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1.   Introduction

Totally real submanifolds are one of the typical classes of submanifolds of Kaehler manifold. In 1974, B. Y. Chen and K. Ogiue ^[10] started the study of the totally real submanifolds from the point of view of their curvatures. Due to its geometrical importance, many geometers studied totally real submanifolds from the different point of views and various results were obtained in different ambient spaces ^[6,9,13,18]. Kaehler product manifold also attracts the attention of geometers toward itself ^[24]. S. Y. Cheng and S. T. Yau ^[11] obtained many well-known results introducing a self-adjoint differential operator $\Box$ defined by

where $f \in C^{2}(\mathcal{N}), (f_{ij})$ is Hessian of $f$, $H$ mean curvature, and $\zeta_{ij}^{n+1}$ is the coefficients of second fundamental form $\zeta$.

Using this differential operator H. Li ^[15] obtained a rigidity result for hypersurfaces in space forms with constant scalar curvature. In 2013, X. Gua and H. Li ^[17] extended the use of the operator for submanifolds and obtained interesting results for submanifolds with constant scalar curvature in a unit sphere.

Motivated by X. Gua and H. Li, we study the totally real submanifolds of Kaehler product manifold with constant scalar curvature using self-adjoint differential operator $\Box$ and obtain a characterization result.

Further, we study $\delta-$invariant totally real submanifolds in same setting and prove some results.

2.   Preliminaries

Let $(\overline{\mathcal{N}}^{m}, J_{m}, g_{m})$ and $(\overline{\mathcal{N}}^{p}, J_{p}, g_{p})$ are Kaehler manifolds of complex dimension $m$ and complex dimension $p$ respectively. Let $J_{m}$ and $g_{m}$ be almost complex structure and metric tensor on $\overline{\mathcal{N}}^{m}$ respectively and $J_{p}$ and $g_{p}$ almost complex structure and metric tensor on $\overline{\mathcal{N}}^{p}$ respectively. Further, let us assume $\overline{\mathcal{N}}^{m}(c_{1})$ and $\overline{\mathcal{N}}^{p}(c_{2})$ are complex space forms with constant holomorphic sectional curvatures $c_{1}$ and $c_{2}$ respectively.

We suppose $\overline{\mathcal{N}}(c_{1}, c_{2}) = \overline{\mathcal{N}}^{m}(c_{1}) \times \overline{\mathcal{N}}^{p}(c_{2})$ the Kaehlerian product manifold with complex dimension $(m + p)$. Let us denote by $\mathcal{P}$ and $\mathcal{Q}$ the projection operators of the tangent space of $\overline{\mathcal{N}}(c_{1}, c_{2})$ to the tangent spaces of $\overline{\mathcal{N}}^{m}(c_{1})$ and $\overline{\mathcal{N}}^{p}(c_{2})$ respectively. Then,

By setting $\digamma = \mathcal{P} - \mathcal{Q}$, it can be easily shown that $\digamma ^{2} = I$. Thus, $\digamma$ is an almost product structure on $\overline{\mathcal{N}}(c_{1}, c_{2})$. Moreover, for a Riemannian metric $g$ on $\overline{\mathcal{N}}(c_{1}, c_{2})$ we have ^[20]

for all vector fields $E, F$ on $\overline{\mathcal{N}}(c_{1}, c_{2})$. We also have

If we assume $JE = J_{m}\mathcal{P}E + J_{p}\mathcal{Q}E$ for any vector field $E$ of $\overline{\mathcal{N}}(c_{1}, c_{2})$. Then from ^[20], we see that

Therefore, $J$ is a Kaehlerian structure on $\overline{\mathcal{N}}(c_{1}, c_{2})$. Let $\overline{R}$ be the Riemannian curvature tensor of a Kaehler product manifold $\overline{\mathcal{N}}(c_{1}, c_{2})$. Then ^[24]

for any vector fields $E, F$ and $G$ on $\overline{\mathcal{N}}(c_{1}, c_{2})$.

