An Optimal Inequality for Warped Product Pointwise Semi-Slant Submanifolds in Complex Space Forms

Abstract

In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized $\delta$-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these inequalities attain equality, providing valuable insights into their geometric and structural significance. Additionally, we also present results involving harmonic and Hessian functions, revealing a broader connection between curvature properties and analytic functions.

1. Introduction

The study of product manifolds plays a crucial role in both physical and geometric contexts, particularly within Hermitian geometry. In physics, spacetime in Einstein’s general relativity is often modeled as the product of three-dimensional space and one-dimensional time, with each component possessing its own metric. Consequently, the topology of spacetime is fundamentally influenced by these metrics. Product manifolds also find applications in several advanced theories, including gauge theory, Kaluza–Klein theory, and brane theory.

In 1969, R. L. Bishop and co-authors [1] introduced a generalized class of Riemannian product manifolds to explore manifolds exhibiting negative sectional curvature, now known as warped product (WP) manifolds. These manifolds are formally defined as follows:

Consider two Riemannian manifolds, $N_1$ of dimension ${\mu }_1$ with metric $g_1$ and $N_2$ of dimension ${\mu }_2$ with metric $g_2$. Let $\zeta$ be a positive differentiable function on $N_1$, and let $N_1\times N_2$ represent their product manifold with projections ${\iota }_1:N_1\times N_2\to N_1$ and ${\iota }_2:N_1\times N_2\to N_2$. The WP manifold $M=N_1{\times }_{\zeta }{N}_2$ is equipped with the metric

$$g(U,V)=g_1({\iota }_{1\ast }U,{\iota }_{1\ast }V)+{(\zeta \circ {\iota }_1)}^2{g}_2({\iota }_{2\ast }U,{\iota }_{2\ast }V),$$

where U and V denote vector fields on M, and ∗ represents the tangent map [2,3].

Separately, Nash’s embedding theorem [4] investigates the conditions under which a Riemannian manifold can be smoothly immersed in a space form. However, intrinsic invariants often constrain the extrinsic properties of submanifolds, limiting the practical application of Nash’s results. To address these challenges, Chen introduced a unified framework that combines intrinsic and extrinsic invariants for submanifolds. In 1993, Chen [5] established a fundamental inequality for submanifolds in real space forms, relating intrinsic invariants such as sectional and scalar curvatures to extrinsic invariants like the squared mean curvature.

This breakthrough laid the foundation for the development of $\delta$-invariants (also referred to as Chen invariants), which have since found extensive use in Riemannian geometry and various related disciplines [6].

Casorati curvature (C-curvature), introduced as the square of the normalized length of the second fundamental form, emerged as an important extrinsic invariant for submanifolds in Riemannian geometry. This concept extends the idea of principal directions for hypersurfaces in a Riemannian manifold [7]. Recent investigations have established several inequalities involving normalized C-curvature and scalar curvature for diverse submanifolds in various ambient spaces [8,9,10,11,12,13,14,15,16].

In 2008, S. Decu, S. Haesen, and L. Verstraelen [17] introduced generalized normalized $\delta$-C-curvatures, denoted by ${\widehat{\delta }}_C(t;\mu -1)$ and ${\delta }_C(t;\mu -1)$. They provided sharp inequalities involving these extrinsic measures and scalar curvature for any real number t such that $0<t<\mu (\mu -1)$. These results have since been extended by numerous researchers [17,18,19,20,21,22,23,24,25,26,27].

In this paper, we derive inequalities for ${\widehat{\delta }}_C(t;\mu -1)$ and ${\delta }_C(t;\mu -1)$ using the constrained extremum method. This approach provides a natural framework for establishing geometric inequalities [28], specifically for WPPSS submanifolds in complex space forms.

2. Some Basics

Let $\overline{M}$ represent an almost Hermitian manifold, characterized by an almost complex structure $\mathsf{\Phi }$ and a Riemannian metric g. Such a manifold is termed a Kähler manifold if the Levi-Civita connection $\overline{\nabla }$ satisfies $\overline{\nabla }\mathsf{\Phi }=0$.

An almost Hermitian manifold $\overline{M}$ with constant holomorphic sectional curvature (CHSC) c is referred to as a complex space form (CSF), denoted by $\overline{M}\left(c\right)$, when its Riemannian curvature tensor $\overline{\Re }$ satisfies the following [23]:

$$\begin{array}{cc}\hfill \overline{\Re }(U,V)Z=& {\displaystyle \frac{c}{4}}\{g(V,Z)U-g(U,Z)V+g(U,\Phi Z)\mathsf{\Phi }V-g(V,\mathsf{\Phi }Z)\mathsf{\Phi }U\hfill \\ \hfill & +2g(U,\mathsf{\Phi }V)\mathsf{\Phi }Z\},\hfill \end{array}$$

(1)

for any vector fields U, V, Z defined on $\overline{M}\left(c\right)$.

Let M be a submanifold of dimension $\mu$ in a CSF $\overline{M}\left(c\right)$ of complex dimension p. The Levi-Civita connections on M and $\overline{M}\left(c\right)$ are denoted by ∇ and $\overline{\nabla }$, respectively. The Weingarten and Gauss formulas are expressed as follows:

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{U}V& ={\nabla }_{U}V+\sigma (U,V),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{U}N& =-A_{N}U+{\nabla }_U^{\perp }N,\hfill \end{array}$$

(3)

where $\sigma$ is the second fundamental form, ${\nabla }^{\perp }$ is the normal connection, and A denotes the shape operator of M. These quantities satisfy the following relation:

$$\begin{array}{c}\hfill g(\sigma (U,V),N)=g(A_{N}U,V),\end{array}$$

(4)

where g plays a dual role as the induced metric on M and the metric on $\overline{M}\left(c\right)$.

We use ℜ to denote the curvature tensor of M and $\overline{\Re }$ for that of $\overline{M}\left(c\right)$.

The Gauss equation relates these as

$$\begin{array}{c}\hfill \overline{\Re }(U,V,Z,W)\\ \hfill & =\Re (U,V,Z,W)+g\left(\sigma \right(U,Z),\sigma (V,W\left)\right)-g\left(\sigma \right(U,W),\sigma (V,Z\left)\right),\hfill \end{array}$$

(5)

for vector fields U, V, Z, W tangent to M.

Suppose M is a Riemannian manifold isometrically immersed into another Riemannian manifold $\overline{M}$. For any vector field X lying in the tangent bundle $TM$, the almost complex structure $\mathsf{\Phi }$ can be expressed as

$$\mathsf{\Phi }X=PX+QX,$$

and in this context, $PX$ corresponds to the tangent part and $QX$ to the normal part of $\mathsf{\Phi }X$ relative to M. When $P=0$, the submanifold is categorized as anti-invariant, and when $Q=0$, it is referred to as invariant.

