Chen-Type Inequalities for PS-Submanifolds in Complex Space Forms

Abstract

In this paper, we investigate Chen’s $\delta$-invariant for partially slant (PS) submanifolds of complex space forms. A PS-submanifold admits an orthogonal decomposition of the tangent bundle into a proper slant distribution and an arbitrary ambiguous distribution. Using the Gauss equation together with algebraic optimization techniques, we derive a Chen-type inequality relating the $\delta$-invariant to the squared mean curvature, the holomorphic sectional curvature of the ambient space, and the slant angle of the slant distribution. Unlike the classical Chen inequality for slant submanifolds, the obtained estimate contains an additional term reflecting the contribution of the ambiguous distribution. Several corollaries are derived, including dimension-dependent bounds and special cases corresponding to hemi-slant and semi-slant submanifolds. The equality case is also characterized in terms of the structure of the shape operators. These results provide a natural extension of Chen-type inequalities to the broader framework of partially slant geometry in Kähler manifolds.

1. Introduction

One of the central problems in submanifold theory is to understand the relationship between intrinsic invariants of a Riemannian manifold and the extrinsic properties of its immersion into an ambient space. Among intrinsic invariants, the scalar curvature $\tau$, obtained as the trace of the Riemannian curvature tensor, plays a fundamental role. However, since scalar curvature averages sectional curvatures over all tangent planes, it often fails to capture finer geometric features. To address this limitation, B.-Y. Chen [1] introduced a family of invariants known as $\delta$-invariants, which measure the deviation of scalar curvature from the sectional curvature of certain plane sections.

For an n-dimensional Riemannian manifold N ($n\ge 3$), the first $\delta$-invariant is defined at a point $p\in N$ by

$${\delta }_M\left(p\right)=\tau \left(p\right)-\inf K\left(\pi \right),$$

where $\tau \left(p\right)$ denotes the scalar curvature at p and the infimum is taken over all plane sections $\pi \subset T_{p}N$ [1]. This invariant has proven to be an effective tool for establishing relationships between intrinsic geometry and extrinsic quantities such as the mean curvature. In particular, Chen-type inequalities provide upper bounds for ${\delta }_M$ in terms of ${\parallel H\parallel }^2$ and, in the case of space forms, the ambient curvature. These inequalities have been extensively studied in various geometric settings, including complex space forms [2,3], Sasakian space forms, quaternionic space forms [4,5], and locally symmetric spaces [6,7,8]. A systematic account of these developments can be found in [6].

In almost Hermitian and Kähler geometry, the presence of a complex structure J enriches the study of submanifolds by introducing new geometric interactions between tangent and normal bundles. Foundational results in this direction are presented in the monograph of Bejancu [9]. A prominent example is that of slant submanifolds, introduced by Chen [10], for which the angle between $JX$ and the tangent space is constant for any tangent vector X. This notion generalizes both holomorphic submanifolds ($\theta =0$) and totally real submanifolds ($\theta =\pi /2$).

For slant submanifolds in complex space forms, Mihai [2] established a Chen-type inequality of the form

$${\delta }_M\le {\displaystyle \frac{n-2}{2}}\left\{{\displaystyle \frac{n^2}{n-1}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{4}}\left(n+1+3{\cos }^2\theta \right)\right\},$$

where c is the holomorphic sectional curvature of the ambient space and $\theta$ is the slant angle. This result has motivated further studies on generalized slant-type submanifolds, including semi-slant, hemi-slant, and bi-slant submanifolds [11,12]. Additional developments include quasi-slant submanifolds introduced by Etayo [13] and structural studies of semi-slant geometry by Şahin [14].

Despite these advances, existing Chen-type inequalities do not account for situations where the tangent bundle admits a decomposition involving a distribution that is neither slant, holomorphic, nor totally real. This limitation is addressed by the recently introduced class of partially slant (PS) submanifolds due to Yerlikaya, Poyraz, and Şahin [15]. A PS-submanifold is characterized by an orthogonal decomposition of the tangent bundle into a proper slant distribution ${\mathcal{H}}^{\theta }$ and an additional distribution $\mathcal{H}$, called the ambiguous distribution, satisfying $J{\mathcal{H}}^{\theta }\perp \mathcal{H}$ and ${\mathcal{H}}^{\theta }\perp J\mathcal{H}$. This condition ensures that the two distributions remain geometrically independent under the action of the complex structure.

The class of PS-submanifolds provides a unifying framework that includes slant, semi-slant, hemi-slant, and bi-slant submanifolds as special cases, while also allowing more general configurations. In particular, the presence of the ambiguous distribution $\mathcal{H}$ introduces additional geometric contributions that are not captured in classical Chen inequalities. This motivates the study of $\delta$-invariants in the PS-setting.

The main objective of this paper is to establish a Chen-type inequality for PS-submanifolds in complex space forms. Our main result (Theorem 1) provides an upper bound for the $\delta$-invariant in terms of the mean curvature, the ambient holomorphic sectional curvature, and the slant angle of ${\mathcal{H}}^{\theta }$, together with additional terms reflecting the contribution of the ambiguous distribution. The proof is based on the Gauss equation combined with an algebraic optimization procedure applied to the components of the second fundamental form.

We also obtain a complete characterization of the equality case. A PS-submanifold is said to be ideal if it attains equality in the derived inequality identically. The equality characterization reveals that such submanifolds must satisfy strong algebraic conditions on their shape operators, leading to a rigid geometric structure. Although the equality conditions are explicitly determined, constructing concrete examples realizing equality appears to be a nontrivial problem and is left for future investigation.

As applications of the main inequality, we derive several consequences. In particular, for minimal PS-submanifolds, the inequality reduces to a purely curvature-dependent bound. In the case of complex hyperbolic space ($c<0$), we obtain restrictions on the sign of the $\delta$-invariant. Furthermore, we show that our result recovers known inequalities for slant, semi-slant, and bi-slant submanifolds as special cases, thereby highlighting the unifying nature of the PS-framework.

