Geometric analysis on warped product semi-slant submanifolds of a locally metallic Riemannian space form

1.   Introduction

A metallic structure is a polynomial structure as defined by Goldberg et al. in ^[1,2], with the structural polynomial $Q(J) = J^2-pJ-qI$. The positive solution of the equation

is named a member of the metallic means family ^[3,4,5], where $p, q$ are positive integers. These numbers are denoted by:

are also called $(p, q)$-metallic numbers.

The esteemed members of the metallic family are elegantly categorized as follows ^[6]:

(1) The golden structure $\sigma = \frac{1+\sqrt{5}}{2}$ for $p = q = 1$, entwined with the ratio of two consecutive classical Fibonacci numbers.

(2) The copper structure $\sigma_{1, 2} = 2$ with $p = 1$ and $q = 2$.

(3) The nickel structure $\sigma_{1, 3} = \frac{1+\sqrt{13}}{2}$ if $p = 1$ and $q = 3$.

(4) The silver structure $\sigma_{2, 1} = 1 + \sqrt{2}$ if $p = 2$ and $q = 1$, enchanted by the ratio of two consecutive Pell numbers.

(5) The bronze structure $\sigma_{3, 1} = \frac{3+\sqrt{13}}{2}$ with $p = 3$ and $q = 1$.

(6) The subtle structure $\sigma_{4, 1} = 2 + \sqrt{5}$ if $p = 4$ and $q = 1$, and so forth.

The members of the metallic means family share an important mathematical property that constitutes a bridge between mathematics and design; e.g., the silver mean has been used in describing fractal geometry ^[7]. Some members of the metallic means family (golden mean and silver mean) appeared already in sacred art of Egypt, Turkey, India, China, and other ancient civilizations ^[8]. The members of the metallic means family are closely related to quasiperiodic dynamics ^[9].

The notion of metallic structure on a Riemannian manifold was introduced in ^[6]. A polynomial structure on a manifold $M$ is called a metallic structure if it is determined by a $(1, 1)$ tensor field $J$, which satisfies the equation

where $p, q$ are positive integers and $I$ is the identity operator on the Lie algebra $\chi(M)$ of the vector fields on $M$. We say that a Riemannian metric $g$ is $J$-compatible if:

for all $X, Y\in \chi(M)$, which means that $J$ is a self-adjoint operator with respect to $g$. This condition is equivalent in our framework with:

A Riemannian manifold $(M, g)$ endowed with a metallic structure $J$ so that the Riemannian metric $g$ is $J$-compatible is named a metallic Riemannian manifolds and $(g, J)$ is called a metallic Riemannian structure on $M$.

A locally metallic Riemannian manifold $(\bar M, g, J)$ is a manifold that has a metallic Riemannian structure such that $J$ is parallel with respect to the Levi–Civita connection $\bar\nabla$ on $\bar M$, that is, $\bar\nabla J = 0$. Hence, we have:

Warped products can be seen as a natural generalization of Cartesian products with distance in one of the factors in skewed. This concept appeared in mathematics starting with Nash's studies ^[10], who proved an embedding theorem that states that every Riemannian manifold can be isometrically embedded into some Euclidean space. Also, Nash's theorem shows that every warped product can be embedded as a Riemannian submanifold in some Euclidean space. Hretcanu and Blaga worked on the existence problem of proper warped product bi-slant submanifolds in locally metallic Riemannian manifolds ^[11]. They investigated the existence of various types of warped products, including warped product CR submanifolds in locally metallic Riemannian manifolds. They proved that there is no proper CR warped product of the form $M_T\times_f M_\perp$ where $M_T$ and $M_\perp$ are invariant and anti-invariant submanifolds, respectively, in a locally metallic Riemannian manifold. In ^[12], Alqahtani et al. derived a relation for the squared norm of the second fundamental form in terms of the components of the gradient of the warping function for CR-submanifolds of the form $M_\perp\times_fM_T$.

In this paper we study warped product semi-slant submanifolds of locally metallic Riemannian manifolds of type $M_\theta\times_f M_T$, where $M_\theta$ and $M_T$ are proper slant and invariant submanifolds, respectively, of locally metallic Riemannian manifolds. We construct a non-trivial example of such type of submanifolds and establish, Chen-type inequality. Similar type inequalities can be found in the work of Mustafa et al. on warped product submanifolds in Kenmotsu manifold ^[13]; Ali et al. ^[14,15] studied on semi slant submanifolds on cosymplectic manifolds. Another work of Ali et al. ^[16] is worth mentioning on Kähler manifolds. Also, Li et al. ^[17,18] studied on generalized Sasakian space forms and in generalized complex space forms are good references for literature.