Let M be a Riemannian manifold isometrically embedded in another Riemannian manifold $(\overline{M},\mathsf{\Phi },g)$. The submanifold M is called a semi-slant submanifold if the tangent bundle $TM$ can be split into two orthogonal distributions, $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^{\theta }$, such that $TM=\mathrm{\Lambda }\oplus {\mathrm{\Lambda }}^{\theta }$, $\mathsf{\Phi }\left(\mathrm{\Lambda }\right)\subseteq \mathrm{\Lambda }$, and the distribution ${\mathrm{\Lambda }}^{\theta }$ is slant, characterized by a slant angle $\theta$, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Expanding on this concept, a pointwise semi-slant submanifold can be defined as follows: Let M be a Riemannian manifold isometrically embedded in $(\overline{M},\mathsf{\Phi },g)$. The submanifold M is termed a pointwise semi-slant submanifold if two orthogonal distributions $\mathrm{\Lambda }$ and ${\mathrm{\Lambda }}^{\theta }$ exist such that $TM=\mathrm{\Lambda }\oplus {\mathrm{\Lambda }}^{\theta }$, $\mathsf{\Phi }\left(\mathrm{\Lambda }\right)\subseteq \mathrm{\Lambda }$, and the distribution ${\mathrm{\Lambda }}^{\theta }$ is pointwise slant, meaning the slant angle $\theta$ depends smoothly on the point in M.

At a point $x\in M$, let $\{e_1,\dots ,e_{\mu }\}$ be an orthonormal basis of the tangent space $T_{x}M$ and $\{e_{\mu +1},\dots ,e_{2p}\}$ an orthonormal basis of the normal space $T_x^{\perp }M$. The mean curvature vector H at x is defined as

$$\begin{array}{c}\hfill H\left(x\right)={\displaystyle \frac{1}{\mu }}{\mathrm{\Xi }}_{i=1}^{\mu }\sigma (e_i,e_i)\end{array}$$

(6)

and the relationship

$$\begin{array}{c}\hfill {\parallel \sigma \parallel }^2={\mathrm{\Xi }}_{i,j=1}^{\mu }g(\sigma (e_i,e_j),\sigma (e_i,e_j))\end{array}$$

(7)

describes the squared norm of the second fundamental form $\sigma$.

The scalar curvature $\tau$ of M at x is defined as

$$\begin{array}{c}\hfill \tau \left(x\right)={\mathrm{\Xi }}_{1\le i<j\le \mu }K(e_i\wedge e_j),\end{array}$$

(8)

and the normalized scalar curvature ℘ is given by

$$\begin{array}{c}\hfill \wp \left(x\right)={\displaystyle \frac{2\tau \left(x\right)}{\mu (\mu -1)}}.\end{array}$$

(9)

The squared mean curvature of M in $\overline{M}\left(4c\right)$ is

$$\begin{array}{c}\hfill {\parallel H\parallel }^2={\displaystyle \frac{1}{{\mu }^2}}{\mathrm{\Xi }}_{u=\mu +1}^{2p}{\left({\mathrm{\Xi }}_{i=1}^{\mu }{\sigma }_{ii}^u\right)}^2,\end{array}$$

(10)

and the C-curvature $\mathcal{C}$ of M is

$$\begin{array}{c}\hfill \mathcal{C}={\displaystyle \frac{1}{\mu }}{\mathrm{\Xi }}_{u=\mu +1}^{2p}{\mathrm{\Xi }}_{i,j=1}^{\mu }{\left({\sigma }_{ij}^u\right)}^2.\end{array}$$

(11)

For an s-dimensional subspace $\varpi \subset T_{x}M$, $s\ge 2$, with orthonormal basis $\{e_1,\dots ,e_s\}$, the scalar curvature $\tau \left(\varpi \right)$ is defined as

$$\begin{array}{c}\hfill \tau \left(\varpi \right)={\mathrm{\Xi }}_{1\le i<j\le s}K(e_i\wedge e_j),\end{array}$$

(12)

and the C-curvature of $\varpi$ is

$$\begin{array}{c}\hfill \mathcal{C}\left(\varpi \right)={\displaystyle \frac{1}{s}}{\mathrm{\Xi }}_{u=\mu +1}^{2p}{\mathrm{\Xi }}_{i,j=1}^s{\left({\sigma }_{ij}^u\right)}^2.\end{array}$$

(13)

The normalized $\delta$-C-curvatures ${\delta }_C(\mu -1)$ and ${\widehat{\delta }}_C(\mu -1)$ are given as follows [29]:

$$\begin{array}{cc}\hfill {\left[{\delta }_C(\mu -1)\right]}_x& ={\displaystyle \frac{1}{2}}{\mathcal{C}}_x+{\displaystyle \frac{\mu +1}{2\mu }}inf\{\mathcal{C}\left(\varpi \right)\mid \varpi :\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\left[{\widehat{\delta }}_C(\mu -1)\right]}_x& =2{\mathcal{C}}_x-{\displaystyle \frac{2\mu -1}{2\mu }}sup\{\mathcal{C}\left(\varpi \right)\mid \varpi :\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}.\hfill \end{array}$$

(15)

The generalized normalized(GN) $\delta$-C-curvatures ${\delta }_C(t;\mu -1)$ and ${\widehat{\delta }}_C(t;\mu -1)$ for $t>0$, $t\ne \mu (\mu -1)$, are defined as follows [29]:

$$\begin{array}{cc}\hfill {\left[{\delta }_C(t;\mu -1)\right]}_x& =t{\mathcal{C}}_x+{\displaystyle \frac{(\mu -1)(\mu +t)({\mu }^2-\mu -t)}{t\mu }}inf\{\mathcal{C}\left(\varpi \right)\mid \varpi :\mathrm{hyperplane}\mathrm{of}{T}_{x}M\},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\left[{\widehat{\delta }}_C(t;\mu -1)\right]}_x& =t{\mathcal{C}}_x-{\displaystyle \frac{(\mu -1)(\mu +t)(t-{\mu }^2+\mu )}{t\mu }}sup\{\mathcal{C}\left(\varpi \right)\mid \varpi :\mathrm{hyperplane}\mathrm{of}{T}_{x}M\}.\hfill \end{array}$$

(17)

These generalized curvatures reduce to ${\delta }_C(\mu -1)$ and ${\widehat{\delta }}_C(\mu -1)$ for specific values of t as follows [29]:

$$\begin{array}{cc}\hfill {\left[{\delta }_C\left({\displaystyle \frac{\mu (\mu -1)}{2}};\mu -1\right)\right]}_x& =\mu (\mu -1){\left[{\delta }_C(\mu -1)\right]}_x,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\left[{\widehat{\delta }}_C\left(2\mu (\mu -1);\mu -1\right)\right]}_x& =\mu (\mu -1){\left[{\widehat{\delta }}_C(\mu -1)\right]}_x.\hfill \end{array}$$

(19)

The equation of Gauss gives us

$$\begin{array}{c}\hfill K(e_i\wedge e_j)=\tilde{K}(e_i\wedge e_j)+{\mathrm{\Xi }}_{u=n+1}^{2p}\left({\sigma }_{ii}^u{\sigma }_{jj}^u-{\left({\sigma }_{ij}^u\right)}^2\right),\end{array}$$

(20)

where $K(e_i\wedge e_j)$ and $\tilde{K}(e_i\wedge e_j)$ denote the sectional curvatures of the submanifold $M^{\mu }$ and the ambient manifold ${\tilde{M}}^{2p}$, respectively.