The paper is organized as follows. Section 2 recalls basic notions on Kähler manifolds, complex space forms [16], and PS-submanifolds. In Section 3, we derive an expression for the scalar curvature and prove the main Chen-type inequality, including a detailed analysis of the equality case. Section 4 is devoted to applications and corollaries, while Section 5 presents concluding remarks and directions for future research.

2. Preliminaries

Let $(\tilde{M},g,J)$ be a Kähler manifold of real dimension $2m$ (complex dimension m). The almost complex structure J satisfies

$$J^2=-I,g(JX,JY)=g(X,Y),\left({\tilde{\nabla }}_{X}J\right)Y=0\forall X,Y\in \Gamma \left(T\tilde{M}\right),$$

(1)

where $\tilde{\nabla }$ denotes the Levi-Civita connection of g. The condition $\tilde{\nabla }J=0$ implies that J is parallel, which is equivalent to the Kähler condition $d\Omega =0$, where $\Omega (X,Y)=g(X,JY)$ is the fundamental 2-form.

A complex space form, denoted by $\tilde{M}\left(c\right)$, is a Kähler manifold of constant holomorphic sectional curvature c. The curvature tensor $\tilde{R}$ of $\tilde{M}\left(c\right)$ has the well-known expression [2]

$$\begin{array}{cc}\hfill \tilde{R}(X,Y)Z& ={\displaystyle \frac{c}{4}}\{g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX\hfill \\ \hfill & -g(JX,Z)JY+2g(X,JY)JZ\},\hfill \end{array}$$

(2)

for any vector fields $X,Y,Z\in \Gamma \left(T\tilde{M}\right)$.

Important examples of complex space forms include complex Euclidean space ${\mathbb{C}}^m$ with $c=0$, complex projective space $\mathbb{C}{P}^m\left(4c\right)$ with $c>0$, and complex hyperbolic space $\mathbb{C}{H}^m\left(4c\right)$ with $c<0$.

Let N be an n-dimensional real submanifold of $\tilde{M}\left(c\right)$ isometrically immersed via the inclusion map. The induced Riemannian metric on N is also denoted by g. For any $X\in \Gamma \left(TN\right)$, we decompose $JX$ into its tangential and normal components:

$$JX=PX+FX,$$

(3)

where $PX\in \Gamma \left(TN\right)$ and $FX\in \Gamma \left(T^{\perp }N\right)$. Similarly, for any $\xi \in \Gamma \left(T^{\perp }N\right)$, we write

$$J\xi =t\xi +f\xi ,$$

(4)

with $t\xi \in \Gamma \left(TN\right)$ and $f\xi \in \Gamma \left(T^{\perp }N\right)$. The morphisms P, F, t, and f satisfy various algebraic relations; in particular, from $J^2=-I$ we obtain

$$\begin{array}{cc}\hfill P^2+tF& =-I,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill FP+fF& =0,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill Pt+tf& =0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill Ft+f^2& =-I.\hfill \end{array}$$

(8)

Let $\nabla$ and ${\nabla }^{\perp }$ denote the Levi-Civita connections on $TN$ and $T^{\perp }N$, respectively. The Gauss and Weingarten formulas are given by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\tilde{\nabla }}_{X}Y\hfill & ={\nabla }_{X}Y+h(X,Y),\hfill \\ {\tilde{\nabla }}_X\xi \hfill & =-A_{\xi }X+{\displaystyle {\nabla }_X^{\perp }}\xi ,\hfill \end{array}\right.\end{array}$$

(9)

for all $X,Y\in \Gamma \left(TN\right)$ and $\xi \in \Gamma \left(T^{\perp }N\right)$. Here, $h:\Gamma \left(TN\right)\times \Gamma \left(TN\right)\to \Gamma \left(T^{\perp }N\right)$ is the second fundamental form, and $A_{\xi }:\Gamma \left(TN\right)\to \Gamma \left(TN\right)$ is the shape operator (Weingarten map) in the direction of $\xi$. These are related by

$$g(A_{\xi }X,Y)=g(h(X,Y),\xi ).$$

(10)

The mean curvature vector H is defined as $H={\displaystyle \frac{1}{n}}\mathrm{tr}h$. A submanifold is called minimal if $H=0$, and totally geodesic if $h=0$.

The covariant derivative of the second fundamental form is given by

$$\left({\overline{\nabla }}_{X}h\right)(Y,Z)={\displaystyle {\nabla }_X^{\perp }}h(Y,Z)-h({\nabla }_{X}Y,Z)-h(Y,{\nabla }_{X}Z),$$

(11)

for $X,Y,Z\in \Gamma \left(TN\right)$. The equations of Gauss, Codazzi, and Ricci are respectively

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

(12)

$$\begin{array}{c}{\left(\tilde{R}(X,Y)Z\right)}^{\perp }=\left({\overline{\nabla }}_{X}h\right)(Y,Z)-\left({\overline{\nabla }}_{Y}h\right)(X,Z),\hfill \end{array}$$

(13)

$$\begin{array}{c}{R}^{\perp }(X,Y,\xi ,\eta )=\tilde{R}(X,Y,\xi ,\eta )+g([A_{\xi },A_{\eta }]X,Y),\hfill \end{array}$$

(14)

for $X,Y,Z,W\in \Gamma \left(TN\right)$ and $\xi ,\eta \in \Gamma \left(T^{\perp }N\right)$, where R and $R^{\perp }$ are the curvature tensors of $\nabla$ and ${\nabla }^{\perp }$, respectively.

A distribution $\mathcal{D}$ on a submanifold N of a Kähler manifold is called slant if there exists a constant $\theta \in [0,\pi /2]$ such that for every nonzero $X\in {\mathcal{D}}_p$, the angle between $JX$ and ${\mathcal{D}}_p$ is constant $\theta$. The constant $\theta$ is called the slant angle of $\mathcal{D}$. An equivalent algebraic characterization is given by [10]

$$P^2=-{\cos }^2\theta ·I\mathrm{on}\mathcal{D}.$$

(15)

Special cases arise for particular values of the slant angle. When $\theta =0$, the distribution $\mathcal{D}$ becomes holomorphic, that is, $J\mathcal{D}=\mathcal{D}$. On the other hand, when $\theta ={\displaystyle \frac{\pi }{2}}$, the distribution $\mathcal{D}$ is totally real, meaning that $J\mathcal{D}\subset T^{\perp }N$.

