Optimal Inequalities Characterizing Totally Real Submanifolds in Quaternionic Space Form

Abstract

In the present paper, we investigate some pinching inequalities on the scalar curvature of a totally real submanifold in quaternionic space form that leads to a topological conclusion of the submanifold. In addition, we construct another inequality which includes the mean curvature and the length of the second fundamental form.

1. Introduction

Exploring the topology of closed submanifolds in space forms presents an intriguing area of study. In contrast to standard techniques in Riemannian geometry, our findings on the topology of these submanifolds are derived by applying specific constraints on the scalar curvature and mean curvatures. In this context, the squared norm of the second fundamental form emerges with significant importance in the study of minimal submanifolds. Therefore, the study of rigidity theorems was essential in the groundbreaking work of Simons [1], Lawson [2], and Chern et al. [3] on minimal submanifolds in spheres. For example, let the square norm of the second fundamental form be presented by $\mathcal{B}$ and a unit sphere by ${\mathbb{S}}^{n+m}$ with codimension m. If there is a compact minimal submanifold ${\mathcal{N}}^n$ in ${\mathbb{S}}^{n+m}$ with pinching condition $0\le \mathcal{B}\le \left({\displaystyle \frac{n}{2-\frac{1}{m}}}\right)$, then either $\mathcal{B}=0\mathrm{or}\mathcal{B}=\left({\displaystyle \frac{n}{2-\frac{1}{m}}}\right)$, and $\mathcal{N}$ is the Clifford hypersurface or the Veronese surface in ${\mathbb{S}}^4$. Later, Li [4] and Chen-Xu [5] improved the pinching number ${\displaystyle \frac{n}{(2-1/m)}}$ to ${\displaystyle \frac{2n}{3}}$. They showed that if $0\le \mathcal{B}\le {\displaystyle \frac{2n}{3}},$ then either $\mathcal{B}=0\mathrm{or}\mathcal{B}={\displaystyle \frac{2n}{3}},$ and $\mathcal{N}$ is the Veronese surface in ${\mathbb{S}}^4.$ This topic has garnered significant attention following the early work by Simons [1] and subsequent developments (e.g., [2,4,5,6]). This study was extended to the rigidity theorems for minimal submanifolds and submanifolds with parallel mean curvature in space forms; see [7,8,9,10,11,12,13], among others.

Space forms are fundamental objects in differential geometry, providing rich settings for studying the relationship between curvature and topology. Over the past two decades, many researchers have investigated the geometry of submanifolds in various types of space forms, particularly through eigenvalue estimates and curvature inequalities. For example, Carriazo et al. [14] established early results on totally real submanifolds in quaternionic space forms. Later, Mutlu and Şentürk [15], as well as Lee and Vîlcu [16], contributed further to the development of curvature inequalities in the quaternionic setting. More recent contributions have appeared in this direction [17,18,19,20].

In complex and contact geometries, Ali et al. [21] studied eigenvalue inequalities for the p-Laplacian on C-totally real submanifolds in Sasakian space forms, and Ali et al. [22] also focused on Lagrangian submanifolds in complex space forms. Li et al. [23] derived Reilly-type inequalities for semi-slant submanifolds, while Alluhaibi and Ali [24] considered totally real submanifolds in generalized complex space forms involving the p-Laplacian operator. More recent work by Mehraj Ahmad [25] and Siddiqi et al. [26] focused on quarter-symmetric and statistical connection settings.

However, rigidity theorems for totally real submanifolds in quaternionic space forms remain relatively underexplored. Motivated by these developments, this paper establishes new rigidity theorems for totally real submanifolds in quaternionic space forms. In particular, we derive optimal inequalities involving scalar curvature, the squared norm of the second fundamental form, and the mean curvature. These results enrich rigidity theory and offer new insights into the interplay between intrinsic and extrinsic geometry in quaternionic settings.

2. Preliminaries

An almost quaternionic structure on $(\mathcal{M},\sigma )$ is defined as having a local canonical basis $\left\{{\mathrm{J}}_1,{\mathrm{J}}_2,{\mathrm{J}}_3\right\}$ on a rank 3-subbundle $\sigma$ of $End\left(T\mathcal{M}\right)$ for a differentiable manifold $\mathcal{M}$ such that where $s, s+1, s+2$ are taken modulo 3, and the following conditions hold:

$$\begin{array}{c}\hfill {\mathrm{J}}_s^2=-Id,{\mathrm{J}}_s{\mathrm{J}}_{s+1}=-{\mathrm{J}}_{s+1}{\mathrm{J}}_s={\mathrm{J}}_{s+2}.\end{array}$$

(1)

and

$$\begin{array}{c}\hfill \tilde{g}\left({\mathrm{J}}_s{X}_1,{\mathrm{J}}_s{Y}_1\right)=\tilde{g}(X_1,Y_1),\end{array}$$

(2)

for all vector fields $X_1,Y_1$ on $\mathcal{M}$, where $\mathcal{M}$ stands for a Riemannian manifold with real dimension $4m$. Moreover, $(\mathcal{M},\sigma ,\tilde{g})$ is said to be an almost quaternionic Hermitian manifold (see [16] for more details). Let there exist 1-form ${\theta }_1,{\theta }_2,{\theta }_3$, and let $\sigma$ be parallel with respect to the Levi–Civita connection $\overline{\nabla }$ satisfying the following:

$$\begin{array}{c}\hfill {\nabla }_{{\mathrm{X}}_1}{\mathrm{J}}_s={\theta }_{s+2}\left({\mathrm{X}}_1\right){\mathrm{J}}_{s+1}-{\theta }_{s+1}\left({\mathrm{X}}_1\right){\mathrm{J}}_{s+2},\end{array}$$