Using (20), the scalar curvature $\tau$ for the submanifold decomposes as

$$\begin{array}{cc}\hfill \tau \left(N_1^{{\mu }_1}\right)& =\tilde{\tau }\left(N_1^{{\mu }_1}\right)+{\mathrm{\Xi }}_{u=\mu +1}^{2p}{\mathrm{\Xi }}_{1\le i<j\le {\mu }_1}\left({\sigma }_{ii}^u{\sigma }_{jj}^u-{\left({\sigma }_{ij}^u\right)}^2\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \tau \left(N_2^{{\mu }_2}\right)& =\tilde{\tau }\left(N_2^{{\mu }_2}\right)+{\mathrm{\Xi }}_{u=\mu +1}^{2p}{\mathrm{\Xi }}_{{\mu }_1+1\le s<t\le \mu }\left({\sigma }_{ii}^u{\sigma }_{jj}^u-{\left({\sigma }_{ij}^u\right)}^2\right).\hfill \end{array}$$

(22)

Finally, a relation established by B.-Y. Chen is given by

$$\begin{array}{c}\hfill {\mathrm{\Xi }}_{1\le i\le {\mu }_1}{\mathrm{\Xi }}_{{\mu }_1+1\le j\le \mu }K(e_i\wedge e_j)={\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}={\mu }_2\left(\mathrm{\Delta }(ln\zeta )-{\parallel \nabla \zeta \parallel }^2\right),\end{array}$$

(23)

where $\mathrm{\Delta }$ denotes the Laplacian operator.

3. Optimal Inequalities

Consider a Riemannian submanifold M of a Riemannian manifold $(\overline{M},g)$ with a differentiable function $f:\overline{M}\to \mathbb{R}$. Taking the limited extremum problem into consideration

$$\begin{array}{c}\hfill \underset{x\in M}{min}f\left(x\right),\end{array}$$

(24)

we obtain the following outcome.

Lemma 1

([28]). If the solution to the problem (24) is $x_0\in M$, then $\left(\mathrm{grad}f\right)\left(x_0\right)\in T_{x_0}^{\perp }M$ and the bilinear form $\mathsf{\Pi }:T_{x_0}M\times T_{x_0}M\to \mathbb{R}$ defined by

$$\mathrm{\Pi }(U,V)={\mathrm{Hess}}_f(U,V)+g(\sigma (U,V),\left(\mathrm{grad}f\right)\left(x_0\right))$$

are positive semi-definite, where $\mathrm{grad}f$ is the gradient of f.

Theorem 1.

Let ${\overline{M}}^{2p}\left(c\right)$ be the CSF and $\phi :M^{\mu }=N_1^{{\mu }_1}{\times }_{\zeta }{N}_2^{{\mu }_2}\to {\overline{M}}^{2p}\left(c\right)$ be an isometric immersion of WPPSS submanifold into ${\tilde{M}}^{2p}\left(c\right)$. Then, we have the following:

(i) 

The GN δ-C-curvature ${\delta }_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{{\delta }_C(t;\mu -1)}{\mu (\mu -1)}}-{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3}{4}}c\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(25)

in the scenario where $0<t<\mu (\mu -1)$, $t\in \mathbb{R}$.

(ii) 

The GN δ-C-curvature ${\widehat{\delta }}_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{\widehat{{\delta }_C}(t;\mu -1)}{\mu (\mu -1)}}-{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3}{4}}c\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(26)

in the scenario where $t>\mu (\mu -1)$, $t\in \mathbb{R}$.

Additionally, if and only if the shape operator for the appropriate tangent orthonormal frame and normal orthonormal frame has the following form, the equality is maintained in (25) and (26):

$$\begin{array}{c}\hfill A_{e_{\mu +1}}=\left(\begin{array}{ccccccccc}{\sigma }_{11}^{\mu +1}& 0& \dots & 0& 0& 0& \dots & 0& 0\\ 0& {\sigma }_{22}^{\mu +1}& \dots & 0& 0& 0& \dots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & {\sigma }_{{\mu }_1{\mu }_1}^{\mu +1}& 0& 0& \dots & 0& 0\\ 0& 0& \dots & 0& {\sigma }_{{\mu }_1+1{\mu }_1+1}^{\mu +1}& 0& \dots & 0& 0\\ 0& 0& \dots & 0& 0& {\sigma }_{{\mu }_1+2{\mu }_1+2}^{\mu +1}& \dots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & 0& 0& 0& ⋮& {\sigma }_{\mu -1\mu -1}^{\mu +1}& 0\\ 0& 0& \dots & 0& 0& 0& ⋮& 0& {\sigma }_{\mu \mu }^{\mu +1}\end{array}\right),\end{array}$$

(27)

$$\begin{array}{cc}\hfill & A_{e_{\mu +2}}=\dots =A_{e_{2p}}=0,\hfill \\ \hfill & {\sigma }_{11}^{\mu +1}=\dots ={\sigma }_{{\mu }_1{\mu }_1}^{\mu +1}={\mu }_1{t}^2{f}_1{e}_{\mu +1},\hfill \\ \hfill & {\sigma }_{{\mu }_1+1{\mu }_1+1}^{\mu +1}=\dots ={\sigma }_{\mu -1\mu -1}^{\mu +1}=({\mu }^2-\mu +{\mu }_{2}t-t)t{f}_1{e}_{\mu +1},\hfill \\ \hfill & {\sigma }_{\mu \mu }=\mu (\mu -1)({\mu }^2-\mu +{\mu }_{2}t-t)f_1{e}_{\mu +1},\hfill \\ \hfill & {\sigma }_{ij}=0,i\ne j,\hfill \end{array}$$

where $f_1$ is a function on $M^{\mu }$.

Proof. 

From the Gauss equation, Equations (1) and (11)–(13), we obtain

$$\begin{array}{cc}\hfill \tau \left(x\right)& ={\mathrm{\Xi }}_{l=1}^{{\mu }_1}{\mathrm{\Xi }}_{r={\mu }_1+1}^{\mu }K(e_l\wedge e_r)+{\mathrm{\Xi }}_{1\le i<j\le {\mu }_1}K(e_i\wedge e_j)+{\mathrm{\Xi }}_{{\mu }_1+1\le s<t\le \mu }K(e_s\wedge e_t)\hfill \\ \hfill & ={\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{{\mu }_1({\mu }_1-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_1+{\displaystyle \frac{{\mu }_2({\mu }_2-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_2{cos}^2\theta \hfill \\ \hfill & +{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{1\le i<j\le {\mu }_1}[{\sigma }_{ii}^{\alpha }{\sigma }_{jj}^{\alpha }-{\left({\sigma }_{ij}^{\alpha }\right)}^2]+{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{{\mu }_1+1\le s<t\le \mu }[{\sigma }_{ss}^{\alpha }{\sigma }_{tt}^{\alpha }-{\left({\sigma }_{st}^{\alpha }\right)}^2].\hfill \end{array}$$

(28)

Moving forward, we define the second-degree polynomial denoted by $\mathbb{G}$ in the components of the second fundamental form as

$$\begin{array}{cc}\hfill \mathbb{G}& =t\mathcal{C}+{\displaystyle \frac{(\mu -1)(\mu +t)({\mu }^2-\mu -t)}{\mu t}}\mathcal{C}\left(\varpi \right)-2\tau \hfill \\ \hfill & +2\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{{\mu }_1({\mu }_1-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_1+{\displaystyle \frac{{\mu }_2({\mu }_2-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_2{cos}^2\theta \right\},\hfill \end{array}$$