Definition 2.1. Let $\mathcal{N}$ be a real $n-$dimensional Riemannian manifold isometrically immersed in a $(m + p)-$ dimensional Kaehlerian product manifold $\overline{\mathcal{N}}(c_{1}, c_{2})$. Then $\mathcal{N}$ is said to be totally real submanifold of $\overline{\mathcal{N}}(c_{1}, c_{2})$ if $JT_{x}(\mathcal{N}) \bot T_{x}(\mathcal{N})$ for each $x \in \mathcal{N}$ where $T_{x}(\mathcal{N})$ denotes the tangent space to $\mathcal{N}$ at $x \in \mathcal{N}$.

Let $g$ be the metric tensor field on $\overline{\mathcal{N}}(c_{1}, c_{2})$ as well as that induced on $\mathcal{N}$. Also, we denote by $\overline{\nabla}$(resp. $\nabla$) the Levi-Civita connection on $\overline{\mathcal{N}}(c_{1}, c_{2})$(resp. $\mathcal{N}$). Then the Gauss and Weingarten formulas are given by

for all $E, F$ tangent to $\mathcal{N}$ and vector field $N$ normal to $\mathcal{N}$, where $\zeta, \nabla_{E}^{\perp}$ and $\Lambda_{N}$ denote the second fundamental form, normal connection and shape operator respectively. The relation between the second fundamental form and the shape operator is given by

We choose a local field of orthonormal frames $e_{1}, \dots, e_{n}; e_{n+1}, \dots, e_{m+p}; e_{1^{\star}} = Je_{1}, \dots, e_{n^{\star}} = Je_{n};$$e_{(n+1)^{\star}} = Je_{n+1}; e_{(m+p)^{\star}} = Je_{m+p}$ in $\overline{\mathcal{N}}(c_{1}, c_{2})$ in such a way that restricted to $\mathcal{N}$, the vectors $e_{1}, \dots, e_{n}$ are tangent to $\mathcal{N}$. With respect to this frame field of $\overline{\mathcal{N}}(c_{1}, c_{2})$, let $\omega^{1}, \dots, \omega^{n}; \omega^{n+1}, \dots, \omega^{m+p}; \omega^{1^{\star}}, \dots, \omega^{n^{\star}}; \omega^{(n+1)^{\star}}, \dots, \omega^{(m+p)^{\star}}$ be the field of dual frames. Unless otherwise stated, we use the following conventions over the range of indices:

Then the mean curvature vector $H$ is defined as

Also, the structure equations of $\overline{\mathcal{N}}(c_{1}, c_{2})$ are given by ^[24]

Restricting these forms to $\mathcal{N}$, we have

Since $0 = d\omega^{\alpha} = - \omega_{i}^{\alpha} \wedge \omega^{i},$ by Cartan's Lemma we get

From (2.6) and (2.11) we find

The covariant derivative of $\zeta_{ij}^{\alpha}$ is given by

The Laplacian $\Delta \zeta_{ij}^{\alpha}$ of $\zeta_{ij}^{\alpha}$ is defined as

where we have put $\zeta_{ijkl}^{\alpha}\omega^{l} = d\zeta_{ijk}^{\alpha} - \zeta_{ljk}^{\alpha}\omega_{i}^{l} - \zeta_{ilk}^{\alpha}\omega_{j}^{l}- \zeta_{ijk}^{\beta}\omega_{\beta}^{\alpha}.$

Now, from ^[17] we have a trace-free linear map $\Theta^{\alpha} : T_{x}\mathcal{N} \rightarrow T_{x}\mathcal{N}$ given by

where $x \in \mathcal{N}$ and the shape operator $\Lambda^{\alpha}$ of $e_{\alpha}$ is given by

and $\Theta$ is a bilinear map $\Theta : T_{x}\mathcal{N} \times T_{x}\mathcal{N} \rightarrow T_{x}^{\perp}\mathcal{N}$ defined by

Then we have $|\Theta|^{2} = |\Lambda|^{2} - nH^{2}$, where $H^{2} = \sum_{\alpha}(H^{\alpha})^{2}$.