(3)

For any vector field ${\mathrm{X}}_1$ on $\mathcal{M}$, and where s, $s+1$, and $s+2$ are taken modulo 3, the triple $(\mathcal{M},\sigma ,\tilde{g})$ denotes a quaternionic Kähler manifold [15]. The curvature tensor $\tilde{\mathrm{R}}$ of a quaternionic space form $\mathcal{M}\left(c\right)$, which has constant quaternionic sectional curvature c [14], is given by:

$$\begin{array}{cc}\hfill \tilde{\mathrm{R}}({\mathrm{X}}_1,{\mathrm{Y}}_1){\mathrm{Z}}_1=& {\displaystyle \frac{c}{4}}\{\tilde{g}({\mathrm{Z}}_1,{\mathrm{Y}}_1){\mathrm{X}}_1-\tilde{g}({\mathrm{X}}_1,{\mathrm{Z}}_1){\mathrm{Y}}_1+\sum _{s=1}^3(\tilde{g}({\mathrm{Z}}_1,{\mathrm{J}}_s{\mathrm{Y}}_1){\mathrm{J}}_s{\mathrm{X}}_1\hfill \\ \hfill & -\tilde{g}\left({\mathrm{Z}}_1,{\mathrm{J}}_s{\mathrm{X}}_1\right){\mathrm{J}}_s{\mathrm{Y}}_1+2\tilde{g}\left({\mathrm{X}}_1,{\mathrm{J}}_s{\mathrm{Y}}_1\right){\mathrm{J}}_s{\mathrm{Z}}_1\left)\right\}\hfill \end{array}$$

(4)

for all vector fields ${\mathrm{X}}_1,{\mathrm{Y}}_1,{\mathrm{Z}}_1$ on $\mathcal{M}$ and any local basis $\left\{{\mathrm{J}}_1,{\mathrm{J}}_2,{\mathrm{J}}_3\right\}$ of $\sigma$.

Assume that $\mathcal{N}$ is a submanifold of an almost Hermitian quaternionic manifold $\mathcal{M}\left(c\right)$ with an induced metric g. Thus, the Weingarten and Gauss formulas are provided by

$$\begin{array}{c}\hfill \left(i\right){\tilde{\nabla }}_{{\mathrm{X}}_2}{\mathrm{Y}}_2={\nabla }_{{\mathrm{X}}_2}{\mathrm{Y}}_2+\mathcal{B}({\mathrm{X}}_2,{\mathrm{Y}}_2),\left(ii\right){\tilde{\nabla }}_{{\mathrm{X}}_2}{\mathrm{Z}}_2^{\perp }=-A_{{\mathrm{Z}}_2^{\perp }}{\mathrm{X}}_2+{\nabla }_{{\mathrm{X}}_2}^{\perp }{\mathrm{Z}}_2^{\perp },\end{array}$$

(5)

for each ${\mathrm{X}}_2,{\mathrm{Y}}_2\in \mathrm{\Gamma }\left(T\mathcal{N}\right)$ and ${\mathrm{Z}}_2^{\perp }\in \mathrm{\Gamma }\left(T^{\perp }\mathcal{N}\right)$, where ∇ and ${\nabla }^{\perp }$ denote the induced connections on the tangent bundle $T\mathcal{N}$ and the normal bundle $T^{\perp }\mathcal{N}$ of $\mathcal{N}$, respectively. The second fundamental form $\mathcal{B}$ and the shape operator $A_{{\mathrm{Z}}_2^{\perp }}$ are associated with the normal vector field ${\mathrm{Z}}_2^{\perp }$ for the immersion of $\mathcal{N}$ into $\mathcal{M}\left(c\right)$. They are related as follows:

$$\begin{array}{c}\hfill g(\mathcal{B}({\mathrm{X}}_2,{\mathrm{Y}}_2),{\mathrm{Z}}_2^{\perp })=g(A_{{\mathrm{Z}}_2^{\perp }}{\mathrm{X}}_2,{\mathrm{Y}}_2),\end{array}$$

(6)

for any ${\mathrm{X}}_2\in \mathrm{\Gamma }\left(T\mathcal{N}\right)$. If the almost complex structure ${\mathrm{J}}_s$ carries each tangent space of the submanifold $\mathcal{N}$ of the quaternion-Kähler manifold $\mathcal{M}$ into its normal space, then the submanifold $\mathcal{N}$ is called a totally real submanifold of $\mathcal{M}$. In this case, $g({\mathrm{J}}_s{\mathrm{X}}_2,{\mathrm{Y}}_2)=0$ for each ${\mathrm{X}}_2,{\mathrm{Y}}_2$ tangent to $\mathcal{N}$, (see [14,15]). Then, (4) is reduced to the following

$$\begin{array}{cc}\hfill g(\mathrm{R}({\mathrm{X}}_2,{\mathrm{Y}}_2){\mathrm{Z}}_2,{\mathrm{W}}_2)& ={\displaystyle \frac{c}{4}}\left\{g({\mathrm{Z}}_2,{\mathrm{Y}}_2)g({\mathrm{X}}_2,{\mathrm{W}}_2)-g({\mathrm{X}}_2,{\mathrm{Z}}_2)g({\mathrm{Y}}_2,{\mathrm{W}}_2)\right\}.\hfill \end{array}$$