(29)

where $\varpi$ is a hyperplane of $T_{x}M$. We can assume that $\varpi$ is spanned by $e_1,\dots ,e_{\mu -1}$ without losing generality. Consequently, (29) indicates that

$$\begin{array}{cc}\hfill \mathbb{G}& ={\displaystyle \frac{t}{\mu }}{\mathrm{\Xi }}_{\alpha }{\mathrm{\Xi }}_{i,j=1}^{\mu }{\left({\sigma }_{ij}^{\alpha }\right)}^2+{\displaystyle \frac{(\mu +t)({\mu }^2-\mu -t)}{\mu t}}{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{i,j=1}^{\mu -1}{\left({\sigma }_{ij}^{\alpha }\right)}^2\hfill \\ \hfill & -2{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{1\le i<j\le {\mu }_1}[{\sigma }_{ii}^{\alpha }{\sigma }_{jj}^{\alpha }-{\left({\sigma }_{ij}^{\alpha }\right)}^2]-2{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{{\mu }_1+1\le s<t\le {\mu }_1}[{\sigma }_{ss}^{\alpha }{\sigma }_{tt}^{\alpha }-{\left({\sigma }_{st}^{\alpha }\right)}^2]\hfill \\ \hfill & \ge {\displaystyle \frac{{\mu }^2+\mu (t-1)-2t}{t}}{\mathrm{\Xi }}_{\alpha }{\mathrm{\Xi }}_{i=1}^{\mu -1}{\left({\sigma }_{ii}^{\alpha }\right)}^2+{\displaystyle \frac{t}{\mu }}{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\left({\sigma }_{\mu \mu }^{\alpha }\right)}^2\hfill \\ \hfill & -2{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{1\le i<j\le {\mu }_1}{\sigma }_{ii}^{\alpha }{\sigma }_{jj}^{\alpha }-2{\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{\mathrm{\Xi }}_{{\mu }_1+1\le s<t\le \mu }{\sigma }_{ss}^{\alpha }{\sigma }_{tt}^{\alpha }\hfill \end{array}$$

(30)

Next, we turn our attention to the quadratic form

$$\begin{array}{c}\hfill f_{\alpha }:{\Re }^{\mu }\to \Re ,\alpha =\mu +1,\mu +2,\dots ,2p,\end{array}$$

defined by

$$\begin{array}{ccc}\hfill f_{\alpha }({\sigma }_{11}^{\alpha },{\sigma }_{22}^{\alpha },\dots ,{\sigma }_{\mu \mu }^{\alpha })& =& {\displaystyle \frac{{\mu }^2+\mu (t-1)-2t}{t}}{\mathrm{\Xi }}_{i=1}^{\mu -1}{\left({\sigma }_{ii}^{\alpha }\right)}^2+{\displaystyle \frac{t}{\mu }}{\left({\sigma }_{\mu \mu }^{\alpha }\right)}^2\hfill \\ & & -2{\mathrm{\Xi }}_{1\le i<j\le {\mu }_1}{\sigma }_{ii}^{\alpha }{\sigma }_{jj}^{\alpha }-2{\mathrm{\Xi }}_{{\mu }_1+1\le s<t\le \mu }{\sigma }_{ss}^{\alpha }{\sigma }_{tt}^{\alpha }.\hfill \end{array}$$

(31)

Then, from (30) and (31), we deduce that

$$\mathbb{G}\ge {\mathrm{\Xi }}_{\alpha =\mu +1}^{2p}{f}_{\alpha }.$$

(32)

We now turn our focus to the topic of extrema

$$\begin{array}{c}\hfill min{f}_{\alpha },\mathrm{subject}\mathrm{to}\mathsf{\Gamma }:{\sigma }_{11}^{\alpha }+{\sigma }_{22}^{\alpha }+\dots +{\sigma }_{\mu \mu }^{\alpha }=r^{\alpha },\end{array}$$

where $r^{\alpha }$ is a real constant.

We calculate the partial derivatives of $f_{\alpha }$ as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial f_{\alpha }}{\partial {\sigma }_{11}^{\alpha }}=\frac{2[{\mu }^2+\mu (t-1)-t]}{t}{\sigma }_{11}^{\alpha }-2{\mathrm{\Xi }}_{i=1}^{{\mu }_1}{\sigma }_{ii}^{\alpha }}\hfill \\ ⋮\hfill \\ {\displaystyle \frac{\partial f_{\alpha }}{\partial {\sigma }_{{\mu }_1{\mu }_1}^{\alpha }}=\frac{2[{\mu }^2+\mu (t-1)-t]}{t}{\sigma }_{{\mu }_1{\mu }_1}^{\alpha }-2{\mathrm{\Xi }}_{i=1}^{{\mu }_1}{\sigma }_{ii}^{\alpha }}\hfill \\ {\displaystyle \frac{\partial f_{\alpha }}{\partial {\sigma }_{{\mu }_1+1{\mu }_1+1}^{\alpha }}=\frac{2[{\mu }^2+\mu (t-1)-t]}{t}{\sigma }_{{\mu }_1+1{\mu }_1+1}^{\alpha }-2{\mathrm{\Xi }}_{s={\mu }_1+1}^{\mu }{\sigma }_{ss}^{\alpha }}\hfill \\ ⋮\hfill \\ {\displaystyle \frac{\partial f_{\alpha }}{\partial {\sigma }_{\mu -1\mu -1}^{\alpha }}=\frac{2[{\mu }^2+\mu (t-1)-t]}{t}{\sigma }_{\mu -1\mu -1}^{\alpha }-2{\mathrm{\Xi }}_{s={\mu }_1+1}{\sigma }_{ss}^{\alpha }}\hfill \\ {\displaystyle \frac{\partial f_{\alpha }}{\partial {\sigma }_{\mu \mu }^{\alpha }}=\frac{2(\mu +t)}{\mu }{\sigma }_{\mu \mu }^{\alpha }-2{\mathrm{\Xi }}_{s={\mu }_1+1}{\sigma }_{ss}^{\alpha }.}\hfill \end{array}\right.$$

(33)

Lemma 1 states that the vector $\mathrm{grad}{f}_{1^{★}}$ is normal to $\mathsf{\Gamma }$ given an optimal solution $({\sigma }_{11}^{1^{★}},{\sigma }_{22}^{1^{★}},\dots ,{\sigma }_{\mu \mu }^{1^{★}})$ of the aforementioned minimization problem. It is collinear with the vector $(1,1,\dots ,1)$, in other words.

