds were defined and studied by Park [12] as a natural generalization of contact CR-submanifolds in terms of slant function. Now, we have the following definition

• The tangent bundle $TM$ can be written as $TM=D⊕Dθ〈η〉,$

• The distribution $D$ is invariant,

• The distribution $Dθ$ is pointwise slant with a slant function $θ$⁠.

As a generalization of the product manifolds, one can consider the warped product of manifolds which are defined as follows.

Let $(C1,gC1)$ and $(C2,gC2)$ be the two Riemannian manifolds with $gC1$ and $gC2$ as their Riemannian metrics resp. and $Φ$ be a positive differentiable function on $C1$⁠. Let $π1:C1×C2→C1$⁠, $π2:C1×C2→C2$ are the projection maps given by $πC1(c1,c2)=c1$ and $πC2(c1,c2)=c2$ for every $(c1,c2)∈C1×C2$⁠. The warped product $M=C1×Φ C2$ [4] is the manifold $C1×C2$ equipped with the Riemannian structure such that

for all $Y1,Y2∈TM$⁠, where $πi∗$ denotes the tangent map corresponding to $πi$ for each $i$⁠. The function $Φ$ is called the warping function of the warped product manifold. If the warping function is constant then the warped product manifold $M$ is said to be trivial warped product.

Let $Y1$ be a vector field on $C1$ and $Y2$ be a vector field on $C2$⁠, then from Lemma 7.3 of [3], we have

where $∇$ is the Levi-Civita connection on $M$⁠. For a warped product $M=C1×Φ C2$ it is easy to observe that

for $Y1∈TC1$ and $Y2∈TC2$⁠.

$∇Φ$ is the gradient of $Φ$ and is defined as

for all $Y∈TM$⁠.

Let $M$ be an $m$-dimensional Riemannian manifold with the Riemannian metric $g$ and let ${e1,e2,…,em}$ be an orthogonal basis of $T M$⁠. Then as a result of (2.23), we get

The Laplacian of $Φ$ is defined by

The Hessian tensor for a differentiable function $Φ$ is symmetric covariant tensor of rank $2$ and is defined as

or we can also write

Now, we state the Hopf’s Lemma.

Hopf’s Lemma [26]. If $M$ is an $m$-dimensional connected compact Riemannian manifold. If $Φ$ is a differentiable function on $M$ s. t. $ΔΦ≥0$ everywhere on $M$ (or $ΔΦ≤0$ everywhere on $M$⁠), then $Φ$ is a constant function.

For a compact orientable Riemannian manifold $M$ with or without boundary and as a consequences of the integration theory of manifolds, we have [27]

where $Φ$ is a function on $M$ and $dV$ is the volume element of $M$⁠.

The Ricci soliton idea was given by Hamilton [28]. It is regarded as the natural generality of Einstein metrics and they are the self similar solution of the Ricci flow $∂∂tg(t)=−2Ric(t)$⁠. If there exists a smooth vector field $Y$ such that the Ricci tensor meets the following condition

for any constant $α$⁠, where $LY$ is the Lie derivative, then the Riemannian metric $g$ on a complete Riemannian manifold $M¯$ is named as Ricci Soliton. If $α<0,α=0$ and $α>0$ then the Ricci soliton is called expanding, steady and Shrinking respectively. If we specify $Y=∇Φ$ for a smooth function $Φ$ defined on $M¯$⁠, then $g$ admits gradient Ricci soliton with the potential function $Φ$⁠. For this case (2.29) takes the form

Since the Laplacian $Δ$ and the gradient $∇2$ are related as $Δ=∇2$⁠. Thus, in terms of Hessian (2.29) can be expressed as

In [12], K. S. Park, proved the existence of the warped product pointwise semi-slant warped product submanifolds of the type $NT×ΦNθ$ of cosymplectic manifolds and achieved the following lemma

and

for any $Y∈T NT$ and $Z,W∈T Nθ$.