The Gauss equation is given by

From (2.1) and Gauss equation, we obtain

where $\tau$ is scalar curvature.

The Codazzi and the Ricci equation are respectively

Then, by using Codazzi equation one can easily see that the operator $\Box$ is self-adjoint. That is

Since we have constant scalar curvature, Eq (2.18) implies that

We can choose a unit normal vector field $e_{n+1}$ which is parallel to $H$. Hence we have ^[16]

Now, we quote the following lemmas for later use.

Lemma 2.2. ^[19] Let $B : R^{n} \rightarrow R^{n}$ be a symmetric linear map such that $trB = 0$, then

where $|B|^{2} = trB^{2}$, and the equality holds if and only if at least $n-1$ eigenvalues of $B$ are equal.

Lemma 2.3. ^[21] Let $C, B : R^{n} \rightarrow R^{n}$ be a symmetric linear map such that $\big[C, B\big] = 0$ and $trC = trB = 0$, then

Lemma 2.4. ^[14] Let $B^{1}, B^{2}, \dots, B^{m}$, be symmetric $(n \times n)-$matrices.

Set $S_{\alpha \beta} = tr(B^{\alpha}B^{\beta}), S_{\alpha} = S_{\alpha \alpha}, S = \sum_{\alpha}S_{\alpha}$, then

3.   Main Theorem

This section is devoted to the proof of main result.

Theorem 3.1. Let $\mathcal{N}^{n}$ be a totally real submanifold in Kaehlerian product manifold $\overline{\mathcal{N}}(c_{1}, c_{2}) = \overline{\mathcal{N}}_{1}^{m}(c_{1}) \times \overline{\mathcal{N}}_{2}^{p}(c_{2})$, $c_{1}, c_{2} > 0,$ with constant scalar curvature. If $tr\digamma$ vanishes, then $\mathcal{N}$ is totally geodesic.

For proving that result, we need to prove the following preliminary Lemmas. Since $\digamma$ is symmetric and $J$ is skew-symmetric, following result is obvious.

Lemma 3.2. Let $\mathcal{N}$ be a totally real submanifold in Kaehler product manifold

$\overline{\mathcal{N}}(c_{1}, c_{2}) = \overline{\mathcal{N}}_{1}^{m}(c_{1}) \times \overline{\mathcal{N}}_{2}^{p}(c_{2})$, then $tr \digamma J = tr J\digamma = 0$.

Lemma 3.3. Let $\mathcal{N}$ be a totally real submanifold in Kaehler product manifold

$\overline{\mathcal{N}}(c_{1}, c_{2}) = \overline{\mathcal{N}}_{1}^{m}(c_{1}) \times \overline{\mathcal{N}}_{2}^{p}(c_{2})$, then

Proof. From ^[12], we have

On the other hand, one has

By using Eqs (2.5), (2.20) and (3.3) in (3.2), we obtain

Using (2.1) and Lemma 3.2, we now compute the values of curvature terms involving $\overline{R}$ of the Eq (3.4) as follows:

Similarly we obtain,

and

Thus, making use of Eqs (3.5)–(3.9) in (3.4), we get (3.1).

Proof of Theorem 3.1. From (1.1) and (2.22), we obtain

Now, using (3.1) in the above equation, we get

which implies

A direct computation gives

Moreover, it is easy to see that

and

where $\zeta_{ij}^{\star} = g(\zeta(e_{i}, e_{j}), e_{t^{\star}})$ and $\zeta_{ij} = \overline{\zeta}_{ij} \oplus \zeta_{ij}^{\star}$.