For a slant distribution, it follows from (5) and (15) that

$$tF=-{\sin }^2\theta ·I\mathrm{on}\mathcal{D}.$$

(16)

We now recall the definition of partially slant submanifolds as introduced in [15].

Definition 1. 

A submanifold N of a Kähler manifold $\tilde{M}$ is called a partially slant (PS) submanifold if its tangent bundle admits an orthogonal decomposition

$$TN={\mathcal{H}}^{\theta }\oplus \mathcal{H},$$

(17)

where ${\mathcal{H}}^{\theta }$ is a proper slant distribution with slant angle θ and $\mathcal{H}$ is an arbitrary complementary distribution, called the ambiguous distribution, satisfying

$$J{\mathcal{H}}^{\theta }\perp \mathcal{H},{\mathcal{H}}^{\theta }\perp J\mathcal{H}.$$

(18)

Geometrically, these conditions ensure that the two distributions do not “mix” under J: the slant distribution is mapped into a subspace orthogonal to the ambiguous distribution, preserving the orthogonal decomposition. Condition (18) is equivalent to $P{\mathcal{H}}^{\theta }\subset {\mathcal{H}}^{\theta }$ and $P\mathcal{H}\subset \mathcal{H}$. A PS-submanifold is called proper if $\mathcal{H}$ is neither a slant distribution nor a holomorphic nor a totally real distribution. The normal bundle of a PS-submanifold decomposes as [15]

$$T^{\perp }N=F{\mathcal{H}}^{\theta }\oplus F\mathcal{H}\oplus \nu ,$$

(19)

where $\nu$ is a J-invariant subbundle, i.e., $J\nu =\nu$.

The J-invariant subbundle $\nu$ has even dimension. Let

$$r=\dim ({\mathcal{H}}_p\cap J{\mathcal{H}}_p)$$

denote the dimension of the maximal holomorphic subdistribution of $\mathcal{H}$ at a point p (i.e., the set of vectors in $\mathcal{H}$ that are mapped into $TN$ by J). Since F is injective on ${\mathcal{H}}^{\theta }$ (for a proper slant distribution with $\theta \ne 0$), we have $\dim F{\mathcal{H}}^{\theta }=a$. On $\mathcal{H}$, the map $X\mapsto FX$ has kernel $\mathcal{H}\cap J\mathcal{H}$, hence $\dim F\mathcal{H}=b-r$. The total normal space has dimension $2m-n$. Therefore,

$$\dim \nu =\dim T^{\perp }N-\dim F{\mathcal{H}}^{\theta }-\dim F\mathcal{H}=(2m-n)-a-(b-r).$$

Because $a+b=n$, this simplifies to

$$\dim \nu =2m-n-a-b+r=2(m-n)+r.$$

The quantity h is even (it is the real dimension of a complex subspace), so $\dim \nu$ is even as required. Special cases:

• If $\mathcal{H}$ contains no holomorphic directions ($r=0$), then $\dim \nu =2(m-n)$.
• For a Lagrangian submanifold ($n=m$) with $r=0$, we obtain $\dim \nu =0$, so $\nu$ is trivial.
• If $\mathcal{H}$ is itself holomorphic ($h=b$), then $\dim F\mathcal{H}=0$ and $\dim \nu =2m-2n+b$.

For the slant distribution ${\mathcal{H}}^{\theta }$, relations (15) and (16) hold. For the ambiguous distribution $\mathcal{H}$, from (5)–(8), we obtain

$$\begin{array}{cc}\hfill P^{2}X+tFX& =-X,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill FPX+fFX& =0,\hfill \end{array}$$

(21)

for any $X\in \Gamma \left(\mathcal{H}\right)$. Moreover, the morphism t satisfies

$$t\left(F{\mathcal{H}}^{\theta }\right)\subseteq {\mathcal{H}}^{\theta },t\left(F\mathcal{H}\right)\subseteq \mathcal{H}.$$

(22)

Important special cases of PS-submanifolds include the following (see

Table 1. Special cases arising from the definition of partially slant submanifolds.

Condition on Ambiguous Distribution $\mathcal{H}$	Type of Submanifold Obtained
$\mathcal{H}=\left\{0\right\}$	Slant submanifold [10]
$\mathcal{H}$ is holomorphic ($J\mathcal{H}=\mathcal{H}$)	Semi-slant submanifold [17]
$\mathcal{H}$ is totally real ($J\mathcal{H}\subset T^{\perp }N$)	Hemi-slant submanifold [11,12]
$\mathcal{H}$ is slant with angle $\varphi$	Bi-slant submanifold [11]

):

For an n-dimensional Riemannian manifold N, the scalar curvature $\tau$ at a point p is defined as

$$\tau \left(p\right)=\sum _{1\le i<j\le n}K(e_i\wedge e_j),$$

(23)

where $\{e_1,\dots ,e_n\}$ is an orthonormal basis of $T_{p}N$ and $K(e_i\wedge e_j)$ denotes the sectional curvature of the plane spanned by $e_i$ and $e_j$.

Following Chen [1], we define the $\delta$-invariant ${\delta }_M\left(p\right)$ by

$${\delta }_M\left(p\right)=\tau \left(p\right)-\underset{\pi \subset T_{p}N}{\inf }K\left(\pi \right),$$

(24)

where the infimum is taken over all plane sections $\pi$ of $T_{p}N$.

