Inequalities for the Generalized Normalized δ-Casorati Curvatures of Submanifolds in Golden Riemannian Manifolds

Abstract

In the present article, we consider submanifolds in golden Riemannian manifolds with constant golden sectional curvature. On such submanifolds, we prove geometric inequalities for the Casorati curvatures. The submanifolds meeting the equality cases are also described.

1. Introduction

In [1], the authors began the study of polynomial structures on a manifold. The golden structure, in particular, was defined (see [2]). In [3], invariant submanifolds in a Riemannian manifold endowed with a golden structure were investigated, and the integrability of such structures was studied [4]. In [5], the authors recently defined and explored the golden sectional curvature and the idea of locally decomposable golden Riemannian manifolds with constant golden sectional curvature was introduced. Additionally, they looked into the submanifolds’ geometry in such manifolds. The idea of a metallic structure was developed by [3], thereby extending the golden structure and [6,7] exploring the curvature of metallic Riemannian manifolds.

Identifying the optimal connections between intrinsic and extrinsic invariants of a Riemannian submanifold is one of the main challenges associated with the geometry of submanifolds. In this context, Chen [8,9] defined the $\delta$-invariants, also referred to as Chen invariants, of Riemannian manifolds. Using the Chen invariants and the mean curvature, the primary extrinsic invariant for Riemannian submanifolds, he established sharp inequality relations that are known as Chen inequalities. Chen invariants and Chen inequalities for various submanifolds in different ambient spaces have been intensively investigated (see [10,11,12,13,14,15,16,17], etc.).

On the other hand, the well-known Gauss curvature can be replaced by the Casorati curvature, due to F. Casorati [18]. In the context of the Casorati curvatures for submanifolds in various ambient spaces, one has established geometric inequalities. F. Casorati motivated the introduction of this curvature by the fact that the Casorati curvature goes away if and only if both principal curvatures of a surface in ${\mathbb{E}}^3$ are zero. This corresponds better to the common intuition of curvature. The Casorati curvatures have been studied by many researchers to obtain sharp inequalities on certain submanifolds in different ambient spaces (see [19,20,21,22,23], etc.).

In the current work, we give sharp inequalities for the generalized normalized $\delta$-Casorati curvatures of submanifolds in locally decomposable golden Riemannian manifolds with constant golden sectional curvature.

We also identify the submanifolds on which the equality signs occur.

2. Preliminaries

2.1. Riemannian Invariants

Let $(\overline{N},g)$ be a Riemannian manifold and N a Riemannian submanifold isometrically immersed in $\overline{N}$. Let $\overline{\nabla }$ and ∇ be the Levi-Civita connections on $\overline{N}$ and N, respectively, and h be the second fundamental form of N.

Then we can recall the Gauss and Weingarten formulae

$$\begin{array}{c}\hfill {\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\end{array}$$

$$\begin{array}{c}\hfill {\overline{\nabla }}_X\xi =-S_{\xi }X+{\nabla }_X^{\perp }\xi ,\end{array}$$

for all $X,Y\in \Gamma \left(TN\right),\xi \in \Gamma \left(T^{\perp }N\right)$; $S_{\xi }$ means the shape operator of N associated to $\xi$ and ${\nabla }^{\perp }$ represents the connection in the normal bundle.

Next, we recall the relation between $S_{\xi }$ and h

$$\begin{array}{c}\hfill g(S_{\xi }X,Y)=g(h(X,Y),\xi ).\end{array}$$

The Gauss equation may be expressed by

$$\begin{array}{c}\hfill \overline{R}(X,Y,Z,W)=R(X,Y,Z,W)-g(h(X,W),h(Y,Z))+g(h(X,Z),h(Y,W)),\end{array}$$

(1)

for any vector fields $X,Y,Z,W$ tangent to N.

Let $\{E_1,\dots ,E_n\}$ be a local orthonormal tangent frame and $\{E_{n+1},\dots ,E_m\}$ a local orthonormal normal frame. The scalar curvature is defined by

$$\begin{array}{c}\hfill \tau =\sum _{1\le i<j\le n}R(E_i,E_j,E_j,E_i),\end{array}$$

and the normalized scalar curvature $\rho$ by

$$\begin{array}{c}\hfill \rho =\frac{2\tau }{n(n-1)}.\end{array}$$

The mean curvature vector H of N is

$$\begin{array}{c}\hfill H=\frac{1}{n}\sum _{i=1}^{n}h(E_i,E_i).\end{array}$$

The components of h are

$$h_{ij}^r=g(h(E_i,E_j),E_r),\forall i,j\in \{1,\dots ,n\},\forall r\in \{n+1,\dots ,m\}.$$