We discover that the structure of the critical point is outlined below based on (33) and Lemma 1.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\sigma }_{11}^{\alpha }=\cdots ={\sigma }_{{\mu }_1{\mu }_1}^{\alpha }={\displaystyle \frac{{\mu }_1{t}^2}{{({\mu }^2-\mu -t-{\mu }_{2}t)}^2+{\mu }_1^2{t}^2}}{r}^{\alpha }\hfill \\ {\sigma }_{{\mu }_1+1{\mu }_1+1}^{\alpha }=\cdots ={\sigma }_{\mu -1\mu -1}^{\alpha }={\displaystyle \frac{({\mu }^2-\mu +{\mu }_{2}t-t)t}{{({\mu }^2-\mu -t-{\mu }_{2}t)}^2+{\mu }_1^2{t}^2}}{r}^{\alpha }\hfill \\ {\sigma }_{\mu \mu }^{\alpha }={\displaystyle \frac{\mu (\mu -1)({\mu }^2-\mu +{\mu }_{2}t-t)}{{({\mu }^2-\mu -t-{\mu }_{2}t)}^2+{\mu }_1^2{t}^2}}{r}^{\alpha }.\hfill \end{array}\right.\end{array}$$

(34)

A random point $x\in \mathrm{\Gamma }$ is now fixed. The appropriate bilinear form $\mathrm{\Pi }:T_x\mathrm{\Gamma }\times T_x\mathrm{\Gamma }\to \Re$ is therefore obtained in accordance with Lemma 1.

$$\begin{array}{c}\hfill \mathrm{\Pi }(U,V)={\mathrm{Hess}}_{f_{\alpha }}(U,V)+\langle {\sigma }^{\prime }(U,V),\left(\mathrm{grad}{f}_{\alpha }\right)\left(x\right)\rangle ,\end{array}$$

In this context, $\langle ·,·\rangle$ stands for the standard inner product on ${\Re }^n$, while ${\sigma }^{\prime }$ refers to the second fundamental form of $\mathrm{\Gamma }$ within ${\Re }^n$.

Based on (33), we conclude

$$\left\{\begin{array}{c}{\displaystyle \frac{{\partial }^2{f}_{\alpha }}{\partial {\left({\sigma }_{ii}^{\alpha }\right)}^2}}={\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}\hfill \\ {\displaystyle \frac{{\partial }^2{f}_{\alpha }}{\partial {\left({\sigma }_{ss}^{\alpha }\right)}^2}}={\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}\hfill \\ {\displaystyle \frac{{\partial }^2{f}_{\alpha }}{\partial {\left({\sigma }_{\mu \mu }^{\alpha }\right)}^2}}={\displaystyle \frac{2t}{\mu }}\hfill \\ {\displaystyle \frac{{\partial }^2{f}_{\alpha }}{\partial {\sigma }_{ii}^{\alpha }\partial {\sigma }_{jj}^{\alpha }}}=-2\hfill \\ {\displaystyle \frac{{\partial }^2{f}_{\alpha }}{\partial {\sigma }_{ss}^{\alpha }\partial {\sigma }_{tt}^{\alpha }}}=-2\hfill \\ {\displaystyle \frac{{\partial }^2{f}_{\alpha }}{\partial {\sigma }_{ii}^{\alpha }\partial {\sigma }_{tt}^{\alpha }}}=0.\hfill \end{array}\right.$$

(35)

Thus, the Hessian matrix of $f_{\alpha }$ is

$$\begin{array}{c}\hfill {\mathrm{Hess}}_{f_{\alpha }}=2\left(\begin{array}{ccccccc}{\displaystyle \frac{{\mu }^2+\mu (t-1)-2t}{t}}& \dots & -1& 0& 0& \dots & 0\\ ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& ⋮\\ -1& \dots & {\displaystyle \frac{{\mu }^2+\mu (t-1)-2t}{t}}& 0& \dots & 0& 0\\ 0& \dots & 0& {\displaystyle \frac{{\mu }^2+\mu (t-1)-2t}{t}}& \dots & -1& -1\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& \dots & 0& -1& \dots & {\displaystyle \frac{{\mu }^2+\mu (t-1)-2t}{t}}& -1\\ 0& \dots & 0& -1& \dots & -1& {\displaystyle \frac{t}{\mu }}\end{array}\right).\end{array}$$

Given that $\mathrm{\Gamma }$ is totally geodesic in ${\Re }^{\mu }$, let $U=(U_1,\dots ,U_{\mu })$ be a tangent vector to $\mathrm{\Gamma }$ at an arbitrary point $x\in \mathrm{\Gamma }$. This implies that U satisfies the condition ${\displaystyle {\mathrm{\Xi }}_{i=1}^{\mu }{U}_i=0}$. Then,

(i)

For $\mu =1$, we have the following possibilities:

• (a) ${\mathrm{Hess}}_{f_{\alpha }}=\left(2t\right)$ (b) ${\mathrm{Hess}}_{f_{\alpha }}=(-2)$.
• Since $U_1=0$, in these two cases, ${\mathrm{Hess}}_{f_{\alpha }}(U,U)=0.$

(ii)

For $\mu =2$, we have the following possibilities:

• (a) ${\mathrm{Hess}}_{f_{\alpha }}=\left(\begin{array}{cc}{\displaystyle \frac{4}{t}}& -2\\ -2& {\displaystyle \frac{4}{t}}\end{array}\right)$; (b) ${\mathrm{Hess}}_{f_{\alpha }}=\left(\begin{array}{cc}{\displaystyle \frac{4}{t}}& 0\\ 0& t\end{array}\right)$;
• (c) ${\mathrm{Hess}}_{f_{\alpha }}=\left(\begin{array}{cc}{\displaystyle \frac{4}{t}}& -2-2t\end{array}\right)$.
• Clearly, for (a) and (c) we have ${\mathrm{Hess}}_{f_{\alpha }}\ge 0$ and for (b) ${\mathrm{Hess}}_{f_{\alpha }}>0$.

(iii)

For $\mu \ge 3$:

• (a) When $n=m$, we have

  $$\begin{array}{cc}\hfill {\mathrm{Hess}}_{f_{\alpha }}& ={\displaystyle \frac{2(\mu +t)(\mu -1)}{t}}\left({\mathrm{\Xi }}_{i=1}^{\mu }{U}_i^2\right)-2{\left({\mathrm{\Xi }}_{i=1}^{\mu }{U}_i\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{2(\mu +t)(\mu -1)}{t}}\left({\mathrm{\Xi }}_{i=1}^{\mu }{U}_i^2\right)\hfill \\ \hfill & \ge 0.\hfill \end{array}$$
• (b) When $n>m$, let $A=\left(\begin{array}{cccc}{\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}& -2& \dots & -2\\ -2& {\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}& \dots & -2\\ ⋮& ⋮& \ddots & ⋮\\ -2& -2& \dots & {\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}\end{array}\right)$ and $B=\left(\begin{array}{ccccc}{\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}& -2& \dots & -2& -2\\ -2& {\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}& \dots & -2& -2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -2& -2& \dots & {\displaystyle \frac{2[{\mu }^2+\mu (t-1)-2t]}{t}}& -2\\ -2& -2& \dots & -2& {\displaystyle \frac{2t}{\mu }}\end{array}\right)$

We notice that $|\lambda E-A|=0$ implies all the eigenvalues of A are greater than 0, i.e., A is positive definite.

Since, $\mu >{\mu }_1$, when $\mu -{\mu }_1={\mu }_2=1$, we obtain

$$B=\left({\displaystyle \frac{2t}{\mu }}\right)$$

That is, B is positive definite.