(7)

for any ${\mathrm{X}}_2,{\mathrm{Y}}_2,{\mathrm{Z}}_2,{\mathrm{W}}_2\in \mathrm{\Gamma }\left(T\mathcal{N}\right).$ The Gauss equation is defined for the curvature tensors $\tilde{\mathrm{R}}$ and $\mathrm{R}$ of ${\mathcal{N}}^m$ and $\mathcal{M}$, respectively.

$$\begin{array}{cc}\hfill \tilde{\mathrm{R}}({\mathrm{X}}_2,{\mathrm{Y}}_2,{\mathrm{Z}}_2,{\mathrm{W}}_2)=& \mathrm{R}({\mathrm{X}}_2,{\mathrm{Y}}_2,{\mathrm{Z}}_2,{\mathrm{W}}_2)+\tilde{g}(\mathcal{B}({\mathrm{X}}_2,{\mathrm{W}}_2),\mathcal{B}({\mathrm{Y}}_2,{\mathrm{Z}}_2))\hfill \\ \hfill & -\tilde{g}(\mathcal{B}({\mathrm{X}}_2,{\mathrm{Z}}_2),\mathcal{B}({\mathrm{Y}}_2,{\mathrm{W}}_2)).\hfill \end{array}$$

(8)

Let us consider an orthonormal frame $\{e_1,\dots ,e_m,e_{m+1},\dots ,e_{4m+h},e_{1^{*}}={\mathrm{J}}_s{e}_1,\dots$, $e_{m^{*}}={\mathrm{J}}_s{e}_m,e_{{(m+1)}^{*}}={\mathrm{J}}_s{e}_{m+1},\dots ,e_{{(4m+h)}^{*}}={\mathrm{J}}_s{e}_{4m+h}\}$ on the ambient space ${\mathcal{M}}^{4m+h}\left(c\right)$, where the codimension is h. On the submanifold ${\mathcal{N}}^m$, we assume that the vectors $e_1,\dots ,e_m$ span the tangent bundle $T{\mathcal{N}}^m$. The indices are assigned accordingly.

$$\begin{array}{cc}\hfill \mathcal{A},\mathcal{B},\mathcal{C}\dots =& 1,\dots ,4m+h,1^{*}\dots ,4m+h^{*},\hfill \\ \hfill i^{*},j^{*},k^{*}=& 4m+1,\dots ,4m+h,1^{*},\dots ,4m+h^{*}.\hfill \\ \hfill & 1\le a,b,c,\dots \le m,\hfill \end{array}$$

Equation (7) is expressed in terms of local coordinates:

$$\begin{array}{cc}\hfill {\tilde{\mathrm{K}}}_{\mathcal{A}\mathcal{B}\mathcal{C}\mathcal{D}}& ={\displaystyle \frac{c}{4}}\left({\delta }_{\mathcal{A}\mathcal{C}}{\delta }_{\mathcal{B}\mathcal{D}}-{\delta }_{\mathcal{A}\mathcal{D}}{\delta }_{\mathcal{B}\mathcal{C}}\right).\hfill \end{array}$$

(9)

Then, the curvature tensor for the submanifold is given by the following:

$$\begin{array}{cc}\hfill R_{abcl}& ={\tilde{K}}_{abcl}+\sum _i\left({\mathcal{B}}_{ac}^i{\mathcal{B}}_{bl}^i-{\mathcal{B}}_{al}^i{\mathcal{B}}_{bc}^i\right),\hfill \end{array}$$

(10)

where $\mathcal{B}$ denotes the second fundamental form of the submanifold.

The Ricci curvature for a totally real submanifold is expressed as follows:

$$\begin{array}{cc}\hfill R_{ab}& ={\displaystyle \frac{c}{4}}(m-1){\delta }_{ab}+\sum _k\left({\mathcal{B}}_{ab}^k\sum _c{\mathcal{B}}_{cc}^k-\sum _c{\mathcal{B}}_{ac}^k{\mathcal{B}}_{cb}^k\right).\hfill \end{array}$$

(11)

Let $\mathrm{\Pi }$ represent the squared length of the second fundamental form $\mathcal{B}$ of the submanifold ${\mathcal{N}}^m$, and it is defined by

$$\begin{array}{c}\hfill \mathrm{\Pi }=\sum _{abk}{\left({\mathcal{B}}_{ab}^k\right)}^2.\end{array}$$

(12)

Correspondingly, the mean curvature of ${\mathcal{N}}^m$ is given by

$$\begin{array}{c}\hfill \zeta ={\displaystyle \frac{1}{m}}\sum _{ak}{\mathcal{B}}_{aa}^k{e}_k\end{array}$$

(13)

From the above, we fix some notation

$$\begin{array}{c}\hfill \mathrm{\Pi }={\parallel \mathcal{B}\parallel }^2,\zeta =\left|\xi \right|,{\zeta }_i={\left({\mathcal{B}}_{ab}^i\right)}_{m\times m}.\end{array}$$

(14)

where $\xi$ is the mean curvature vector.

