Some rigidity theorems for totally real submanifolds in complex space forms

1.   Introduction

A key challenge in differential geometry is exploring the relationship between the geometry and topology of Riemannian manifolds. In submanifold theory, an exciting question is how the pinching conditions on intrinsic or extrinsic curvature invariants affect the geometry and topology of submanifolds in space forms. Simons first established a key result on minimal submanifolds of spheres with a sufficiently pinched second fundamental form in his seminal paper ^[18]. Later, Chern, et al. ^[6] proved a well-known rigidity theorem, which has since motivated numerous significant advances in the study of pinching conditions. The study of rigidity theorems is crucial in the theory of minimum submanifolds. Some pioneering work and substantial research on rigidity theorems for minimum submanifolds in spheres have been conducted by Lawson ^[8]. Let the square norm of the second fundamental form be represented by $\sigma$ and a unit sphere be represented by $\mathbb{S}^{n+m}$ with the codimension $m$. If a compact minimal submanifold $\mathcal{N}^n$ in $\mathbb{S}^{n+m}$ with the following pinching condition:

then either

and $\mathcal{N}$ is the Clifford hypersurface or the Veronese surface in $\mathbb{S}^4$. Later, Li ^[9] and Chen ^[5] improved the pinching number $\frac{n}{(2-1/m)}$ to $\frac{2n}{3}$. They showed that if

then either

and $\mathcal{N}$ is the Veronese surface in $\mathbb{S}^4.$

After initial motivation by Simons ^[18] and preliminary developments (for example, ^{[5,8,12,16,17]}), this topic has received much attention. These underlying works reveal, in particular, several similarities between the free boundary minimal surfaces in a Euclidean unit ball and closed minimal surfaces in the sphere. In this respect, the classical results and tactics for obtaining rigidity results, in conclusion, may indicate the direction of interest in exploring similar progress in the free boundary case. This study was motivated by the rigidity theorems for minimal submanifolds and submanifolds with parallel mean curvature in space forms see ^[9,11,19,20], etc.

On the other hand, the space forms are useful for understanding the geometric analysis. Several authors constructed the first eigenvalues for submanifolds in different space forms such as in $C$-totally real submanifolds in Sasakian space forms ^[1], Lagrangian submanifolds in complex space forms ^[2], slant submanifolds of Sasakian space form ^[13,15], semi-slant submanifolds of Sasakian space forms ^[14] and totally real submanifolds in generalized complex space forms ^[3] that contain a $p$-laplacian operator. It should be noted that little work has been done on the rigidity theorems for totally real submanifolds in space form geometry. Therefore, motivated by some previous work, we constructed the rigidity for a totally real submanifold in complex space form and discuss their consequences in the present paper.

2.   Basic formulas and definitions

Assume that $\mathcal{\widetilde{N}}^{n}$ is a complex space form of constant holomorphic sectional curvature $4\kappa,$ denoted $\mathcal{\widetilde{N}}^n(4\kappa)$. The curvature tensor $\mathrm{\widetilde{R}}$ of $\mathcal{\widetilde{N}}^n(4\kappa)$ can be expressed as:

for all $\mathrm{V}_1, \mathrm{V}_2, \mathrm{V}_3\in\Gamma(T\mathcal{\widetilde{N}})$. Based on the cases, $\kappa < 0, \, \, \kappa = 0, \, \ \text {and}\ \kappa > 0$, $\mathbb{\widetilde{N}}^n(4\kappa)$ is the complex hyperbolic space $\mathbb{C}H^n$, the complex Euclidean space $\mathbb{C}^n$ and the complex projective space $\mathbb{C}P^n$. We call an $m$-dimensional Riemannian submanifold $\mathcal{N}^m$ of $\mathbb{\widetilde{N}}^n(4\kappa)$ as totally real if the standard complex structure $\mathrm{J}$ of $\mathcal{\widetilde{N}}^n(4\kappa)$ maps any tangent space of $\mathcal{N}^m$ into its corresponding normal space ^[4].