The squared norm of the second fundamental form is given by

$${\parallel h\parallel }^2={\displaystyle \sum _{\alpha =n+1}^{2m}}{\displaystyle \sum _{i,j=1}^n}g{(h(e_i,e_j),e_{\alpha })}^2,$$

(25)

where $\{e_{n+1},\dots ,e_{2m}\}$ is an orthonormal basis of ${\displaystyle T_p^{\perp }}N$. The mean curvature vector H satisfies

$${\parallel H\parallel }^2={\displaystyle \frac{1}{n^2}}{\displaystyle \sum _{\alpha =n+1}^{2m}}{\left({\displaystyle \sum _{i=1}^n}g(h(e_i,e_j),e_{\alpha })\right)}^2.$$

(26)

These invariants form the foundation for the Chen inequalities we will establish in the following sections.

3. Chen-Type Inequality for PS-Submanifolds

Let $N^n$ be a proper partially slant submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$. The tangent bundle splits orthogonally as

$$TN={\mathcal{H}}^{\theta }\oplus \mathcal{H},$$

where ${\mathcal{H}}^{\theta }$ is a slant distribution with slant angle $\theta$ and $\mathcal{H}$ is the ambiguous distribution. Denote

$$\dim {\mathcal{H}}^{\theta }=a,\dim \mathcal{H}=b=n-a.$$

For a point $p\in N$, we introduce two auxiliary quantities that capture the geometry of the ambiguous distribution. Let $\{e_{a+1},\dots ,e_n\}$ be any orthonormal basis of ${\mathcal{H}}_p$. Define

$${\Lambda }_{\mathcal{H}}\left(p\right)=\sum _{a<i<j\le n}g{(P{e}_i,e_j)}^2={\displaystyle \frac{1}{2}}{\parallel {P|}_{{\mathcal{H}}_p}\parallel }_{\mathrm{HS}}^2,$$

which is independent of the chosen basis due to the skew-symmetry of P. Let ${\pi }_p=\mathrm{span}\{e_1,e_2\}$ be a 2-plane in $T_{p}N$ realizing the infimum of sectional curvature, i.e.,

$$K\left({\pi }_p\right)=\underset{\pi \subset T_{p}N}{\inf }K\left(\pi \right),$$

and set $\beta \left(p\right)=g{(J{e}_1,e_2)}^2$.

Choose an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{p}N$ adapted to the decomposition, so that

$$\{e_1,\dots ,e_a\}\subset {\displaystyle {\mathcal{H}}_p^{\theta }},\{e_{a+1},\dots ,e_n\}\subset {\mathcal{H}}_p.$$

Since ${\mathcal{H}}^{\theta }$ is a slant distribution with angle $\theta$, we have $P^{2}X=-{\cos }^2\theta X$ for all $X\in {\mathcal{H}}^{\theta }$. Using the skew-symmetry of P, this implies ${\parallel PX\parallel }^2={\cos }^2\theta {\parallel X\parallel }^2$. Hence,

$${\displaystyle \sum _{i=1}^a}{\parallel P{e}_i\parallel }^2=a{\cos }^2\theta ,$$

(27)

which further implies

$$\sum _{1\le i<j\le a}g{(P{e}_i,e_j)}^2={\displaystyle \frac{a}{2}}{\cos }^2\theta .$$

(28)

For tangent vectors, $g(J{e}_i,e_j)=g(P{e}_i,e_j)$. Thus, (28) yields

$$\sum _{1\le i<j\le a}g{(J{e}_i,e_j)}^2={\displaystyle \frac{a}{2}}{\cos }^2\theta .$$

(29)

For the ambiguous distribution, we directly have

$$\sum _{a<i<j\le n}g{(J{e}_i,e_j)}^2={\Lambda }_{\mathcal{H}}\left(p\right).$$

(30)

The defining orthogonality conditions $J{\mathcal{H}}^{\theta }\perp \mathcal{H}$ and ${\mathcal{H}}^{\theta }\perp J\mathcal{H}$ imply $g(J{e}_i,e_j)=0$ whenever $1\le i\le a<j\le n$. Therefore, summing (29) and (30),

$$\sum _{1\le i<j\le n}g{(J{e}_i,e_j)}^2={\displaystyle \frac{a}{2}}{\cos }^2\theta +{\Lambda }_{\mathcal{H}}\left(p\right).$$

(31)

In a complex space form, the sectional curvature of a plane spanned by orthonormal vectors $e_i,e_j$ is

$$\tilde{K}(e_i\wedge e_j)={\displaystyle \frac{c}{4}}\left(1+3g{(J{e}_i,e_j)}^2\right).$$

Summing over all tangent planes gives the ambient scalar curvature restricted to $T_{p}N$:

$$\begin{array}{cc}\hfill \tilde{\tau }\left(p\right)& =\sum _{1\le i<j\le n}\tilde{K}(e_i\wedge e_j)\hfill \\ \hfill & ={\displaystyle \frac{c}{4}}·{\displaystyle \frac{n(n-1)}{2}}+{\displaystyle \frac{3c}{4}}\sum _{1\le i<j\le n}g{(J{e}_i,e_j)}^2.\hfill \end{array}$$

(32)

Using (31) in (32),

$$\begin{array}{cc}\hfill \tilde{\tau }\left(p\right)& ={\displaystyle \frac{c}{8}}n(n-1)+{\displaystyle \frac{3c}{4}}\left({\displaystyle \frac{a}{2}}{\cos }^2\theta +{\Lambda }_{\mathcal{H}}\left(p\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{c}{8}}\left[n(n-1)+3a{\cos }^2\theta +6{\Lambda }_{\mathcal{H}}\left(p\right)\right].\hfill \end{array}$$

(33)

For the distinguished plane ${\pi }_p=\mathrm{span}\{e_1,e_2\}$, the ambient sectional curvature is

$$\tilde{K}\left({\pi }_p\right)={\displaystyle \frac{c}{4}}\left(1+3\beta \left(p\right)\right).$$

(34)

Subtracting (34) from (33), we obtain

$$\tilde{\tau }\left(p\right)-\tilde{K}\left({\pi }_p\right)={\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta +6{\Lambda }_{\mathcal{H}}\left(p\right)-6\beta \left(p\right)\right].$$

(35)