Also we have

Using Lemma 2.4 in (3.15), we get

Taking into account the Eq (2.24), we derive

Taking Lemma 2.2 and Eq (3.17) into account, we have

Which by virtue of Lemma 2.3 and (3.18), yields

Now, substituting (3.12)–(3.14), (3.16) and (3.19) in (3.11), we find

It is known that ^[17],

Therefore, from (3.20) we have

Since, $c_{1}, c_{2} > 0$ and $tr\digamma = 0$. Then the above inequality implies the following inequality

From (2.21) we have $\int_{\mathcal{N}}\Box(nH) dv = 0$. Thus we have following two cases:

Case 1:

which yields $\Lambda = 0$ and $H = 0$. Thus, the submanifold is totally geodesic.

Case 2:

which implies that $\Lambda = 0$ and $H = 0$. It is again totally geodesic.

Hence, we have our assertion.

Now, we give an example in the support of the Theorem 3.1.

Example. It is known that the real projective space $\mathbb{R}P^{n}(1)$ is totally geodesic submanifold of the complex projective space $\mathbb{C}P^{n}(4)$ ^[3]. Also from ^[23] we know that, if $N_{1}$ is any submanifolds of Kaehler manifold $M_{1}$ and $N_{2}$ is any submanifold of Kaehler manifold $M_{2}$, then the natural product $N = N_{1} \times N_{2}$ is a submanifold of the Kaehler product manifold $M = M_{1} \times M_{2}$. Hence, $\mathbb{R}P^{n}(1) \times \mathbb{R}P^{n}(1)$ is a submanifolds of the Kaehler product manifold $\mathbb{C}P^{n}(4)\times \mathbb{C}P^{n}(4)$, which satisfies all the hypothesis of the Theorem 3.1 and indeed totally geodesic.

Remark 3.4. In the above example, it can be noticed that $tr\digamma$ vanishes, due to the fact that the projection operators $\mathcal{P}$ and $\mathcal{Q}$ coincide.

4.   $\delta-$invariant totally real submanifold in Kaehler product manifold

Let $\mathcal{N}$ be a Riemannian manifold and $K(\pi)$ denotes the sectional curvature of $\mathcal{N}$ of the plane section $\pi \subset T_{x}\mathcal{N}$ at a point $x\in \mathcal{N}$. Let $\{e_{1}, \dots, e_{n}\}$ and $\{e_{n+1}, \dots, e_{2(m+p)}\}$ be the orthonormal basis of $T_{x}\mathcal{N}$ and $T_{x}^{\perp}\mathcal{N}$ at any $x \in \mathcal{N}$, then the scalar curvature $\tau$ at that point is given by

If we consider that $L$ is an $r$-dimensional subspace of $T\mathcal{N}$, $r \geq 2$, and $\{e_{1}, e_{2}, \dots, e_{r}\}$ is an orthonormal basis of $L$. Then the scalar curvature of the $r$-plane section $L$ is given as

for $n \geq 3$ and $k \geq 1$. Let us assume $\mathfrak{S}(n, k)$ the finite set consisting of $k-$tuples $(n_{1}, \dots, n_{k})$ of integers satisfying

Also denote by $\mathfrak{S}(n)$ the union $\bigcup_{k \geq 1}\mathfrak{S}(n, k)$.

For each $(n_{1}, \dots, n_{k}) \in \mathfrak{S}(n)$ and each point $x \in \mathcal{N}$, B. Y. Chen ^[8] introduced a Riemannian invariant $\delta(n_{1}, \dots, n_{k})(x)$ defined by

where $L_{1}, \dots, L_{k}$ run over all $k$ mutually orthogonal subspaces of $T_{x}\mathcal{N}$ such that $dim L_{j} = \sum n_{j}$, $j = 1, \dots, k.$

We recall the following Lemma ^[7]:

Lemma 4.1. Let $a_{1}, \dots, a_{n}, a_{n+1}$ be $n + 1$ real numbers such that

Then $2a_{1}a_{2} \geq a_{n+1}$, with equality holding if and only if $a_{1} + a_{2} = a_{3} = \dots = a_{n}$.