Let us assume that $e_{m+1}$ is parallel to $\xi$; then, we have

$$\begin{array}{c}\hfill tr{\zeta }_{m+1}=m\zeta ,tr{\zeta }_i=0,i\ne m+1\end{array}$$

(15)

where tr stands for the trace of the matrix ${\zeta }_i=\left({\mathcal{B}}_{ab}^i\right).$ Taking (9), (10), and (15) into account, we have scalar curvature as follows:

$$\begin{array}{c}\hfill \mathrm{R}={\displaystyle \frac{1}{4}}m(m-1)c+m^2{\zeta }^2-\mathrm{\Pi }\end{array}$$

(16)

where $\zeta$ is the mean curvature of ${\mathcal{N}}^m$. Since $\zeta$ is constant, it can be concluded that the scalar curvature $\mathrm{R}$ is constant if and only if $\mathrm{\Pi }$ is constant according to (16). Let ${\mathcal{B}}_{abc}^k$ denote the second covariant derivative of ${\mathcal{B}}_{ab}^k$; we have

$$\begin{array}{c}\hfill \sum _c{\mathcal{B}}_{abc}^i{\theta }_c=\mathrm{d}{\mathcal{B}}_{ab}^i-\sum _c{\mathcal{B}}_{cb}^i{\theta }_{ca}-\sum _c{\mathcal{B}}_{ac}^i{\theta }_{cb}-\sum _j{\mathcal{B}}_{ab}^j{\theta }_{ji}\end{array}$$

(17)

where $\left\{{\theta }_a\right\}$ is the dual frame of ${\mathcal{N}}^m$. Taking the exterior derivative of the above equation, we obtain

$$\begin{array}{cc}\hfill \sum _l{\mathcal{B}}_{abcl}^i{\theta }_l=& \mathrm{d}{\mathcal{B}}_{abc}^i-\sum _l{\mathcal{B}}_{abc}^i{\theta }_{la}+\sum _j{\mathcal{B}}_{abc}^j{\theta }_{ji}\hfill \\ \hfill & -\sum _l{\mathcal{B}}_{alc}^i{\theta }_{lb}-\sum _l{\mathcal{B}}_{abl}^i{\theta }_{lc}.\hfill \end{array}$$

(18)

Moreover, the Laplacian of ${\mathcal{B}}_{ab}^i$ is

$$\begin{array}{cc}\hfill \mathrm{\Delta }{\mathcal{B}}_{ab}^i=& \sum _c{\mathcal{B}}_{abcc}^i=\sum _c{\mathcal{B}}_{ccab}^i+\sum _{cd}\left({\mathcal{B}}_{cd}^i{\mathrm{R}}_{dabc}+{\mathcal{B}}_{da}^i{\mathrm{R}}_{dcbc}\right)\hfill \\ \hfill & -\sum _{jc}{\mathcal{B}}_{ca}^j{\mathrm{R}}_{ijbc}.\hfill \end{array}$$

(19)

The DDVV conjecture, also known as the normal scalar curvature conjecture, establishes a pointwise inequality that relates the scalar curvature, the normal scalar curvature, and the mean curvature of a submanifold in a real space form. This conjecture has been confirmed and is now known as the DDVV inequality, stated as follows:

Lemma 1 

([27,28]). If we let ${\mathrm{T}}_1,\dots ,{\mathrm{T}}_n$ be $(m\times m)$-symmetric matrices, then

$$\begin{array}{c}\hfill \sum _{r,s=1}^n{\parallel [{\mathrm{T}}_r,{\mathrm{T}}_s]\parallel }^2\le {\left(\sum _{r=1}^n{\parallel {\mathrm{T}}_r\parallel }^2\right)}^2\end{array}$$

such that equality holds if and only if the following matrices are satisfied:

$${\mathrm{T}}_r=P\left(\begin{array}{ccc}0& \mu & 0\dots 0\\ \mu & 0& 0\dots 0\\ 0& 0& 0\dots 0\\ ⋮& ⋮& ⋮\ddots ⋮\\ 0& 0& 0\dots 0\end{array}\right)P^t,{\mathrm{T}}_s=P\left(\begin{array}{ccc}\mu & 0& 0\dots 0\\ 0& -\mu & 0\dots 0\\ 0& 0& 0\dots 0\\ ⋮& ⋮& ⋮\ddots ⋮\\ 0& 0& 0\dots 0\end{array}\right)P^t$$

where P is an orthogonal $(m\times m)$-matrix, and $[{\mathrm{T}}_r,{\mathrm{T}}_s]={\mathrm{T}}_r{\mathrm{T}}_s-{\mathrm{T}}_s{\mathrm{T}}_r$ is the commutator of the matrices ${\mathrm{T}}_r,{\mathrm{T}}_s$.

Lemma 2 

([4]). Let ${\mathcal{T}}_1,{\mathcal{T}}_2,\dots {\mathcal{T}}_n$ be $(m\times m)$-symmetric matrices. Then,

$$\begin{array}{cc}\hfill -2\sum _{i,j=1}^n\left(tr\left({\mathcal{T}}_i^2{\mathcal{T}}_j^2\right)-tr{\left({\mathcal{T}}_i{\mathcal{T}}_j\right)}^2\right)\ge & \sum _{i,j=1}^n{\left[tr\left({\mathcal{T}}_i{\mathcal{T}}_j\right)\right]}^2\hfill \\ \hfill & -{\displaystyle \frac{3}{2}}{\left(\sum _{i=1}^{n}tr\left({\mathcal{T}}_i^2\right)\right)}^2.\hfill \end{array}$$

(20)

3. Main Results

In this section, we present our main results concerning totally real submanifolds in quaternionic space forms.

Theorem 1. 