When $\mu -{\mu }_1={\mu }_2\ge 21$, we find

$$\begin{array}{cc}\hfill 0& =|\lambda E-B|\hfill \\ \hfill & ={\left[\lambda -{\displaystyle \frac{2(\mu +t)(\mu -1)}{t}}\right]}^{{\mu }_2-2}\left[{\lambda }^2-\left({\displaystyle \frac{2t}{\mu }}+{\displaystyle \frac{2(\mu +t)(\mu -1)-2({\mu }_2-1)t}{t}}\right)\lambda +{\displaystyle \frac{4(\mu +t)}{t}}\times {\displaystyle \frac{{\mu }_{1}t}{\mu }}\right]\hfill \end{array}$$

Since

$$\begin{array}{c}\hfill {\lambda }^2-\left({\displaystyle \frac{2t}{\mu }}+{\displaystyle \frac{2(\mu +t)(\mu -1)-2({\mu }_2-1)t}{t}}\right)\lambda +{\displaystyle \frac{4(\mu +t)}{t}}\times {\displaystyle \frac{{\mu }_{1}t}{\mu }}=0.\end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill & {\lambda }_{\mu -1}·{\lambda }_{\mu }={\displaystyle \frac{4(\mu +t)}{\mu }}\times {\displaystyle \frac{{\mu }_{1}t}{\mu }}\ge 0,\hfill \\ \hfill & {\lambda }_{\mu -1}+{\lambda }_{\mu }={\displaystyle \frac{2t}{\mu }}+{\displaystyle \frac{2(\mu +t)(\mu -1)+2{\mu }_{1}t}{t}}>0\hfill \end{array}$$

That is, either ${\lambda }_{{\mu }_1}>0,{\lambda }_{\mu }\ge 0$ or ${\lambda }_{{\mu }_1}\ge 0,{\lambda }_{\mu }>0$.

Hence, all the eigenvalues of B are $\ge 0$, i.e., B is positive definite. Thus, it is clear that ${\mathrm{Hess}}_{f_{\alpha }}(U,U)\ge 0$. This implies $\mathrm{\Pi }(U,U)\ge 0$.

Hence, from (34), the point $({\sigma }_{11}^{\alpha },{\sigma }_{22}^{\alpha },\dots ,{\sigma }_{\mu \mu }^{\alpha })$ is a global minimum point. Therefore, using (34) in $f_{\alpha }$, a lengthy but straightforward computation yields

$$\begin{array}{c}\hfill f_{\alpha }\ge {\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)}{{({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\left(r^{\alpha }\right)}^2.\end{array}$$

(36)

As $0<t<\mu (\mu -1)$ or $t>\mu (\mu -1)$, we now split the theorem’s proof into two major situations.

Case (i): $0<t<\mu (\mu -1)$. In light of this, using (32) in (36), we find

$$\begin{array}{cc}\hfill \mathbb{G}& \ge {\mathrm{\Xi }}_{\alpha }{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)}{{({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\left(r^{\alpha }\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)}{{({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\mathrm{\Xi }}_{\alpha }{\left(r^{\alpha }\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t){\mu }^2}{{({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2.\hfill \end{array}$$

(37)

Combining (29) and (37), we derive

$$\begin{array}{cc}\hfill t\mathcal{C}& +{\displaystyle \frac{(\mu -1)(\mu +t)({\mu }^2-\mu -t)}{\mu t}}\mathcal{C}\left(\varpi \right)-2\tau \hfill \\ \hfill & +2\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{{\mu }_1({\mu }_1-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_1+{\displaystyle \frac{{\mu }_2({\mu }_2-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_2{cos}^2\theta \right\}\hfill \\ \hfill & \ge {\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t){\mu }^2}{{({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2.\hfill \end{array}$$

(38)

Calculating the infimum of all tangent hyperplanes $\varpi$ of $T_{x}M$ in (38), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\delta }_C(t;\mu -1)}{\mu (\mu -1)}}& \ge \wp +{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{{\mu }_1({\mu }_1-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_1+{\displaystyle \frac{{\mu }_2({\mu }_2-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_2{cos}^2\theta \right\},\hfill \end{array}$$

(39)

and (39) instantly leads to inequality (25).

Case (ii): $t>\mu (\mu -1)$. Then, with the same method the following inequality is evidently true:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\widehat{\delta }}_C(t;\mu -1)}{\mu (\mu -1)}}& \ge \wp +{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{{\mu }_1({\mu }_1-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_1+{\displaystyle \frac{{\mu }_2({\mu }_2-1)c}{8}}+{\displaystyle \frac{3c}{4}}{n}_2{cos}^2\theta \right\}.\hfill \end{array}$$

(40)

□

Equalities hold in (39) and (40) at a point if and only if inequalities (30) and (36) become equalities. Thus, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\sigma }_{11}^{\alpha }=\cdots ={\sigma }_{{\mu }_1{\mu }_1}^{\alpha }={\displaystyle \frac{{\mu }_1{t}^2}{{({\mu }^2-\mu -t-{\mu }_{2}t)}^2+{\mu }_1^2{t}^2}}{r}^{\alpha }\hfill \\ {\sigma }_{{\mu }_1+1{\mu }_1+1}^{\alpha }=\cdots ={\sigma }_{\mu -1\mu -1}^{\alpha }={\displaystyle \frac{({\mu }^2-\mu +{\mu }_{2}t-t)t}{{({\mu }^2-\mu -t-{\mu }_{2}t)}^2+{\mu }_1^2{t}^2}}{r}^{\alpha }\hfill \\ {\sigma }_{\mu \mu }^{\alpha }={\displaystyle \frac{\mu (\mu -1)({\mu }^2-\mu +{\mu }_{2}t-t)}{{({\mu }^2-\mu -t-{\mu }_{2}t)}^2+{\mu }_1^2{t}^2}}{r}^{\alpha }\hfill \\ {\sigma }_{ij}^{\alpha }=0,i\ne =0.\hfill \end{array}\right.\end{array}$$

(41)

By selecting an orthonormal basis that aligns $e_{\mu +1}$ with the mean curvature vector, we obtain

$$\begin{array}{cc}\hfill & \sigma (e_1,e_1)=\cdots =\sigma (e_{{\mu }_1},e_{{\mu }_1})={\mu }_1{t}^2{f}_1{e}_{\mu +1}\hfill \\ \hfill & \sigma (e_{{\mu }_1+1},e_{{\mu }_1+1})=\cdots =\sigma (e_{\mu -1},e_{\mu -1})=({\mu }^2-\mu +{\mu }_{2}t-t)t{f}_1{e}_{\mu +1}\hfill \\ \hfill & \sigma (e_{\mu },e_{\mu })=\mu (\mu -1)({\mu }^2-\mu +{\mu }_{2}t-t)f_1{e}_{\mu +1}\hfill \\ \hfill & \sigma (e_i,e_j)=0,i\ne j,\hfill \end{array}$$

where $f_1={\displaystyle \frac{r^{\mu +1}}{{({\mu }^2-\mu +{\mu }_{2}t-t)}^2+{\mu }_1^2{t}^2}}$ is a function of $M^{\mu }$.