In this section we state and prove the following.

Theorem 4.2. Let $\mathcal{N}$ be a totally real submanifold in Kaehler product manifold $\overline{\mathcal{N}}(c_{1}, c_{2}) = \overline{\mathcal{N}}_{1}^{m}(c_{1}) \times \overline{\mathcal{N}}_{2}^{p}(c_{2})$ and if $tr\mathcal{P}$ coincides with $tr\mathcal{Q}$, then

and the equality holds in $(4.4)$ if and only if at a point $x \in \mathcal{N}$ there exists an orthonormal basis $e_{1}, \dots, e_{2(m+p)}$ at $x$ such that the shape operator of $\mathcal{N}$ in $\overline{\mathcal{N}}(c_{1}, c_{2})$ at $x$ takes the forms:

where O is a null matrix, $I$ is an identity matrix and each $\Lambda_{j}^{r}$ is a symmetric $n_{j} \times n_{j}$ submatrix such that

Proof. We put

By combining (2.18) and (4.7), we obtain

With respect to the orthonormal basis, the last equation can be written as

which implies that

Now, let us set

and

Then Eq (4.9) is equivalent to

where $\alpha_{2}, \beta_{2} \in \Delta_{2}, \dots, \alpha_{k}, \beta_{k} \in \Delta_{k}$.

Using Lemma 4.1 in (4.10) yields

where $\alpha_{j}, \beta_{j} \in \Delta_{j}, j = 1, \dots, k.$

Furthermore, combining (4.1) with the Gauss equation, we obtain

Combining (4.11) and (4.12) gives

where $\Delta = \Delta_{1} \cup \dots \cup \Delta_{k}, \Delta^{2} = (\Delta_{1} \times \Delta_{1}) \cup \dots \cup (\Delta_{k} \times \Delta_{k}).$

Thus, Eqs (4.2), (4.7) and (4.13) imply (4.4).

Moreover, equality in (4.11) and (4.13) holds at a point $x$, if it holds for (4.4) at a point $x$. In this case from Lemma 4.1 and Eqs (4.10)–(4.13), we have (4.5) and (4.6). A straightforward computation yields the converse part.

Example. Due to the fact that the real hyperbolic space $H^{n}(1)$ can be isometrically embedded in the complex hyperbolic space $\mathbb{C}H^{n}(-4)$ as a totally real totally geodesic submanifold of minimal codimension ^[22]. It follows that $N = RP^{n}(1)\times RP^{n}(1)$ is a totally real submanifold of $M = HP^{n}(4)\times HP^{n}(4)$. This submanifold satisfies all hypotheses of Theorem 4.2. In this case the inequality is satisfied with equality at all points.

Theorem 4.2 yields the following obstruction result.

Corollary 4.3. Let $\mathcal{N}$ be a totally real submanifold in Kaehler product manifold $\overline{\mathcal{N}}(c_{1}, c_{2}) = \overline{\mathcal{N}}_{1}^{n}(c_{1}) \times \overline{\mathcal{N}}_{2}^{p}(c_{2})$ and if $tr\digamma$ vanishes, then for $c_{1} + c_{2} = 0$, $\mathcal{N}$ can not be minimally immersed in $\overline{\mathcal{N}}(c_{1}, c_{2}).$

5.   Conclusions

We characterized totally real submanifold using self-adjoint differential operator. The self-adjoint differential operators are mainly used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Drac-Von Neumann formulation of quantum mechanics in which momentum, angular momentum and spin are represented by self-adjoint operators on Hilbert space. A self-adjoint differential operator is an important class of unbounded operators. Therefore, we can use such operator for infinite dimensional cases and we resemble the finite dimensional case. Thus, use of the operator for such characterization may open a new path to link results in differential geometry with quantum mechanics as well as well with functional analysis.

Acknowledgments

This work is supported by Taif University Researchers Supporting Project number (TURSP-2020/223), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare that there is no conflict of interests.