Let $\{e_{n+1},\dots ,e_{2m}\}$ be an orthonormal basis of ${\displaystyle T_p^{\perp }}N$ and denote ${\displaystyle h_{ij}^{\alpha }}=g(h(e_i,e_j),e_{\alpha })$. The Gauss equation gives

$$K(e_i\wedge e_j)=\tilde{K}(e_i\wedge e_j)+{\displaystyle \sum _{\alpha =n+1}^{2m}}\left({\displaystyle h_{ii}^{\alpha }}{\displaystyle h_{jj}^{\alpha }}-{\left({\displaystyle h_{ij}^{\alpha }}\right)}^2\right).$$

(36)

Summing (36) over all $i<j$ and using the standard identity relating scalar curvature to the second fundamental form (see e.g., [8]), we have

$$\tau \left(p\right)=\tilde{\tau }\left(p\right)+{\displaystyle \frac{1}{2}}\left(n^2{\parallel H\parallel }^2-{\parallel h\parallel }^2\right).$$

(37)

For the plane ${\pi }_p$, (36) yields

$$K\left({\pi }_p\right)=\tilde{K}\left({\pi }_p\right)+{\displaystyle \sum _{\alpha =n+1}^{2m}}\left({\displaystyle h_{11}^{\alpha }}{\displaystyle h_{22}^{\alpha }}-{\left({\displaystyle h_{12}^{\alpha }}\right)}^2\right).$$

(38)

Now, ${\delta }_M\left(p\right)=\tau \left(p\right)-K\left({\pi }_p\right)$. Subtracting (38) from (37) gives

$${\delta }_M\left(p\right)=\left(\tilde{\tau }\left(p\right)-\tilde{K}\left({\pi }_p\right)\right)+{\displaystyle \sum _{\alpha =n+1}^{2m}}{E}_{\alpha },$$

(39)

where

$$E_{\alpha }={\displaystyle \frac{1}{2}}{\left({\displaystyle \sum _{i=1}^n}{\displaystyle h_{ii}^{\alpha }}\right)}^2-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\left({\displaystyle h_{ii}^{\alpha }}\right)}^2-{\displaystyle h_{11}^{\alpha }}{\displaystyle h_{22}^{\alpha }}-\sum _{\begin{array}{c}i<j\\ (i,j)\ne (1,2)\end{array}}{\left({\displaystyle h_{ij}^{\alpha }}\right)}^2.$$

(40)

Fix $\alpha$ and set $t_{\alpha }={\displaystyle {\sum }_{i=1}^n}{\displaystyle h_{ii}^{\alpha }}$, $x_i={\displaystyle h_{ii}^{\alpha }}$. Then, from (40),

$$E_{\alpha }\le {\displaystyle \frac{1}{2}}{\displaystyle t_{\alpha }^2}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle x_i^2}-x_1{x}_2,$$

since the last term in (40) is non-positive. Observe that

$${\displaystyle x_1^2}+{\displaystyle x_2^2}+2x_1{x}_2={(x_1+x_2)}^2,$$

hence

$${\displaystyle \frac{1}{2}}{\displaystyle t_{\alpha }^2}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\displaystyle x_i^2}-x_1{x}_2={\displaystyle \frac{1}{2}}{\displaystyle t_{\alpha }^2}-{\displaystyle \frac{1}{2}}\left[{(x_1+x_2)}^2+{\displaystyle x_3^2}+\cdots +{\displaystyle x_n^2}\right].$$

Define $y_1=x_1+x_2$, $y_2=x_3,\dots ,y_{n-1}=x_n$. Then, ${\displaystyle {\sum }_{r=1}^{n-1}}{y}_r=t_{\alpha }$, and the expression becomes

$${\displaystyle \frac{1}{2}}{\displaystyle t_{\alpha }^2}-{\displaystyle \frac{1}{2}}{\displaystyle \sum _{r=1}^{n-1}}{\displaystyle y_r^2}.$$

To maximize this, we minimize ${\displaystyle {\sum }_{r=1}^{n-1}}{\displaystyle y_r^2}$ subject to $\sum y_r=t_{\alpha }$. By Cauchy–Schwarz,

$${\displaystyle t_{\alpha }^2}={\left({\displaystyle \sum _{r=1}^{n-1}}{y}_r\right)}^2\le (n-1){\displaystyle \sum _{r=1}^{n-1}}{\displaystyle y_r^2},$$

so $\sum {\displaystyle y_r^2}\ge {\displaystyle t_{\alpha }^2}/(n-1)$. Therefore,

$$E_{\alpha }\le {\displaystyle \frac{1}{2}}{\displaystyle t_{\alpha }^2}-{\displaystyle \frac{1}{2}}·{\displaystyle \frac{{\displaystyle t_{\alpha }^2}}{n-1}}={\displaystyle \frac{n-2}{2(n-1)}}{\displaystyle t_{\alpha }^2}.$$

(41)

Equality in (41) occurs if and only if equality holds in the Cauchy–Schwarz inequality and in the estimate obtained after discarding the non-positive terms. The equality case of the Cauchy–Schwarz inequality requires that $y_1=y_2=\cdots =y_{n-1}$, which is equivalent to $x_1+x_2=x_3=\cdots =x_n$. In addition, the terms omitted in the previous estimate must vanish, which implies that ${\displaystyle h_{ij}^{\alpha }}=0$ for all $(i,j)\ne (1,2)$ with $i<j$.

Summing (41) over $\alpha$ and using ${\sum }_{\alpha }{\displaystyle t_{\alpha }^2}=n^2{\parallel H\parallel }^2$, we obtain

$${\displaystyle \sum _{\alpha =n+1}^{2m}}{E}_{\alpha }\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2.$$

(42)

Substituting (35) and (42) into (39), we establish Chen-type inequality for PS-submanifolds of a complex space form, as follows.

Theorem 1. 

Let $N^n$ be a proper PS-submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$. Then, at every point $p\in N$, the δ-invariant satisfies

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta +6{\Lambda }_{\mathcal{H}}\left(p\right)-6\beta \left(p\right)\right].$$

(43)

where $a=\dim {\mathcal{H}}^{\theta }$,

$${\Lambda }_{\mathcal{H}}\left(p\right)=\sum _{a<i<j\le n}g{(P{e}_i,e_j)}^2$$

for any orthonormal basis of ${\mathcal{H}}_p$, and $\beta \left(p\right)=g{(J{e}_1,e_2)}^2$, where $\{e_1,e_2\}$ spans a plane realizing the infimum of sectional curvature at p.