We considered an orthonormal frame

in $\mathcal{\widetilde{N}}^{m+h}(4\kappa)$ restricted to $\mathcal{N}^m$, $e_1\cdots e_m$ is tangent to $\mathcal{N}^m$. We provided the indices as follows:

Let ${\Pi}$ denote the squared length of the second fundamental form $\zeta$ of $\mathcal{N}^m,$ which is defined by

Similarly, the mean curvature of $\mathcal{N}^m$ is calculated as:

If $\mathsf{H} = 0$ in (2.3), then $\mathcal{N}^m$ is minimal. From (2.1), we get the following equation for submanifold in complex space form:

where $\mathrm{\widetilde{K}}$ is the sectional curvature of $\mathbb{\widetilde{N}}^n(4\kappa)$. The curvature tensor of indices for the submanifold is

We define the Ricci curvature for a totally real submanifold:

From the above, we can establish some notation

Let us assume that $e_{m+1}$ is parallel to $\mathsf{H}$ in which case we have

where tr stands for the trace of the matrix $\mathsf{H}_\alpha = (\zeta_{ab}^\alpha).$ Taking account of (2.5) and (2.8), we have the scalar curvature as

where $\mathsf{H}$ stands for the mean curvature vector of $\mathcal{N}^m$. Since $\mathsf{H}$ is constant, it can be concluded that the scalar curvature $\mathrm{R}$ is constant if and only if ${\Pi}$ is constant by (2.9). Let $\zeta_{abc}^k$ denote the second covariant derivative of $\zeta_{ab}^k$ in which case we have

where $\{\omega_a\}$ is the dual frame of $\mathcal{N}^m$. Taking the exterior derivative of the equation (2.10), we obtain

Moreover, the Laplacian of $\zeta_{ab}^\alpha$ is

Lemma 2.1. ^[10] Let $\mathrm{T}_1, \cdots, \mathrm{T}_n$ be symmetric $(m\times m)$-matrices, in which case

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

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 2.2. Let us $\mathcal{T}_1, \mathcal{T}_2, \cdots\mathcal{T}_m(n\geq 2)$ be symmetric $(m\times m)$-matrices, in this case

We can now estimate our first main result, which is as follows.

Theorem 2.1. If the mean curvature vector of an $m$-dimensional compact totally real submanifold $\mathcal{N}^m$ in complex space form $\mathcal{\widetilde{N}}^{m+h}(4\kappa)$ is parallel and satisfies the following inequality

then $\mathcal{N}^m$ is a totally umbilical sphere $\mathbb{S}^m\Big(\frac{1}{\sqrt{\kappa+\mathsf{H}^2}}\Big)$, where $\mathsf{H}$ denotes the mean curvature of $\mathcal{N}^m$.

Proof. Assume that $\mathcal{N}^m$ is a totally real submanifold of complex space form $\mathcal{\widetilde{N}}^{m+h}(4\kappa)$ with the parallel mean curvature vector $\mathsf{H}$. Consider an $e_{m+1}$ that it is parallel to $\mathsf{H}$ and

We assume that the mean curvature vector $\mathsf{H}$ is parallel, so we have

where $\mathrm{D}$ is a Levi-Civita connection. From the structure equation and (2.16), we derive

If we consider, from (2.12), that $\mathcal{N}^m$ has as parallel mean curvature vector and $\sum_{c}\mathsf{H}_{ccab}^\alpha = 0,$ we can derive

Let $\mathrm{R}_{\mathcal{N}}(p, \pi)$ the represent the sectional curvature of $\mathcal{N}^m$ for the $2$-plane $\pi\subset T_p\mathcal{N}$ at the point $p\in \mathcal{N}^m$. Then set

Therefore, the orthonormal fields are $\{e_i\}$ such that $\zeta^{m+1}_{ab} = \lambda_{i}\delta_{ab}$, where $\lambda_{i}$ represents the eigenvalues; hence, we get