Let ${\mathcal{N}}^m$ be an m-dimensional, compact, totally real submanifold ${\mathcal{N}}^m$ of a quaternionic space forms ${\mathcal{M}}^{4m+h}\left(c\right)$. If the mean curvature vector ξ of ${\mathcal{N}}^m$ is parallel and the the mean curvature ζ of ${\mathcal{N}}^m$ satisfies the inequality

$$\begin{array}{c}\hfill {\mathrm{R}}_{\mathcal{N}}\ge \left({\displaystyle \frac{m+2h-1}{2(m+2h)}}\right)\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right),\end{array}$$

(21)

then, under these conditions, ${\mathcal{N}}^m$ is a totally umbilical sphere ${\mathbb{S}}^m\left({\displaystyle \frac{1}{\sqrt{\frac{c}{4}+{\zeta }^2}}}\right)$.

Proof. 

Assume that ${\mathcal{N}}^m$ is a totally real submanifold of quaternionic space form $\mathcal{M}\left(c\right)$ with parallel mean curvature vector $\xi$. Consider $e_{m+1}$ such that it is parallel to $\xi$. Then, from (15), we have

$$\begin{array}{c}\hfill {\nabla }^{\perp }\xi =\mathrm{d}\zeta e_{m+1}+\zeta {\nabla }^{\perp }{e}_{m+1}=\mathrm{d}\zeta e_{m+1}+\zeta \sum _j{\theta }_{m+1j}{e}_j=0\end{array}$$

(22)

From the structure equation and (22), we have

$$\begin{array}{cc}\hfill \mathrm{d}{\theta }_{m+1j}=& -\sum _k{\theta }_{m+1k}\wedge {\theta }_{kj}+{\displaystyle \frac{1}{2}}\sum _{cl}{\mathrm{R}}_{m+1jcl}{\theta }_c\wedge {\theta }_l\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\sum _{cl}{\mathrm{R}}_{m+1jcl}{\theta }_c\wedge {\theta }_l=0\hfill \end{array}$$

(23)

From Equation (19), and considering that ${\mathcal{N}}^m$ has a parallel mean curvature vector and satisfies ${\sum }_c{\zeta }_{ccab}^i=0$, we obtain the following:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }{\mathrm{\Pi }}_{\zeta }=& \sum _{abc}{\left({\mathcal{B}}_{abc}^{m+1}\right)}^2+\sum _{ab}{\mathcal{B}}_{ab}^{m+1}\mathrm{\Delta }{\mathcal{B}}_{ab}^{m+1}\hfill \\ \hfill =& \sum _{abc}{\left({\mathcal{B}}_{abc}^{m+1}\right)}^2+\sum _{abcl}{\mathcal{B}}_{ab}^{m+1}\left({\mathcal{B}}_{cl}^{m+1}{R}_{labc}+{\mathcal{B}}_{la}^{m+1}{\mathrm{R}}_{lcbc}\right).\hfill \end{array}$$

(24)

If 2-plane $\pi \subset T_p\mathcal{N}$ at the point $p\in {\mathcal{N}}^m$, the sectional curvature is denoted by ${\mathrm{R}}_{\mathcal{N}}(p,\pi )$. Then, the following is fixed:

$${\mathrm{R}}_{min}\left(p\right)=\underset{\pi \subset T_p\mathcal{N}}{min}{\mathrm{R}}_{\mathcal{N}}(p,\pi ).$$

Therefore, we consider $\left\{e_i\right\}$ the orthonormal fields such that ${\mathcal{B}}_{ab}^{m+1}={\lambda }_a{\delta }_{ab}$ for non-zero eigenvalues ${\lambda }_i$. Then, we obtain

$$\begin{array}{cc}\hfill \sum _{abcl}{\mathcal{B}}_{ab}^{m+1}\left({\mathcal{B}}_{cl}^{m+1}{\mathrm{R}}_{labc}+{\mathcal{B}}_{la}^{m+1}{\mathrm{R}}_{lcbc}\right)=& {\displaystyle \frac{1}{2}}\sum _{ab}{({\lambda }_a-{\lambda }_b)}^2{\mathrm{R}}_{abab}\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}\sum _{ab}{\left({\lambda }_a-{\lambda }_b\right)}^2{\mathrm{R}}_{min}.\hfill \end{array}$$

(25)

Taking (24) and (25) into account, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }{\mathrm{\Pi }}_{\zeta }\ge \sum _{abc}{\left({\mathcal{B}}_{abc}^{m+1}\right)}^2+{\displaystyle \frac{1}{2}}\sum _{ab}{({\lambda }_a-{\lambda }_b)}^2{\mathrm{R}}_{min}.\end{array}$$

(26)

It follows from ${\mathrm{R}}_{\mathcal{N}}\ge {\displaystyle \frac{m+2h-1}{2(m+2h)}}({\displaystyle \frac{c}{4}}+{\zeta }^2)$ with Hopf’s lemma that ${\mathrm{\Pi }}_{\zeta }$ is a constant, and we derive

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\sum _{ab}{({\lambda }_a-{\lambda }_b)}^2{\mathrm{R}}_{min}=0.\end{array}$$

(27)

It is implied that ${\lambda }_a={\lambda }_b$. Then, ${\mathcal{N}}^m$ is pseudo-umbilical.