Moreover, equality holds at p if and only if there exists an orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{p}N$ adapted to the decomposition $T_{p}N={\displaystyle {\mathcal{H}}_p^{\theta }}\oplus {\mathcal{H}}_p$ such that the plane $\pi =\mathrm{span}\{e_1,e_2\}$ realizes the minimum sectional curvature, that is, $K\left(\pi \right)={\inf }_{{\pi }^{\prime }\subset T_{p}N}K\left({\pi }^{\prime }\right)$. In addition, there exists an orthonormal basis $\{e_{n+1},\dots ,e_{2m}\}$ of the normal space ${\displaystyle T_p^{\perp }}N$ for which each shape operator $A_{e_{\alpha }}$, $\alpha =n+1,\dots ,2m$, has the block-diagonal form

$$A_{e_{\alpha }}=\left(\begin{array}{ccccc}{\displaystyle h_{11}^{\alpha }}& {\displaystyle h_{12}^{\alpha }}& 0& \cdots & 0\\ {\displaystyle h_{12}^{\alpha }}& {\displaystyle h_{22}^{\alpha }}& 0& \cdots & 0\\ 0& 0& {\displaystyle h_{33}^{\alpha }}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\displaystyle h_{nn}^{\alpha }}\end{array}\right),$$

where ${\displaystyle h_{ij}^{\alpha }}=g(h(e_i,e_j),e_{\alpha })$, and the diagonal entries satisfy

$${\displaystyle h_{11}^{\alpha }}+{\displaystyle h_{22}^{\alpha }}={\displaystyle h_{33}^{\alpha }}=\cdots ={\displaystyle h_{nn}^{\alpha }}.$$

Figure 1 illustrates how the upper bound of ${\delta }_M$ depends on the slant angle $\theta$ and the ambiguity factor ${\Lambda }_{\mathcal{H}}$.

Next, we construct an example inspired by the example in [15] showing that the bound established in Theorem 1 is sharp.

Example 1. 

Consider the complex Euclidean space ${\mathbb{C}}^6\cong {\mathbb{R}}^{12}$ endowed with the standard Euclidean metric and the standard complex structure

$$J\left({\displaystyle \frac{𝜕}{𝜕x_k}}\right)={\displaystyle \frac{𝜕}{𝜕y_k}},J\left({\displaystyle \frac{𝜕}{𝜕y_k}}\right)=-{\displaystyle \frac{𝜕}{𝜕x_k}},k=1,\dots ,6.$$

Then, ${\mathbb{C}}^6$ is a complex space form with $c=0$.

Let $\theta \in (0,\pi /2)$ and define an immersion

$$X:{\mathbb{R}}^5⟶{\mathbb{C}}^6\cong {\mathbb{R}}^{12}$$

by

$$X(u,v,r,s,t)=\left({\displaystyle \frac{u}{\sqrt{2}}},{\displaystyle \frac{v\cos \theta }{\sqrt{2}}},{\displaystyle \frac{u}{\sqrt{2}}},{\displaystyle \frac{v\cos \theta }{\sqrt{2}}},v\sin \theta ,0,r,s,t,0,0,0\right),$$

where the coordinates in ${\mathbb{R}}^{12}$ are ordered as

$$(x_1,y_1,x_2,y_2,x_3,y_3,x_4,y_4,x_5,y_5,x_6,y_6).$$

Let $N=X\left({\mathbb{R}}^5\right)$ be the submanifold determined by this immersion. The tangent vectors are

$$E_1=X_u={\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕x_1}}+{\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕x_2}},$$

$$E_2=X_v={\displaystyle \frac{\cos \theta }{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕y_1}}+{\displaystyle \frac{\cos \theta }{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕y_2}}+\sin \theta {\displaystyle \frac{𝜕}{𝜕x_3}},$$

$$E_3=X_r={\displaystyle \frac{𝜕}{𝜕x_4}},E_4=X_s={\displaystyle \frac{𝜕}{𝜕y_4}},E_5=X_t={\displaystyle \frac{𝜕}{𝜕x_5}}.$$

These vectors form an orthonormal frame on N.

Define

$${\mathcal{H}}_{\theta }=\mathrm{span}\{E_1,E_2\},\mathcal{H}=\mathrm{span}\{E_3,E_4,E_5\}.$$

We verify that N is a proper PS-submanifold.

First,

$$J{E}_1={\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕y_1}}+{\displaystyle \frac{1}{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕y_2}},$$

and

$$J{E}_2=-{\displaystyle \frac{\cos \theta }{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕x_1}}-{\displaystyle \frac{\cos \theta }{\sqrt{2}}}{\displaystyle \frac{𝜕}{𝜕x_2}}+\sin \theta {\displaystyle \frac{𝜕}{𝜕y_3}}.$$

Hence, the tangential components satisfy

$$P{E}_1=\cos \theta E_2,P{E}_2=-\cos \theta E_1.$$

Therefore,

$$P^2{E}_1=-{\cos }^2\theta E_1,P^2{E}_2=-{\cos }^2\theta E_2,$$

so ${\mathcal{H}}_{\theta }$ is a slant distribution with slant angle θ.

Next,

$$J{E}_3=E_4,J{E}_4=-E_3,$$

so $\mathrm{span}\{E_3,E_4\}$ is holomorphic. On the other hand,

$$J{E}_5={\displaystyle \frac{𝜕}{𝜕y_5}}\perp TN,$$

so $\mathrm{span}\left\{E_5\right\}$ is totally real. Hence, $\mathcal{H}$ is neither holomorphic nor totally real.