2.   Preliminaries

Let $\bar M$ be a smooth manifold of dimension $m$. The metallic structure $J$ is a $(1, 1)$ tensor field defined by the equation

where $p, q\in \mathbb N$ and $I$ is the identity operator on the space of all vector fields on $\bar M$, denoted by $\Gamma(TM)$. Let $M$ be an isometrically immersed submanifold in a metallic Riemannian manifold $(\bar M, \bar g, J)$. Let $T_x M$ be the tangent space of $M$ at a point $x\in M$, and $T^\perp_x M$ is the normal space of $M$ at $x$. The tangent space $T_x\bar M$ can be decomposed into the direct sum $T_x\bar M = T_xM \oplus T_x^\perp M$, for any $x\in M$. Let $i_*$ be the differential of the immersion $i: M\rightarrow \bar M$. Then the induced Riemannian metric $g$ on $M$ is given by $g(X, Y) = \bar g (i_*X, i_*Y)$, for any $X, Y\in \Gamma(TM)$. For the simplification of the notations, in the rest of the paper we shall denote by $X$ the vector field $i_*X$, for any $X\in \Gamma(TM)$.

Let $TX = (JX)^T$ and $NX = (JX)^\perp$, respectively, be the tangential and normal components of $JX$, for any $X\in \Gamma(TM)$ and $tV = (JV)^T, nV = (JV)^\perp$ be the tangential and normal components of $JV$, for any $V\in \Gamma(T^\perp M)$. Then we get:

for any $X\in \Gamma(TM), V\in\Gamma(T^\perp M)$.

The maps $T$ and $n$ are $\bar g$-symmetric:

and

for any $X, Y\in \Gamma(TM)$ and $U, V\in \Gamma(T^\perp M)$.

If $M$ is a submanifold in a metallic Riemannian manifold $(\bar M, \bar g, J)$, then:

for any $X\in \Gamma(TM)$ and $V\in \Gamma(T^\perp M)$.

If $p = q = 1$ then $M$ is said to be a submanifold of golden Riemannian manifold $(\bar M, \bar g, J)$.

Let $\bar \nabla$ and $\nabla$ be the Levi–Civita connections on $(\bar M, \bar g)$ and on its submanifold $(M, g)$, respectively. The Gauss and Weingarten formulas are given by:

for any $X, Y\in \Gamma (TM)$ and $V\in \Gamma(T^\perp M)$, where $h$ is the second fundamental form and $A_V$ is the shape operator, which are related by:

A $(1, 1)$-tensor field $F$ on an $m$-dimensional Riemannian manifold $(\bar M, g)$ is known as an almost product structure ^[19] if it satisfies $F^2 = I$ and $F\neq \pm I$. Furthermore, $(\bar M, g)$ is said to be an almost product Riemannian manifold when the almost product structure $F$ agrees with $g(FX, Y) = g(X, FY),$ for any $X, Y\in\Gamma(T\bar M)$.

By means of any metallic structure $J$ on $\bar M$ one obtains two almost product structures on $\bar M$ ^[6]

Also, any almost product structure $F$ on $\bar M$ determines two metallic structures

Let $(\bar M = M_1(c_1)\times M_2(c_2), F)$ be a locally Riemannian product manifold, where $M_1$ and $M_2$ have constant sectional curvatures $c_1$ and $c_2$, respectively. Then, the Riemannian curvature tensor $R$ of $\bar M = M_1(c_1)\times M_2(c_2)$ is

In view of the above and the expressions for $F_1$ and $F_2$, we achieve

Definition 2.1. A submanifold $M$ in a metallic Riemannian manifold $(\bar M, \bar g, J)$ is called a slant submanifold if the angle $\theta(X_x)$ between $JX_x$ and $T_xM$ is constant for any $x\in M$ and $X_x\in T_xM$. In such a case, $\theta = \theta(X_x)$ is called the slant angle of $M$ in $\bar M$ and it verifies,

The immersion $i:M\rightarrow \bar M$ is named the slant immersion of $M$ in $\bar M$.

Remark 2.1. The invariant and anti-invariant submanifolds in metallic Riemannian manifolds $(\bar M, \bar g, J)$ are particular cases of slant submanifolds with the slant angles $\theta = 0$ and $\theta = \frac{\pi}{2}$, respectively. A slant submanifold $M$ in $\bar M$, which is neither invariant nor anti-invariant, is called a proper slant submanifold, and the immersion $i$ is called a proper slant immersion.

Moreover, if $M$ is a slant submanifold of a metallic Riemannian manifold $(\bar M, g, J)$ with a slant angle $\theta$, the following relationships hold:

and

for all $X, Y\in \Gamma(TM)$.