Taking (2.18) and (2.19), we have

It follows from $\mathrm{R}_{\mathcal{N}}\geq \frac{m+2h-1}{2(m+2h)}(\kappa+\mathsf{H}^2)$ and the lemma of Hopf that $\Pi_{\mathsf{H}}$ is a constant ^[21], and we derive

It is implied that $\lambda_a = \lambda_b$. In this case, $\mathcal{N}^m$ is pseudo-umbilical. Again, from (2.12), $\sum_{c}\mathsf{H}_{ccab}^\alpha = 0$ and mean curvature of $\mathcal{N}^m$ is parallel; we can constructs

where $\tau$ is the scalar curvature of $\mathcal{N}^m$. From (2.5) and (2.15), we get

Again (2.5), we derive

Inserting (2.24) and (2.23) into (2.22), we obtain

For a fixed $\alpha$, we choose the orthonormal frame field $\{e_a\}$ such that $\zeta_{ab}^\alpha = \lambda_a^\alpha\delta_{ab}$. From (2.15), we get

which implies that

In the implementation of DDVV (a conjecture for the Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken), demonstrated by the article ^[7], inequality of Lemma 2.2, we construct the following:

On the other hand, we have

Setting $a^\prime = \frac{m+2h-1}{m+2h+1}$ in (2.25), and combining (2.26)–(2.28), we derive

If our assumption (2.14) is satisfied, then we have

by Hopf's lemma. This concludes that $\Delta\tau = 0$. Hence, we get the following:

For the first case, $\tau = 0$, and thus $\mathcal{N}^m$ is totally umbilical. For the second case, on the basis of (2.5), we derive

and conclude that $\mathcal{N}^m$ is a totally umbilical sphere

Moreover, all inequalities (2.26)–(2.29) changed to equalities if

Now, we will show that the second case can not occur. For this, we consider the equality (2.27) implies that either all $\mathsf{H}_\alpha's$ are zero or two of the $\mathsf{H}_\alpha's$ are nonzero $\alpha\not = m+1$. We estimate the following if the inequality in (2.28) and (2.29) converts into equalities:

Thus $\mathcal{N}^m$ is totally umbilical with $\mathrm{R}_{abab} = \kappa+\mathsf{H}^2,$ as the $\mathsf{H}^{'s}_\alpha$ are zero ($\alpha\not = m+1).$ This leads to a contradiction. This completes the proof of the theorem. □

In the results below, we have the following:

Theorem 2.2. Let $\mathrm{J}\xi$ be normal to an $(m\geq2)$-dimensional totally real submanifold $\mathcal{N}^m$ in the complex space form $\mathcal{\widetilde{N}}^{m+h}(4\kappa)$. Then either $\mathcal{N}^m$ is totally umbilical or it satisfies the inequality

where the scalar curvature, the mean curvature, and the mean curvature vector are represented by $\rho, \mathsf{H},$ and $\xi$, respectively.

Proof. Let $\mathrm{J}\xi$ be normal to $\mathcal{N}^m$. We can consider $e_{m+1}$ is parallel to $\xi,$ and we have

From (2.12), we have

From (2.5), (2.31), and if $\mathcal{N}^m$ is totally umbilical, we obtain

In view of (2.31), and the pseudo-umbilical condition such that $\zeta_{ab}^{m+1} = \mathsf{H}\delta_{ab}$, we derive

By Lemma 2.2 and pseudo-umbilical condition $\zeta_{ab}^{m+1} = \mathsf{H}\delta_{ab}$, we have

Substituting (2.33)–(2.38) into (2.32), we have

By the same argument as in ^[16], we conclude that either $M^n$ is totally umbilical or

This completes the proof of the theorem. □

Theorem 2.3. Let $\mathrm{J}\xi$ be normal to an $(m\geq2)$-dimensional compact totally real submanifold $\mathcal{N}^m$ in the complex space form $\mathcal{\widetilde{N}}^{m+h}(4\kappa)$. Then we have the inequality

where $\mathsf{H}$ and ${\Pi}$ denote the mean curvature of $\mathcal{N}^m$ and the squared norm of the length of the second fundamental form of $\mathcal{N}^m$, respectively.