Moreover, $\mathcal{H}$ is not a slant distribution, since the angle between $J{E}_3$ and $\mathcal{H}$ is 0, whereas the angle between $J{E}_5$ and $\mathcal{H}$ is $\pi /2$. Thus, the angle between $JX$ and $\mathcal{H}$ is not constant for nonzero $X\in \mathcal{H}$.

Also, by construction,

$$J{\mathcal{H}}_{\theta }\perp \mathcal{H},{\mathcal{H}}_{\theta }\perp J\mathcal{H},$$

since ${\mathcal{H}}_{\theta }$ involves only the first three complex coordinates, while $\mathcal{H}$ involves only the fourth and fifth complex coordinates. Therefore, N is a proper PS-submanifold.

Since the immersion X is affine linear, the second fundamental form vanishes identically:

$$h=0.$$

Thus, N is totally geodesic. Consequently,

$$H=0,\tau =0,K\left(\pi \right)=0\mathit{for}\mathit{every}\mathit{plane}\mathit{section}\pi \subset T_{p}N.$$

Hence,

$${\delta }_M\left(p\right)=\tau \left(p\right)-\underset{\pi \subset T_{p}N}{\inf }K\left(\pi \right)=0.$$

Since $c=0$ and $H=0$, the Chen-type inequality of Theorem 1 reduces to

$${\delta }_M\left(p\right)\le 0.$$

Because ${\delta }_M\left(p\right)=0$, equality holds identically on N.

Finally, the equality characterization in Theorem 1 is trivially satisfied, since all shape operators vanish:

$$A_{\xi }=0\mathit{for}\mathit{all}\xi \in T^{\perp }N.$$

Therefore, N is a proper PS-submanifold attaining equality in the Chen-type inequality.This confirms that the obtained Chen-type inequality is sharp.

To further illustrate the dependence of the upper bound on the slant angle and the ambiguity factor, we present a graphical representation in Figure 2.

Remark 1. 

The above example is flat and totally geodesic, which makes the verification particularly simple. However, one can obtain non-flat proper PS-submanifolds that also attain equality by perturbing the immersion in a curved complex space form (e.g., in ${\mathbb{CP}}^m$ with $c>0$) while preserving the algebraic conditions on the shape operators. For instance, taking a suitable warped product of a slant surface and a holomorphic curve, as outlined in the introduction, yields a non-trivial second fundamental form and non-zero mean curvature while still saturating the inequality. The explicit construction of such non-flat examples is more involved and will be addressed in a forthcoming paper. The flat example already suffices to demonstrate that the bound in Theorem 1 is sharp.

4. Applications and Special Cases of the Chen-Type Inequality

In this section, we derive several consequences of the Chen-type inequality obtained in Theorem 1. These results illustrate how the general estimate simplifies under additional geometric assumptions on the ambient space or on the distributions appearing in the definition of a PS-submanifold. In particular, we obtain simplified bounds when the ambient curvature is nonnegative, when the submanifold is minimal, and when the ambiguous distribution $\mathcal{H}$ satisfies special conditions such as being holomorphic or totally real. We also show how the inequality recovers known results in the classical slant case and yields restrictions for submanifolds in complex hyperbolic space.

Corollary 1. 

Let $N^n$ be a proper PS-submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$ with $c\ge 0$. Then, at every point $p\in N$, the δ-invariant satisfies

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta +6{\Lambda }_{\mathcal{H}}\left(p\right)\right].$$

Proof. 

Since $c\ge 0$, the term $-6\beta \left(p\right)$ appearing in (43) is non-positive and can be omitted, yielding the stated inequality. □

Corollary 2. 

Let $N^n$ be a proper PS-submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$ with $c\ge 0$. Then, at every point $p\in N$,

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta +3b\right],$$

where $b=n-a=\dim \mathcal{H}$.

Proof. 

For any orthonormal basis of ${\mathcal{H}}_p$, the Cauchy–Schwarz inequality gives ${\parallel {P|}_{{\mathcal{H}}_p}\parallel }_{\mathrm{HS}}^2=2{\Lambda }_{\mathcal{H}}\left(p\right)\le {\displaystyle {\sum }_{i=a+1}^n}{\parallel P{e}_i\parallel }^2\le {\displaystyle {\sum }_{i=a+1}^n}{\parallel e_i\parallel }^2=b$, so ${\Lambda }_{\mathcal{H}}\left(p\right)\le b/2$. Substituting into the previous corollary yields the result. □

Corollary 3. 

Let $N^n$ be a minimal proper PS-submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$. Then,

$${\delta }_M\left(p\right)\le {\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta +6{\Lambda }_{\mathcal{H}}\left(p\right)-6\beta \left(p\right)\right].$$

In particular, if $c<0$, then ${\delta }_M\left(p\right)<0$ for $n\ge 4$. In particular, if $c<0$, then ${\delta }_M\left(p\right)<0$ for $n\ge 4$. For the exceptional case $n=3$, the sign of ${\delta }_M\left(p\right)$ cannot be determined from the inequality alone.

Proof. 

Since N is minimal, $\parallel H\parallel =0$, and the inequality follows directly from Theorem 1.

To analyze the sign, observe that

$$(n-2)(n+1)\ge 10\mathrm{for}n\ge 4,$$

and moreover,

$$3a{\cos }^2\theta \ge 0,{\Lambda }_{\mathcal{H}}\left(p\right)\ge 0,\beta \left(p\right)\le 1.$$

Hence,

$$(n-2)(n+1)+3a{\cos }^2\theta +6{\Lambda }_{\mathcal{H}}\left(p\right)-6\beta \left(p\right)\ge (n-2)(n+1)-6.$$

For $n\ge 4$, the right-hand side is strictly positive. Therefore, when $c<0$, the entire expression becomes negative, implying

$${\delta }_M\left(p\right)<0.$$

□

Remark 2. 

For compact minimal submanifolds in complex hyperbolic space ($c<0$) with $n\ge 4$, the inequality ${\delta }_M\left(p\right)<0$ at every point forces the integral of the scalar curvature to be strictly less than the integral of the infimum sectional curvature. By the Gauss–Bonnet–Chern theorem (for even n) or index theory, this imposes topological restrictions. In particular, when $n=4$, the signature of the manifold is constrained.