4. Some Consequences

Let $\varphi$ be a $C^{\infty }$-differentiable function that is positive. Consequently, the symmetric 2-covariant tensor field on $M^{\mu }$ that is the Hessian tensor of function $\varphi$ is defined by

$$\begin{array}{c}\hfill {\mathbb{H}}^{\varphi }:\mathfrak{U}\left(M\right)\times \mathfrak{U}\left(M\right)\to \mathcal{F}\left(M\right)\end{array}$$

(42)

such that

$$\begin{array}{c}\hfill {\mathbb{H}}^{\varphi }(U,V)={\mathbb{H}}_{ij}^{\varphi }{U}^i{V}^j,\end{array}$$

(43)

for any $U,V\in \mathfrak{U}\left(M\right)$, where ${\mathbb{H}}_{ij}^{\varphi }$ can be expressed as

$$\begin{array}{c}\hfill {\mathbb{H}}_{ij}^{\varphi }={\displaystyle \frac{{\partial }^2\varphi }{\partial x_i\partial x_j}}-{\mathrm{\Gamma }}_{ij}^{\mu }{\displaystyle \frac{\partial \varphi }{\partial x_{\mu }}}.\end{array}$$

(44)

Let us assume that $\varphi =ln\zeta$.

We next draw the following conclusion as a result of the aforesaid relation and Theorem 1.

Corollary 1.

Let ${\overline{M}}^{2p}\left(c\right)$ be the CSF and $\phi :M^{\mu }=N_1^{{\mu }_1}{\times }_{\zeta }{N}_2^{{\mu }_2}\to {\overline{M}}^{2p}\left(c\right)$ be an isometric immersion of the WPPSS submanifold into ${\tilde{M}}^{2p}\left(c\right)$. Then, we have the following:

(i) 

The GN δ-C-curvature ${\delta }_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{{\delta }_C(t;\mu -1)}{\mu (\mu -1)}}-{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{trace{\mathbb{H}}^{\varphi }}{\zeta }}+{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3c}{4}}\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(45)

in the scenario where $0<t<\mu (\mu -1)$, $t\in \mathbb{R}$.

(ii) 

The GN δ-C-curvature ${\widehat{\delta }}_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{\widehat{{\delta }_C}(t;\mu -1)}{\mu (\mu -1)}}-{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{trace{\mathbb{H}}^{\varphi }}{\zeta }}+{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3c}{4}}\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(46)

in the scenario where $t>\mu (\mu -1)$, $t\in \mathbb{R}$.

Further, if the warping function $\zeta$ is a harmonic function, then we have $\mathrm{\Delta }\zeta =0$. Hence, we obtain the following.

Corollary 2.

Let ${\overline{M}}^{2p}\left(c\right)$ be the CSF, $\phi :M^{\mu }=N_1^{{\mu }_1}{\times }_{\zeta }{N}_2^{{\mu }_2}\to {\overline{M}}^{2p}\left(c\right)$ be an isometric immersion of the WPPSS submanifold into ${\tilde{M}}^{2p}\left(c\right)$, and ζ be a harmonic function. Then,

(i) 

The GN δ-C-curvature ${\delta }_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{{\delta }_C(t;\mu -1)}{\mu (\mu -1)}}-{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3c}{4}}\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(47)

in the scenario where $0<t<\mu (\mu -1)$, $t\in \mathbb{R}$.

(ii) 

The GN δ-C-curvature ${\widehat{\delta }}_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{\widehat{{\delta }_C}(t;\mu -1)}{\mu (\mu -1)}}-{\displaystyle \frac{{\mu }_{1}t({\mu }^2-\mu +{\mu }_{2}t-t)\mu }{(\mu -1){({\mu }^2-\mu -t+{\mu }_{2}t)}^2{\mu }_1^2{t}^2}}{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3c}{4}}\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(48)

in the scenario where $t>\mu (\mu -1)$, $t\in \mathbb{R}$.

Finally, it is known that if a submanifold is minimal, then the mean curvature vector $H=0$. Thus, we conclude the following result.

Corollary 3.

Let M be a μ-dimensional WPPSS minimal immersion of a CSF ${\overline{M}}^{2p}\left(c\right)$. Then, we have the following:

(i) 

The GN δ-C-curvature ${\delta }_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{{\delta }_C(t;\mu -1)}{\mu (\mu -1)}}\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3c}{4}}\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(49)

in the scenario where $0<t<\mu (\mu -1)$, $t\in \mathbb{R}$.

(ii) 

The GN δ-C-curvature ${\widehat{\delta }}_C(t;\mu -1)$ satisfies

$$\begin{array}{cc}\hfill \wp & \le {\displaystyle \frac{\widehat{{\delta }_C}(t;\mu -1)}{\mu (\mu -1)}}\hfill \\ \hfill & +{\displaystyle \frac{2}{\mu (\mu -1)}}\left\{{\mu }_2{\displaystyle \frac{\mathrm{\Delta }\zeta }{\zeta }}+{\displaystyle \frac{\mu (\mu -1)-2{\mu }_1{\mu }_2}{8}}c+{\displaystyle \frac{3c}{4}}\left(n_1+n_2{cos}^2\theta \right)\right\}\hfill \end{array}$$

(50)

in the scenario where $t>\mu (\mu -1)$, $t\in \mathbb{R}$.

5. Examples

In this section, we present two non-trivial examples of WPPSS in CSF in light of [30] to illustrate and support our article.

Example 1.

Consider the immersion $\phi :M^3\to {\mathbb{R}}^6$ given by

$$\phi (u,\beta ,\gamma )=\left(ucos\gamma ,usin\gamma ,0,\beta ,u+\beta ,u-\beta \right),$$

where $u\ne \beta \ne 0$ and $\gamma \in (0,\pi /2)$ with the following almost complex structure:

$$\begin{array}{cc}\hfill \mathsf{\Phi }{\displaystyle \frac{\partial }{\partial x_{2i-1}}}& ={\displaystyle \frac{\partial }{\partial x_{2i}}},\mathsf{\Phi }{\displaystyle \frac{\partial }{\partial x_{2i}}}=-{\displaystyle \frac{\partial }{\partial x_{2i-1}}},\mathrm{for}i=1,2,3.\hfill \end{array}$$

A spanning set for the tangent space of M consists of the following vectors:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{Z}_1=cos\gamma {\displaystyle \frac{\partial }{\partial x_1}}+sin\gamma {\displaystyle \frac{\partial }{\partial x_2}}+{\displaystyle \frac{\partial }{\partial y_2}}+{\displaystyle \frac{\partial }{\partial y_3}},\hfill \\ Z_2={\displaystyle \frac{\partial }{\partial x_1}}+{\displaystyle \frac{\partial }{\partial x_2}}-{\displaystyle \frac{\partial }{\partial y_3}},\hfill \\ Z_3=-ucos\gamma {\displaystyle \frac{\partial }{\partial x_1}}+usin\gamma {\displaystyle \frac{\partial }{\partial x_2}}.\hfill \end{array}\right.\end{array}$$

