**1. Introduction**

There have been several studies in the past to demonstrate the geometries of submanifolds in the settings of almost Hermitian (A-H) and almost contact metric (A-C M) manifolds. By the operation of the almost complex structure *J*, the tangent space of a submanifold of an almost Hermitian manifold can be classified into holomorphic and totally real submanifolds. The notion of CR-submanifolds was introduced and studied by A. Bejancu [1] in 1981 as a generalization of holomorphic and totally real submanifolds. Thus, as to have a more profound knowledge of the geometry of CR-submanifolds of almost Hermitian "AH" manifolds, Chen [2] further explored these submanifolds and provided many fundamental results. In 1990 Chen [3] instigated a generalized class of submanifolds, namely, slant submanifolds. Moreover, advances in the geometry of CR-submanifolds and slant submanifolds stimulated various authors to search for the class of submanifolds which unifies the properties of all previously discussed submanifolds. In this context, N. Papaghuic [4] introduced the notion of semi-slant submanifolds in the framework of almost-Hermitian manifolds and showed that submanifolds belonging to this class enjoy many of the desired properties. Later, the contact variant of semi-slant submanifolds was studied by Cabrerizo et al. [5]. Recently, B. Sahin [6] investigated another class of submanifolds in the setting of almost Hermitian manifolds and he called these submanifolds Hemi-slant submanifolds. This class includes the CR-submanifolds and slant submanifolds.

In 1990, Ronsse [7] started the study of skew CR-submanifolds in the setting of almost Hermitian manifolds. Skew CR-submanifolds contain the classes of CR-submanifolds, semi-slant submanifolds and Hemi-slant submanifolds.

The acknowledgment of warped product manifolds appeared after the methodology of Bishop and O'Neill [8] on the manifolds of non positive curvature. By analyzing the way that a Riemannian product of manifolds cannot have non positive curvature, they represented warped product (W-P) manifolds for the class of manifolds of non-positive curvature which is characterized as follows:

Let (*<sup>S</sup>*1,,1) and (*<sup>S</sup>*2,,2) be two Riemannian manifolds with Riemannian metrics ,<sup>1</sup> and ,<sup>2</sup> respectively and *g* be a smooth positive function on *S*1. If *π* : *S*1 × *S*2 → *S*1 and *η* : *S*1 × *S*2 → *S*2 are the projection maps given by *<sup>π</sup>*(*<sup>x</sup>*, *y*) = *x* and *η*(*<sup>x</sup>*, *y*) = *y* for every (*<sup>x</sup>*, *y*) ∈ *S*1 × *S*2, then the *W-P manifold* is the product manifold *S*1 × *S*2 holding the Riemannian structure such that

$$
\langle \mathcal{U}\_1, \mathcal{U}\_2 \rangle = \langle \pi\_\* \mathcal{U}\_1, \pi\_\* \mathcal{U}\_2 \rangle\_1 + (\mathcal{g} \circ \pi)^2 \langle \eta\_\* \mathcal{U}\_1, \eta\_\* \mathcal{U}\_2 \rangle\_{2, 1}
$$

for all *U*1, *U*2 ∈ *TS*. The function *g* is called the *warping function* (W-F) of the warped product (W-P) manifold. If the W-F is constant, then the W-P is a trivial, i.e., simply Riemannian product. Further, if *U*1 ∈ *TS*1 and *U*2 ∈ *TS*2, then from Lemma 7.3 of [8], we have the following well-known result

$$D\_{lI\_1} \mathcal{U}\_2 = D\_{lI\_2} \mathcal{U}\_1 = (\frac{\mathcal{U}\_1 \mathcal{g}}{\mathcal{g}}) \mathcal{U}\_{2\prime} \tag{1}$$

where *D* is the Levi-Civita connection on *S*. In the light of the fact that W-P manifolds have various uses in physics and the theory of relativity [9], this has been a subject of broad interest. The idea of displaying the space-time close to black holes admits the W-P manifolds [10]. Schwartzschild space-time *T* <sup>×</sup>*k S*2, is a model of W-P, wherein the base *T* = *R* × *R*<sup>+</sup> is a half plane *k* > 0 and the fiber *S*2 is the unit sphere. A cosmological model to show the universe as space-time, known as the Robertson–Walker model, is a W-P manifold [11].

Some common properties of W-P manifolds were concentrated on in [8]. B.-Y. Chen [12] played out an outward investigation of W-P submanifolds in a Kaehler manifold. From that point forward, numerous geometers have investigated W-P manifolds in various settings such as almost complex and almost contact manifolds, and different existence results have been researched (see the survey article [13–16]). Recently, B. Sahin [17] contemplated SCR W-P submanifolds in Kaehler manifolds and go<sup>t</sup> some essential outcomes. Further, these submanifolds were explored by Haidar and Thakur in the context of cosymplectic manifolds [18].

In 1999, Chen [19] discovered a relationship between Ricci curvature and a squared mean curvature vector for a discretionary Riemannian manifold. More precisely, Chen proved the following theorem

**Theorem 1.** *Let φ* : *St* → *<sup>S</sup>*¯*m*(*c*) *be an isometric immersion of a t*− *dimensional Riemannian manifold into a Riemannian space form S* ¯ *<sup>m</sup>*(*c*).

*1. For each unit tangent vector χ* ∈ *TpS<sup>t</sup>*, *we have*

$$\|\|\Pi\|^2(p) \ge \frac{4}{t^2} \{ \mathbb{R}^S(\chi) - (t-1)c \},$$

*where* -Π-<sup>2</sup>(*p*) *is the squared mean curvature and <sup>R</sup><sup>S</sup>*(*χ*) *the Ricci curvature of St at χ.*


Theorem 1 was generalized for semi-slant submanifolds in Sasakian space form by Cioroboiu and Chen [20]. Further, D. W. Yoon [21] studied Chen Ricci inequality for slant submanifols in the framework of cosymplectic space forms. Motivated by Chen [19], Mihai and Ozgur [ ¨ 22] studied Chen Ricci inequality for real space forms with semi-symmetric connections. In [23] M. M. Tripathi formulated an improved relationship between Ricci curvature and squared mean curvature. More recently, Ali et al. [24] generalized Chen Ricci inequality for warped product submanifolds in spheres and provided some applications in mechanics and mathematical physics.

The class of SCR W-P submanifolds is rich in its geometric behavior; it contains classes of CR-warped product submanifolds, semi-slant warped product submanifolds and hemi-slant warped product submanifolds. In the literature it was found that Ricci curvature for these warped product submanifolds in complex space forms has not been studied. In other words, we can say that Theorem 1 is an open problem for skew CR-warped product submanifolds in the setting of complex space forms.

In this study our point is to establish a connection between Ricci curvature and squared mean curvature for SCR W-P submanifolds in the setting of complex space forms.