Furthermore, we have the additional relation

Now, the warped product of two Riemannian manifolds $(M_1, g_1)$ and $(M_2, g_2)$ with warping function $f$ is denoted as $M_1\times_f M_2$ and is the product manifold $M_1\times M_2$ equipped with the metric $g = g_1+f^2g_2$.

3.   Example of semi-slant submanifolds of metallic Riemannian manifold

Let $M = \mathbb R^2$ be a submanifold of $\bar M = \mathbb R^4$ defined by the immersion $i$ as follows:

where $\sigma$ is the metallic number defined as $\sigma = \frac{p+\sqrt{p^2+4q}}{2}$. We also consider the metallic structure $J$ of $\bar M$ as:

where $\bar \sigma = \frac{p-\sqrt{p^2+4q}}{2}$ and $J^2 = pJ+qI$.

It is straightforward to compute that the tangent bundle of $M$ is spanned by the vectors $(Z_1, Z_2)$, where

Now, we define two vector spaces, $D_\theta$ and $D_T$, where $D_\theta = Span\{Z_1\}$ is the proper slant distribution with slant angle $\cos\theta = \frac{(\sigma+p)}{\sqrt {3}\sqrt{\sigma^2+p^2+2q}}$ and $D_T = Span\{Z_2\}$ is the invariant distribution, which are preserved by the action of $J$. Hence, the Riemannian metric of the warped product semi-slant submanifold $M$ is given by the following:

Thus, $M$ is a warped product semi-slant submanifold of type $M_\theta \times_f M_T$ with a warping function $f = \alpha_1$.

4.   Main results

Let $M$ be a Riemannian (or pseudo-Riemannian) manifold. The Ricci curvature $Ric$ and scalar curvature $\rho$ of $M$ are defined as:

where $K(E_q\wedge E_s)$ is the sectional curvature of the plane spanned by $E_q$ and $E_s$. We also have

where $G_k$ is the $k$-plane section of $TM$ spanned by the orthonormal basis $\{E_1, E_2, \cdots, E_k\}$.

In this section, we study warped product submanifolds of locally metallic Riemannian space form $\bar M(c)$ which is of the form $M_\theta\times_f M_T$, where $M_\theta$ and $M_T$ are proper slant and invariant submanifolds of $\bar M(c)$, respectively, with $dim\bar M = m, dim M = n, dim M_\theta = 2d_1 = n_1, dim M_T = 2d_2 = n_2$. We also assume $D_\theta$ and $D_T$ are the corresponding distributions.

We consider the basis of $D_\theta$ and $D_T$ as follows:

Let $i:M_\theta^{n_1}\times_fM_T^{n_2}$ be an isometric immersion of a warped product $M_\theta\times_fM_T$ into a Riemannian space form $\bar M(c)$, then

for unit vectors $X\in \Gamma(M_1)$ and $Z\in \Gamma(M_2)$.

If we consider the local orthonormal frame $\{e_1, e_2, \cdots, e_n\}$ such that $\{e_1, e_2, \cdots, e_{n_1}\}$ and $\{e_{n_1+1}, e_{n_1+2}, \cdots, e_n\}$ are tangents to $M_\theta$ and $M_T$, respectively, then we have

For the submanifold $M$, the Gauss equation is defined as

for any $U, V, Z, W\in \Gamma(TM)$, where $\bar R$ and $R$ are the curvature tensors on $\bar M$ and $M$, respectively.

Theorem 4.1. Let $i:M = M_\theta\times_fM_T\rightarrow \bar{M}$ be an $M_\theta$-minimal isometric immersion from an $n$-dimensional warped product semi-slant submanifold $M$ into an $m$-dimensional locally metallic product space form

Then

The equality holds if and only if $M_\theta$ is totally geodesic and $M_T$ is totally umbilical in $\bar M$.

Proof. The proof is similar to [14, Theorem 5.1]. □

Using the result of the above theorem, we construct a Chen-type inequality for warped product semi-slant submanifolds of locally metallic Riemannian manifolds as follows:

Theorem 4.2. Let $i:M = M_\theta\times_fM_T\rightarrow \bar{M}$ be an $M_\theta$-minimal isometric immersion from an $n$-dimensional warped product semi-slant submanifold $M$ into an $m$-dimensional locally metallic product space form

Then

where

The equality holds if and only if $M_\theta$ is totally geodesic and $M_T$ is totally umbilical in $\bar M$.