Again from (19), ${\sum }_c{\zeta }_{ccab}^i=0$, and the mean curvature vector of ${\mathcal{N}}^m$ is parallel; thus, one constructs

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\tau =& \sum _{i\ne m+1}\sum _{abc}{\left({\mathcal{B}}_{abc}^i\right)}^2+\sum _{i\ne m+1}\sum _{abcl}{\mathcal{B}}_{ab}^i\left({\mathcal{B}}_{cl}^i{\mathrm{R}}_{labc}+{\mathcal{B}}_{la}^i{\mathrm{R}}_{lcbc}\right)\hfill \\ \hfill & -\sum _{i\ne m+1}\sum _{jabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ca}^j{\mathrm{R}}_{ijbc}\hfill \end{array}$$

(28)

where $\tau$ is a scalar curvature in ${\mathcal{N}}^m.$ From (10) and (15), we have

$$\begin{array}{cc}\hfill \sum _{i\ne m+1}\sum _{abcl}{\mathcal{B}}_{ab}^i({\mathcal{B}}_{cl}^i{\mathrm{R}}_{labc}& +{\mathcal{B}}_{la}^i{\mathrm{R}}_{lcbc})\hfill \\ \hfill =& m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\tau +\sum _{ij\ne m+1}\left(\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right)\hfill \\ \hfill & -\sum _{ij\ne m+1}{\left(\mathrm{tr}\left({\zeta }_i{\zeta }_j\right)\right)}^2\hfill \end{array}$$

(29)

From (10), we derive

$$\begin{array}{c}\hfill \sum _{i\ne m+1}\sum _{jabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ca}^j{\mathrm{R}}_{ijbc}=-\sum _{i}tr{\zeta }_{i^{*}}^2-\sum _{i,j\ne m+1}\left(tr{\left({\zeta }_i{\zeta }_j\right)}^2-tr\left({\zeta }_i^2{\zeta }_j^2\right)\right)\end{array}$$

(30)

Inserting (30) and (29) into (28), for any real number $b^{\prime }$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\tau =& \sum _{i\ne m+1}\sum _{abc}{\left({\mathcal{B}}_{abc}^i\right)}^2+\sum _{i}tr{\zeta }_{i^{*}}^2-b^{\prime }m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\tau \hfill \\ & +(1+b^{\prime })\sum _{i\ne m+1}\sum _{abcl}{\mathcal{B}}_{ab}^i({\mathcal{B}}_{cl}^i{R}_{labc}+{\mathcal{B}}_{la}^i{R}_{lcbc})+b^{\prime }\sum _{ij\ne m+1}{\left(\mathrm{tr}\left({\zeta }_i{\zeta }_j\right)\right)}^2\hfill \\ & +(1-b^{\prime })\sum _{ij\ne m+1}\left(\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right)\hfill \end{array}$$

(31)

For a fixed index i, we consider an orthonormal frame field $\left\{e_i\right\}$ satisfying ${\mathcal{B}}_{ab}^i={\lambda }_a^i{\delta }_{ab}$. From the above assumption, together with Equation (15), it follows that

$$\begin{array}{cc}\hfill \sum _{abcl}{\mathcal{B}}_{ab}^i\left({\mathcal{B}}_{cl}^i{\mathrm{R}}_{labc}+{\mathcal{B}}_{la}^i{\mathrm{R}}_{lcbc}\right)=& {\displaystyle \frac{1}{2}}\sum _{ab}{({\lambda }_a^i-{\lambda }_b^i)}^2{\mathrm{R}}_{abab}\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2}}\sum _{ab}{({\lambda }_a^i-{\lambda }_b^i)}^2{\mathrm{R}}_{min}\hfill \\ \hfill =& m\mathrm{tr}{\zeta }_i^2{\mathrm{R}}_{min}\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \sum _{i\ne m+1}\sum _{abcl}{\mathcal{B}}_{ab}^i\left({\mathcal{B}}_{cl}^i{\mathrm{R}}_{labc}+{\mathcal{B}}_{la}^i{\mathrm{R}}_{lcbc}\right)\ge m\tau {\mathrm{R}}_{min}.\end{array}$$

(32)

By applying the DDVV inequality stated in Lemma 1, we derive the following:

$$\begin{array}{cc}\hfill \sum _{ij\ne m+1}\left\{\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)-\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2\right\}& ={\displaystyle \frac{1}{2}}\sum _{ij\ne m+1}\mathrm{tr}{({\zeta }_i{\zeta }_j-{\zeta }_j{\zeta }_i)}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\left(\sum _{i\ne m+1}\mathrm{tr}{\zeta }_i^2\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\tau }^2.\hfill \end{array}$$

(33)

Also, we have

$$\begin{array}{c}\hfill \sum _{ij\ne m+1}{\left(\mathrm{tr}\left({\zeta }_i{\zeta }_j\right)\right)}^2\ge {\displaystyle \frac{{\tau }^2}{m+2h-1}}.\end{array}$$

(34)

Choosing $b^{\prime }={\displaystyle \frac{m+2h-1}{m+2h+1}}$ in (31) and making use of (32)–(34) leads to the following result:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\tau \ge \left\{-\left({\displaystyle \frac{m+2h-1}{m+2h+1}}\right)\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)+\left({\displaystyle \frac{2m+4h}{m+2h+1}}\right){\mathrm{R}}_{min}\right\}m\tau \end{array}$$

(35)

Under the assumption given by (21), the following result is obtained:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\tau \ge 0\end{array}$$

According to Hopf’s lemma. This leads to the conclusion that $\mathrm{\Delta }\tau =0$. Hence, we obtain the following:

$$\begin{array}{c}\hfill \tau =0\mathit{or}{\mathrm{R}}_{\mathcal{N}}=\left({\displaystyle \frac{m+2h-1}{2(m+2h)}}\right)\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right).\end{array}$$

In the case where $\tau =0$, the submanifold ${\mathcal{N}}^m$ becomes totally umbilical. On the other hand, when $\tau \ne 0$, from (10), we derive

$$\begin{array}{c}\hfill {\mathrm{R}}_{abab}={\displaystyle \frac{c}{4}}+{\zeta }^2,\end{array}$$

which leads to the conclusion that ${\mathcal{N}}^m$ is a totally umbilical sphere ${\mathbb{S}}^m\left({\displaystyle \frac{1}{\sqrt{\frac{c}{4}+{\zeta }^2}}}\right)$.