Applying the almost complex structure Φ to these vectors, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathsf{\Phi }{Z}_1=-sin\gamma {\displaystyle \frac{\partial }{\partial x_1}}-cos\gamma {\displaystyle \frac{\partial }{\partial x_2}}-{\displaystyle \frac{\partial }{\partial x_3}}+{\displaystyle \frac{\partial }{\partial x_4}},\hfill \\ \mathsf{\Phi }{Z}_2=-{\displaystyle \frac{\partial }{\partial x_1}}+{\displaystyle \frac{\partial }{\partial x_2}}-{\displaystyle \frac{\partial }{\partial x_4}},\hfill \\ \mathsf{\Phi }{Z}_3=-usin\gamma {\displaystyle \frac{\partial }{\partial x_1}}-ucos\gamma {\displaystyle \frac{\partial }{\partial x_2}}.\hfill \end{array}\right.\end{array}$$

We observe that the distribution $\mathrm{\Lambda }=Span\{Z_1,Z_3\}$ is an invariant distribution, while ${\mathrm{\Lambda }}^{\theta }=Span\left\{Z_2\right\}$ is a proper slant distribution with slant angle given by

$$\theta ={cos}^{-1}\left({\displaystyle \frac{1}{3}}\right)=70.{53}^{\circ }.$$

Thus, M is a poinwise semi-slant submanifold of ${\mathbb{R}}^6$.

Since the distributions Λ and ${\mathrm{\Lambda }}^{\theta }$ satisfy the integrability condition, they define the integral manifolds $N_1$ and $N_2$, respectively. Accordingly, the metric g of the product manifold M is given by

$$g=3d{u}^2+3d{\beta }^2+u^{2}d{\gamma }^2.$$

Hence, we conclude that M is a WPPSS of ${\mathbb{R}}^6$ of the type $N_1{\times }_{\zeta }{N}_2$ with warping function $\zeta =u$.

Example 2.

Consider the immersion $\beta :M^4\to {\mathbb{R}}^8$ given by

$$\phi (u,v,\gamma ,\beta )=(5u+v,u-\sqrt{5}v,ucos\gamma ,usin\gamma ,v,3u,ucos\beta ,usin\beta ),$$

where $u,v\ne 0$ and $\gamma ,\beta \in (0,\pi /2)$ with the following almost complex structure:

$$\begin{array}{cc}\hfill \mathsf{\Phi }{\displaystyle \frac{\partial }{\partial x_{2i-1}}}& ={\displaystyle \frac{\partial }{\partial x_{2i}}},\mathsf{\Phi }{\displaystyle \frac{\partial }{\partial x_{2i}}}=-{\displaystyle \frac{\partial }{\partial x_{2i-1}}},\mathit{for}i=1,2,3,4.\hfill \end{array}$$

A spanning set for the tangent space of M consists of the following vectors:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{Z}_1\hfill & =5{\displaystyle \frac{\partial }{\partial x_1}}+{\displaystyle \frac{\partial }{\partial x_2}}+cos\gamma {\displaystyle \frac{\partial }{\partial x_3}}+sin\gamma {\displaystyle \frac{\partial }{\partial x_4}}+3{\displaystyle \frac{\partial }{\partial y_2}}+sin\beta {\displaystyle \frac{\partial }{\partial y_3}}+cos\beta {\displaystyle \frac{\partial }{\partial y_4}},\hfill \\ Z_2\hfill & ={\displaystyle \frac{\partial }{\partial x_1}}-\sqrt{5}{\displaystyle \frac{\partial }{\partial x_2}}+{\displaystyle \frac{\partial }{\partial y_1}},\hfill \\ Z_3\hfill & =-usin\gamma {\displaystyle \frac{\partial }{\partial x_3}}+ucos\gamma {\displaystyle \frac{\partial }{\partial x_4}},\hfill \\ Z_4\hfill & =-usin\beta {\displaystyle \frac{\partial }{\partial y_3}}+ucos\beta {\displaystyle \frac{\partial }{\partial y_4}}.\hfill \end{array}\right.\end{array}$$

Applying the almost complex structure Φ, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathsf{\Phi }{Z}_1\hfill & =-{\displaystyle \frac{\partial }{\partial x_1}}+5{\displaystyle \frac{\partial }{\partial x_2}}-sin\gamma {\displaystyle \frac{\partial }{\partial x_3}}+cos\gamma {\displaystyle \frac{\partial }{\partial x_4}}-3{\displaystyle \frac{\partial }{\partial x_5}}-cos\beta {\displaystyle \frac{\partial }{\partial x_7}}+sin\beta {\displaystyle \frac{\partial }{\partial x_8}},\hfill \\ \mathsf{\Phi }{Z}_2\hfill & =\sqrt{5}{\displaystyle \frac{\partial }{\partial x_1}}+{\displaystyle \frac{\partial }{\partial x_2}}+{\displaystyle \frac{\partial }{\partial x_6}},\hfill \\ \mathsf{\Phi }{Z}_3\hfill & =-ucos\gamma {\displaystyle \frac{\partial }{\partial x_3}}-usin\gamma {\displaystyle \frac{\partial }{\partial x_4}},\hfill \\ \mathsf{\Phi }{Z}_4\hfill & =-ucos\beta {\displaystyle \frac{\partial }{\partial x_7}}-usin\beta {\displaystyle \frac{\partial }{\partial x_8}}.\hfill \end{array}\right.\end{array}$$

We observe that the distribution $\mathrm{\Lambda }=Span\{Z_3,Z_4\}$ is an invariant distribution, while ${\mathrm{\Lambda }}^{\theta }=Span\{Z_1,Z_2\}$ is a pointwise slant distribution with slant angles given by

$${\theta }_1={cos}^{-1}\left(\sqrt{{\displaystyle \frac{28}{37}}}\right),$$

$${\theta }_2={cos}^{-1}\left(\sqrt{{\displaystyle \frac{6}{7}}}\right).$$

Thus, M is a piontwise semi-slant submanifold of ${\mathbb{R}}^8$.

Since the distributions Λ and ${\mathrm{\Lambda }}^{\theta }$ satisfy the integrability condition, they define the integral manifolds $N_1$ and $N_2$, respectively. Accordingly, the metric g of the product manifold M is given by:

$$g=37d{u}^2+7d{v}^2+u^2(d{\gamma }^2+d{\beta }^2).$$

Hence, we conclude that M is a WPPSS of ${\mathbb{R}}^8$ of the type $N_1{\times }_{\zeta }{N}_2$ with warping function $\zeta =u$.

6. Conclusions and Future Directions

In this work, we obtain two inequalities concerning the GN $\delta$-C-curvatures of WPPSS submanifolds over CSF. Using the optimization method, we determined exactly when these inequalities are equal, providing insight into both their geometric and structural consequences. In addition, the introduction of harmonic and Hessian functions allowed the interactions between curvature and analytic/associated functions to be viewed from a more general angle, and offered a deeper understanding of the interplay between geometry and analysis.

This work opens several promising avenues for future research. There is one promising generalization of such findings to more general families of submanifolds, including bi-slant submanifolds, in more general ambient manifolds, such as nearly Kähler or Sasakian manifolds. Furthermore, investigating the effects of curvature inequalities on the global topology of submanifolds may yield another source of information. Research into higher order curvature invariants and their relationship to physical theories, e.g., general relativity [31,32] or string theory, also raises some interesting opportunities.