Now let $M=NT×Φ Nθ$ be a warped product pointwise semi-slant submanifold of a cosymplectic manifold $M¯$ and we consider the vector field $η$ tangent to $NT$⁠. If $D$ is invariant distribution and $Dθ$ is proper point wise slant distribution with the slant function $θ$⁠, then the tangent bundle $TM$ and $T⊥M$ are decomposed (resp.) as follows

where $μ$ is the orthogonal complementary distribution of $FDθ$ in $T⊥M$⁠. It is easy to see that $μ$ is an invariant subbundle of $T⊥M$ with respect to $Ψ$⁠.

In view of the above direct decomposition, the second fundamental form $σ$ can be written as

for $U1,U2∈TM$⁠, where $σFDθ(U1,U2)$ and $σμ(U1,U2)$ are the components of $σ(U1,U2)$ in the normal sub-bundles $FDθ$ and $μ$ respectively. Moreover if ${V1,V2,…,Vq}$ be a local orthonormal frame of vector fields of $Dθ$⁠, then

where

To ensure the existence, we construct an example of a warped product pointwise semi-slant submanifold of the type $M=NT×Φ Nθ$ in cosymplectic manifold with $η$ tangent to $NT$⁠.

The tangent space $T M$ is spanned by the vector fields $X1$⁠, $X2$⁠, $X3$⁠, $X4$⁠, $X5$ such that

Then $Dθ=span{X4,X5}$ is a pointwise slant distribution with slant function $cos−1(1u2+v2+1)$ and invariant distribution $D=span{X1,X2,X3}$⁠. Thus $M5=NT×Φ Nθ$ is a non-trivial warped product pointwise semi-slant submanifold of $M11$ with the warping function $Φ=(u2+v2+1)$⁠.

In this section, for convention, we denote by $Y,X∈TNT$ and $V,U∈TNθ$ as the vector fields of respective tangent bundles of $NT$ and $Nθ$⁠. At first, some initial results to be proved.

• $ηlnΦ=0$,

• $g(σ(ΨY,V),FV)=YlnΦ‖V‖2$,

• $g(σ(ΨY,V),Ψσ(Y,V))=‖σμ(Y,V)‖2+cos2 θ(YlnΦ)2‖V‖2$,

for all $Y∈TNT$ and$V∈TNθ$⁠, where$σμ$ is the$μ$ component of the second fundamental form $σ$⁠.

Now using (2.22), the above equation can be written as

Comparing the normal parts

taking inner product with $Ψσ(Y,V)$⁠, we get

Calculating the last term of above equation by using (2.6), (2.3), and (2.20) as follows

Utilizing part (ii), we get

using in (3.6), we get the required result.

for all $Y∈TNT$⁠, $V∈TNθ$⁠.

for any $Y∈TNT$ and $V,U∈TNθ$⁠. Replacing $U$ by $PV∈Dθ$ and using the fact that $V$ and $PV$ are orthogonal, we have

By using (2.12), (2.14), and (2.22), we have

Now taking inner product with $U∈TNθ$ in the above equation, we have

Interchanging $V$ and $U$ and subtracting the resultant from the above equation, we get

In particular, replacing $U$ by $PV∈Dθ$⁠, the above equation yields

Using (3.7), we obtain

where ${e0=η,e1,e2,…, ep,Ψe1,Ψe2, …,Ψep}$ and ${e1,e2,…,eq,eq+1=sec θPe1, eq+2=sec θPe2,…,e2q=sec θPeq}$ are the frames of the orthonormal vector fields on $T NT$ and $T Nθ$ respectively.

Using part (ii) of Lemmas 3.2, 3.3 and utilizing (2.24), we get

Replacing $ei$ by $Ψei$ in above equation, we get

Subtracting the above two findings, we get the required result. $□$

Now, we prove the following characterization.

and

where$σμ$denotes the component of$σ$in$μ$, and $(2p+1)$and$2q$are the dimensions of$NT$, and$Nθ$respectively.

On the other hand by Codazzi equation

Now, we compute the values of the terms involved in (3.11). First, we have