Moreover, the inequalities (32)–(35) all become equalities when the sectional curvature satisfies

$$\begin{array}{c}\hfill tr{\zeta }_i^2=tr{\zeta }_j^2(i,j\ne m+1),\mathrm{and}\sum _{a}tr{\zeta }_{a^{*}}^2=0.\end{array}$$

Next, we demonstrate that the second case cannot occur. For this, we must examine the case when equality holds in (33), which indicates either that all components ${\zeta }_i$ vanish or that exactly two of them are nonzero, with $i\ne m+1$.

Now, assume that the inequalities (34) and (35) also hold as equalities. Then, we must have

$$\begin{array}{c}\hfill \mathrm{tr}{\zeta }_i^2=\mathrm{tr}{\zeta }_j^2(i,j\ne m+1),\mathrm{and}\sum _a\mathrm{tr}{\zeta }_{a^{*}}^2=0.\end{array}$$

(36)

These conditions imply that ${\mathcal{N}}^m$ must be totally umbilical, satisfying

$$\begin{array}{c}\hfill {\mathrm{R}}_{abab}={\displaystyle \frac{c}{4}}+{\zeta }^2,\end{array}$$

(37)

since all ${\zeta }_i$ with $i\ne m+1$ are zero. However, this leads to a contradiction. This completes the proof of the theorem. □

In the subsequent results, we have the following.

Theorem 2. 

Let ${\mathcal{N}}^m$ be an m-dimensional, totally real submanifold of a quaternionic space form ${\mathcal{M}}^{4m+h}\left(c\right)$. If $\mathrm{J}\xi$ has a normal relation to ${\mathcal{N}}^m$, then ${\mathcal{N}}^m$ is totally umbilical or satisfies the inequality

$$\begin{array}{c}\hfill inf\rho \le m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\left(m-{\displaystyle \frac{5}{3}}\right)\end{array}$$

(38)

where ρ is the scalar curvature.

Proof. 

Let $\mathrm{J}\xi$ be normal to ${\mathcal{N}}^m$, and we can consider $e_{m+1}$ such that it is parallel to $\xi$, so we have

$$\begin{array}{c}\hfill tr{\zeta }_{m+1}=m\zeta ,tr{\zeta }_i=0,i\ne m+1\end{array}$$

(39)

From (19), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\mathrm{\Pi }=& \sum _{iijk}{\left({\mathcal{B}}_{abc}^i\right)}^2+\sum _{iabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ccab}^i\hfill \\ \hfill & +\sum _{iabcl}{\mathcal{B}}_{ab}^i\left({\mathcal{B}}_{cl}^i{\mathrm{R}}_{labc}+{\mathcal{B}}_{la}^i{\mathrm{R}}_{lcbc}\right)-\sum _{ijabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ca}^j{\mathrm{R}}_{ijbc}.\hfill \end{array}$$

(40)

From (10), (39) and ${\mathcal{N}}^m$ are totally umbilical, and we obtain

$$\begin{array}{cc}\hfill \sum _{iabcl}{\mathcal{B}}_{ab}^i\left({\mathcal{B}}_{cl}^i{\mathrm{R}}_{labc}+{\mathcal{B}}_{la}^i{\mathrm{R}}_{lcbc}\right)=& m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\mathrm{\Pi }-m^2{\zeta }^2\hfill \\ \hfill & +\sum _{ij}\left\{\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right\}\hfill \\ \hfill & -\sum _{ij}{\left(\mathrm{tr}\left({\zeta }_i{\zeta }_j\right)\right)}^2,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \sum _{ijabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ca}^j{\mathrm{R}}_{ijbc}=& -\sum _{ij}\left\{\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right\}\hfill \\ \hfill & -\sum _a\mathrm{tr}{\zeta }_{a^{*}}^2.\hfill \end{array}$$

(42)

In view of (39) and the pseudo-umbilical such that ${\mathcal{B}}_{ab}^{m+1}=\zeta {\delta }_{ab}$, we derive

$$\begin{array}{cc}\hfill \sum _{iabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ccab}^i=& m\zeta \mathrm{\Delta }\zeta ,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \sum _{iabc}{\left({\mathcal{B}}_{abc}^i\right)}^2\ge \sum _{ac}{\left({\mathcal{B}}_{aac}^{m+1}\right)}^2=& m\sum _a{\left({\nabla }_a\zeta \right)}^2,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }{\zeta }^2=& \zeta \mathrm{\Delta }\zeta +\sum _a{\left({\nabla }_a\zeta \right)}^2.\hfill \end{array}$$

(45)