Proof. Without loss of generality, we consider $e_{1^*}$ such that it is parallel to $\xi$ and $tr\mathsf{H}_{1^*} = m\mathsf{H}.$ In this case $tr\mathsf{H}_\alpha = 0,$ for $\alpha\not = 1^*,$ and $\mathrm{J}\xi$ is normal to $\mathcal{N}^m.$ Taking this together with (2.5), we have

By the same argument as in Theorem 2.2, we conclude that

The boundary of $\mathcal{N}^m$ is compact, by Stokes's theorem, we obtain

which implies (2.40). This completes the proof of the theorem. □

Remark 2.1. Note that if $\kappa = 1$ in (2.36), then the complex space form $\mathcal{\widetilde{N}}^{m+h}(4\kappa)$ turns to a complex projective space with constant curvature of $1.$

From the hypothesis, Theorem 2.1, we rewrite that

Theorem 2.4. Let $\mathcal{N}^m$ be an $m$-dimensional compact totally real submanifold in complex projective space $\mathbb{C}P^{m+h}$ with parallel mean curvature. In this case, we have

and thus $\mathcal{N}^m$ is a totally umbilical sphere $\mathbb{S}^m\bigg(\frac{1}{\sqrt{1+\mathsf{H}^2}}\bigg)$, where $\mathsf{H}$ and ${\Pi}$ denote the mean curvature of $\mathcal{N}^m$ and the squared norm of the length of the second fundamental form of $\mathcal{N}^m$, respectively. Moreover, $\mathbb{C}P^{m+h}$ has the constant sectional curvature 1.

Theorem 2.5. Let $\mathcal{N}^m$ be an $(m\geq2)$-dimensional totally real submanifold in complex projective spaces $\mathbb{C}P^{m+h}$. If $\mathrm{J}\xi$ is normal to $\mathcal{N}^m$, then either $\mathcal{N}^m$ is totally umbilical or satisfies the following inequality:

Similarly, Theorem 2.3 can be written as if the mean curvature is minimal from Theorem 2.4

Corollary 2.1. Let $\mathcal{N}^m$ be an $(m\geq2)$-dimensional compact totally real submanifold in complex projective spaces $\mathbb{C}P^{m+h}$. If $\mathrm{J}\xi$ is normal to $\mathcal{N}^m$, then we have the following inequality:

Remark 2.2. Using Remark 2.1 in Theorems 2.4 and 2.5, then Theorems 2.4 and 2.5 coincided with Theorems 1 and 2 in ^[21].

3.   Conclusions

The study of totally real submanifolds in complex space forms is a rich area in differential geometry, with deep connections to complex geometry, curvature theory, and minimal submanifold theory. Their extrinsic curvature properties (like second fundamental form, mean curvature) are deeply influenced by the complex structure of the ambient space. This leads to classification results that help understand the geometric landscape of complex manifolds. In theoretical physics (e.g., string theory), totally real submanifolds relate to real slices of complexified spaces. In all these, totally real submanifolds offer tools to probe the nature of curvature, symmetry, and submanifold geometry.

Author contributions

Fatimah Alghamdi: Conceptualization, Investigation; Fatemah Mofarreh: Conceptualization, Methodology, Writing-review and editing Writing-original draft preparation, Investigation, Funding acquisition; Akram Ali: Conceptualization, Methodology, Investigation, Writing-review and editing; Mohamed Lemine Bouleryah: Methodology, Writing-original draft preparation, Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgment

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through a Large Research Project under grant number R.G.P.2/22/45. The author, Fatemah Mofarreh, expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no competing interests.