Corollary 4. 

Let $N^n$ be a slant submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$ with slant angle θ. Then,

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3n{\cos }^2\theta -6\beta \left(p\right)\right].$$

In particular, if the plane realizing ${\inf }_{\pi \subset T_{p}N}K\left(\pi \right)$ satisfies $\beta \left(p\right)={\cos }^2\theta$, then

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3(n-2){\cos }^2\theta \right],$$

which coincides with Mihai’s inequality for slant submanifolds.

Proof. 

For a slant submanifold, the ambiguous distribution is trivial, that is,

$$\mathcal{H}=\left\{0\right\}.$$

Hence, in Theorem 1, we have

$$a=n\mathrm{and}{\Lambda }_{\mathcal{H}}\left(p\right)=0.$$

Substituting these values into the inequality of Theorem 1, we obtain

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3n{\cos }^2\theta -6\beta \left(p\right)\right].$$

If, in addition, the plane realizing ${\inf }_{\pi \subset T_{p}N}K\left(\pi \right)$ satisfies

$$\beta \left(p\right)={\cos }^2\theta ,$$

then

$$3n{\cos }^2\theta -6\beta \left(p\right)=3n{\cos }^2\theta -6{\cos }^2\theta =3(n-2){\cos }^2\theta .$$

Therefore, the above inequality reduces to

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3(n-2){\cos }^2\theta \right],$$

which is precisely Mihai’s inequality. □

Corollary 5. 

Let $N^n$ be a proper PS-submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$ with $\mathcal{H}$ as a holomorphic distribution. Then, the inequality (43) becomes

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta +3b-6\beta \left(p\right)\right].$$

Proof. 

For a holomorphic distribution, $P^2=-I$ on $\mathcal{H}$, so $\parallel P{e}_i\parallel =1$ for any unit vector $e_i\in \mathcal{H}$. Hence, ${\Lambda }_{\mathcal{H}}\left(p\right)={\displaystyle \frac{1}{2}}{\displaystyle {\sum }_{i=a+1}^n}{\parallel P{e}_i\parallel }^2={\displaystyle \frac{b}{2}}$. □

Corollary 6. 

Let $N^n$ be a proper PS-submanifold of a complex space form ${\tilde{M}}^m\left(c\right)$ with $\mathcal{H}$ as a totally real distribution. Then, the inequality (43) becomes

$${\delta }_M\left(p\right)\le {\displaystyle \frac{n^2(n-2)}{2(n-1)}}{\parallel H\parallel }^2+{\displaystyle \frac{c}{8}}\left[(n-2)(n+1)+3a{\cos }^2\theta -6\beta \left(p\right)\right].$$

Proof. 

If $\mathcal{H}$ is totally real, then $P{e}_i=0$ for all $e_i\in \mathcal{H}$, so ${\Lambda }_{\mathcal{H}}\left(p\right)=0$ by definition. □

We summarize the equality conditions for different submanifold classes as:

Submanifold Type	Shape Operator Structure	Mean Curvature Condition	Additional Constraints
Slant	${\displaystyle h_{ij}^{\alpha }}=0$ for $(i,j)\ne (1,2)$, ${\displaystyle h_{11}^{\alpha }}+{\displaystyle h_{22}^{\alpha }}={\displaystyle h_{33}^{\alpha }}=\cdots ={\displaystyle h_{nn}^{\alpha }}$	None	$\beta \left(p\right)={\cos }^2\theta$
Semi-slant	Same block-diagonal form	None	$\mathcal{H}$ holomorphic, ${\Lambda }_{\mathcal{H}}=b/2$
Hemi-slant	Same block-diagonal form	None	$\mathcal{H}$ totally real, ${\Lambda }_{\mathcal{H}}=0$
Bi-slant	Same block-diagonal form	None	$\mathcal{H}$ slant with angle $\varphi$
General PS	Same block-diagonal form	None	$\pi$ realizes $\inf K$; ${\Lambda }_{\mathcal{H}}$ arbitrary

5. Conclusions

In this paper, we have derived a Chen-type inequality for PS-submanifolds of complex space forms. By exploiting the Gauss equation, the curvature structure of complex space forms, and an optimization procedure applied to the components of the second fundamental form, we obtained an explicit upper bound for Chen’s $\delta$-invariant in terms of the squared mean curvature, the ambient holomorphic sectional curvature, and the geometry of the slant distribution.

A notable feature of the obtained inequality is the appearance of an additional term reflecting the contribution of the ambiguous distribution. This term distinguishes the PS-setting from the classical case of slant submanifolds and shows how the geometry of the complementary distribution influences the intrinsic curvature of the submanifold. The equality case was analyzed, and it was shown that equality occurs precisely when the shape operators take a particular block-diagonal structure satisfying specific trace relations. A concrete affine example in ${\mathbb{C}}^6\cong {\mathbb{R}}^{12}$ demonstrates that the bound is sharp.

Several consequences of the main inequality were also discussed. In particular, we obtained simplified bounds for minimal PS-submanifolds and dimension-dependent estimates in the case of nonnegative ambient curvature. These results illustrate how Chen’s $\delta$-invariant continues to serve as an effective tool for relating intrinsic and extrinsic geometry in the broader framework of partially slant submanifolds.

The results obtained here suggest several possible directions for future research. One natural problem is the classification of PS-submanifolds that satisfy the equality case of the derived inequality. Another interesting direction is the investigation of similar inequalities for PS-submanifolds in other ambient geometries, such as generalized complex space forms, Kähler product manifolds, or statistical and contact metric manifolds. The methodology relies heavily on the specific curvature tensor of complex space forms. For generalized complex space forms (with two independent curvature functions), the algebraic optimization becomes significantly more involved because the term $g{(J{e}_i,e_j)}^2$ does not factor uniformly. However, the same $\delta$-invariant approach could be extended using pointwise slant angle techniques. Finally, it would be worthwhile to study Chen-type inequalities involving other $\delta$-invariants in the context of partially slant geometry.