According to the pseudo-umbilical condition with Lemma 2, ${\mathcal{B}}_{ab}^{m+1}=\zeta {\delta }_{ab}$, we have

$$\begin{array}{cc}\hfill 2\sum _{ij}\{\mathrm{tr}{\left({\zeta }_{\mathrm{i}}{\zeta }_{\mathrm{j}}\right)}^2& -\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\}-\sum _{ij}\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2\hfill \\ \hfill & =2\sum _{ij\ne m+1}\left\{\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right\}\hfill \\ \hfill & -\sum _{ij\ne m+1}tr{\left({\zeta }_i{\zeta }_j\right)}^2-{\left(\mathrm{tr}{\mathit{H}}_{m+1}^2\right)}^2\hfill \\ \hfill & \ge -{\displaystyle \frac{3}{2}}{\tau }^2-m^2{\zeta }^4\hfill \\ \hfill & =-{\displaystyle \frac{3}{2}}{(\mathrm{\Pi }-m{\zeta }^2)}^2-m^2{\zeta }^4.\hfill \end{array}$$

(46)

Substituting (41)–(46) into (40), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\mathrm{\Pi }\ge & {\displaystyle \frac{1}{2}}m\mathrm{\Delta }{\zeta }^2+m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\mathrm{\Pi }-{\displaystyle \frac{3}{2}}{\left(\mathrm{\Pi }-m{\zeta }^2\right)}^2-m^2{\zeta }^4-m^2{\zeta }^2\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}m\mathrm{\Delta }{\zeta }^2+\left(\mathrm{\Pi }-m{\zeta }^2\right)\left\{m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)-{\displaystyle \frac{3}{2}}\left(\mathrm{\Pi }-m{\zeta }^2\right)\right\}\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}m\mathrm{\Delta }{\zeta }^2+\tau \left\{m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)-{\displaystyle \frac{3}{2}}\tau \right\}.\hfill \end{array}$$

(47)

Applying a similar argument as that in [10], we deduce that either ${\mathcal{N}}^m$ is totally umbilical or

$$\begin{array}{c}\hfill \inf \rho \le m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\left(m-{\displaystyle \frac{5}{3}}\right).\end{array}$$

This completes the proof of the theorem. □

Theorem 3. 

Let $\mathrm{J}\xi$ be normal to an m-dimensional compact totally real submanifold ${\mathcal{N}}^m$ in a quaternionic space form ${\mathcal{M}}^{4m+h}\left(c\right)$. Then, the following inequality holds

$$\begin{array}{c}\hfill \int \left\{2m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\mathrm{\Pi }-3{\mathrm{\Pi }}^2-5m^2{\zeta }^4-4m^2{\zeta }^2+2m{\zeta }^2\right\}\mathrm{dV}\le 0\end{array}$$

(48)

Proof. 

Assume that $\mathrm{J}\xi$ is normal to the submanifold ${\mathcal{N}}^m$, and, without loss of generality, let the vector $e_{1^{*}}$ be chosen to be parallel to the mean curvature vector $\xi$. This choice implies that $tr{\zeta }_{1^{*}}=m\zeta .$ Furthermore, for all indices $i\ne 1^{*}$, we have $tr{\zeta }_i=0$. Under these assumptions, and in view of Equation (10), we obtain the following:

$$\begin{array}{cc}\hfill \sum _{ijabc}{\mathcal{B}}_{ab}^i{\mathcal{B}}_{ca}^j{\mathrm{R}}_{ijbc}=& m^2{\zeta }^2-\sum _a\mathrm{tr}{\zeta }_{a^{*}}^2-\sum _{ij}\left\{\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right\}\hfill \\ \hfill \le & m^2{\zeta }^2-\mathrm{tr}{\zeta }_{1^{*}}^2-\sum _{ij}\left\{\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right\}\hfill \\ \hfill =& m^2{\zeta }^2-m{\zeta }^2-\sum _{ij}\left\{\mathrm{tr}{\left({\zeta }_i{\zeta }_j\right)}^2-\mathrm{tr}\left({\zeta }_i^2{\zeta }_j^2\right)\right\}.\hfill \end{array}$$

(49)

Applying a similar argument as in Theorem 2, we deduce that

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}\mathrm{\Delta }\mathrm{\Pi }& \ge {\displaystyle \frac{1}{2}}m\mathrm{\Delta }{\zeta }^2+m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\mathrm{\Pi }\hfill \\ \hfill & -{\displaystyle \frac{3}{2}}{(\mathrm{\Pi }-m{\zeta }^2)}^2-m^2{\zeta }^4-2m^2{\zeta }^2+m{\zeta }^2.\hfill \end{array}$$

Since the boundary of ${\mathcal{N}}^m$ is compact, it follows from Stokes’ theorem that

$$\begin{array}{c}\hfill \int \left\{m\left({\displaystyle \frac{c}{4}}+{\zeta }^2\right)\mathrm{\Pi }-{\displaystyle \frac{3}{2}}{(\mathrm{\Pi }-m{\zeta }^2)}^2-m^2{\zeta }^4-2m^2{\zeta }^2+m{\zeta }^2\right\}\le 0\end{array}$$

which implies (48). This completes the proof of the theorem. □

Remark 1. 

It has been noted that Theorems 1–3 are extended versions of Theorems 1 and 2 in [29].

4. Conclusions

The study of rigidity theorems in space forms, particularly for totally real submanifolds, has been an area of limited exploration. This work extends previous studies on totally real submanifolds by investigating scalar curvature pinching conditions in quaternionic space forms. We derived refined inequalities involving the scalar curvature, mean curvature, and the second fundamental form. These results enhance our understanding of the rigidity and geometry of such submanifolds and provide a basis for further exploration of their topological and analytical properties.