Proof. Now, for metallic Riemannian space form $\bar M = (\bar M_1(c_1)\times \bar M_2(c_2), g, \phi)$, we have

Putting $X = W = e_i$ and $Y = Z = e_j$ and taking summation over $1\leq i\neq j \leq n$, we obtain

taking summation over $1\leq i\neq j \leq n_1$, we obtain

taking summation over $n_1+1\leq i\neq j \leq n$, we obtain

we assume that,

So we have

□

5.   Application to Euler–Lagrange equation

In the potential theory, Dirichlet energies have significant use. If $f:M\rightarrow \mathbb R$ is a smooth function, then the Dirichlet energy is defined as:

where $E(f)$ and $dV$ are Dirichlet energy and volume element, respectively. In Lagrangian mechanics, the Lagrangian $L$ of a mechanical system is $T-V$, where $T$ is the kinetic energy and $V$ is the system's potential energy, respectively. As a generalization to smooth manifolds, the Lagrangian of the smooth function $f$, is determined by

The Euler–Lagrange equation for a Lagrangian $L$ is $\Delta f = 0$. Following are a few useful results that will be needed to prove the next results:

Lemma 5.1. (^[14]). Let $M$ be a compact, connected Riemannian manifold without boundary and $f$ be a smooth function on $M$ such that $\Delta f\geq 0(\Delta f\leq 0)$. Then $f$ is a constant function.

Theorem 5.1. Let $M = M_\theta\times_f M_T$ be a compact (without boundary) and connected warped product semi-slant submanifold in locally metallic Riemannian manifold $\bar M(c)$. If the warping function $f$ is a solution of the Euler–Lagrange equation, then $M$ is simply a Riemannian product with $||h||^2\geq \alpha$.

Proof. Suppose, $\Delta f = 0$, then we have $\Delta (ln f) = -\frac{1}{f^2}|\nabla f|^2\le 0.$ By Lemma 1 we have $f =$ constant, which proves that $M$ is simply a Riemannian product. Further, from Eq (4.1), we have $\Delta \ln f\le 0\implies ||h||^2\ge \alpha$. □

6.   Application to gradient Ricci soliton

Gradient Ricci solitons are important in understanding the Hamilton's Ricci flow. They are self-similar solutions of the Ricci flows and arise often as singularity models of the Ricci flow.

A complete $n$-dimensional Riemannian manifold $(M, g)$ is said to be a gradient shrinking Ricci soliton if there exists a smooth function $f$ on $M$ such that the equation

holds for some positive constant $\lambda\in \mathbb R$ ^[20], where $Ric$ is the Ricci tensor and $\nabla^2 f$ is the Hessian of the function $f$.

The function $f$ is called the potential function of the gradient-shrinking Ricci soliton. Similarly, a gradient Ricci soliton is called steady and expanding if the real number $\lambda$ is $0$ and negative, respectively.

We have

Now, if $M$ admits a shrinking gradient Ricci soliton, then

for

If $\lambda < 0$, then $||h||^2\geq \alpha+2n_2||\nabla lnf||^2+\sum_{t = 1}^n Ric(e_t, e_t).$ So we have the following:

Theorem 6.1. Let $M = M_\theta\times_fM_T$ be a warped product semi-slant submanifold in $\bar M(c)$ admitting a shrinking gradient Ricci soliton. Then

Proof. Proof follows from the above discussion. □

7.   Conclusions

In conclusion, this research has delved deeply into the examination of warped product semi-slant submanifolds situated within a locally metallic Riemannian manifold. Through this study, we have crafted a Chen type inequality that is uniquely suited for these submanifolds, offering a specialized insight into their geometric properties. Additionally, we have showcased a non-trivial example belonging to this specific class of submanifolds, underscoring the richness and diversity within this geometric framework. Our exploration extends beyond theoretical considerations, as we have uncovered a multitude of practical applications stemming from the implications of the derived inequality. This work not only contributes to the theoretical understanding of these submanifolds but also lays the groundwork for further research and applications in the broader field of geometric analysis.The results established in this paper can be extended to other classes of submanifolds of locally metallic Riemannian manifolds. One may also consider the applications of the above result to other soliton structures like, Ricci-Yamabe soliton, Ricci -Bourguignon soliton or Bach soliton etc. for further investigation.

Author contributions

Conceptualization, S.K.H and B.B.; methodology, B.B and T.P. (Assistant Teacher); validation, S.K.H., M.A.K. and T.P.; formal analysis, B.B. and T.P.; investigation, B.B. and S.K.H.; writing–original draft preparation, B.B. and M.A.K.; writing–review and editing, B.B. and M.A.K.; supervision, S.K.H.; project administration, S.K.H. and M.A.K. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

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

Conflict of interest

The authors declare no conflicts of interest.