Then

$$\begin{array}{c}\hfill {\left|\right|H\left|\right|}^2=\frac{1}{n^2}\sum _{r=n+1}^m{\left(\sum _{i=1}^n{h}_{ii}^r\right)}^2\end{array}$$

and

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2=\sum _{r=n+1}^m\sum _{i,j=1}^n{\left(h_{ij}^r\right)}^2.\end{array}$$

The Casorati curvature $\mathcal{C}$ of N is defined by

$$\begin{array}{c}\hfill \mathcal{C}=\frac{1}{n}{\left|\right|h\left|\right|}^2.\end{array}$$

Assume $p\in N$ and L is a t-dimensional subspace of $T_{p}N$, $t\ge 2$. For an orthonormal basis $\{E_1,\dots ,E_t\}$, the scalar curvature of L can be written as

$$\begin{array}{c}\hfill \tau \left(L\right)=\sum _{1\le i<j\le t}R(E_i,E_j,E_j,E_i).\end{array}$$

One defines

$$\begin{array}{c}\hfill \mathcal{C}\left(L\right)=\frac{1}{t}\sum _{r=n+1}^m\sum _{i,j=1}^t{\left(h_{ij}^r\right)}^2.\end{array}$$

Let L be a hyperplane of $T_{p}N$. Then, the normalized $\delta$-Casorati curvatures ${\delta }_c(n-1)$ and ${\widehat{\delta }}_c(n-1)$ are expressed by

$$\begin{array}{c}\hfill {\left[{\delta }_c(n-1)\right]}_p=\frac{1}{2}{\mathcal{C}}_p+\frac{n+1}{2n}\inf \left\{\mathcal{C}\left(L\right)\right\},\end{array}$$

$$\begin{array}{c}\hfill {\left[{\widehat{\delta }}_c(n-1)\right]}_p=2{\mathcal{C}}_p-\frac{2n-1}{2n}sup\left\{\mathcal{C}\left(L\right)\right\}.\end{array}$$

For any real number $r>0$, the generalized normalized $\delta$-Casorati curvatures of N have the following expressions.

If $0<r<n(n-1)$,

$${\left[{\delta }_c(r;n-1)\right]}_p=r{\mathcal{C}}_p+\frac{1}{rn}.A_{1}inf\left\{\mathcal{C}\left(L\right)\right\},$$

and, if $r>n(n-1),$

$${\left[{\widehat{\delta }}_c(r;n-1)\right]}_p=r{\mathcal{C}}_p+\frac{1}{rn}.A_{1}sup\left\{\mathcal{C}\left(L\right)\right\},$$

with $A_1=(n-1)(n+r)(n^2-n-r).$

A submanifold N of $\overline{N}$ is invariantly quasi-umbilical [24] iff there exist $m-n$ orthonormal normal vectors $\{E_{n+1},\dots ,E_m\}$ such that with regard to all of them, the shape operators have an eigenvalue of multiplicity $n-1$ and the distinguished eigendirection remains the same for all $E_r$, $r=n+1,...,m$.

2.2. Golden Riemannian Manifolds

An endomorphism $\phi$ of the tangent bundle $T\overline{N}$ is said to be a structure of golden type on a manifold $\overline{N}$ [1,11,25] if

$$\begin{array}{c}\hfill {\phi }^2-\phi -I=0;\end{array}$$

The pairing $(\overline{N},\phi )$ is a golden manifold. Furthermore, a golden manifold $(\overline{N},\phi )$ endowed with a Riemannian metric g such that

$$\begin{array}{ccc}\hfill g(\phi Y_1,Y_2)& =& g(Y_1,\phi Y_2),\hfill \end{array}$$

(2)

for all vector fields $Y_1,Y_2$, is called a golden Riemannian manifold [2,25].

Substituting $Y_1$ by $\phi Y_1$, (2) implies

$$\begin{array}{cc}& g(\phi Y_1,\phi Y_2)=g(\phi Y_1,Y_2)+g(Y_1,Y_2).\end{array}$$

If $\phi$ is parallel with respect to the Levi-Civita connection $\overline{\nabla }$, i.e.,

$$\overline{\nabla }\phi =0,$$

then a golden Riemannian manifold $(\overline{N},\phi ,g)$ is a locally decomposable golden Riemannian manifold.

The concept of golden sectional curvature on a locally decomposable golden Riemannian manifold $(\overline{N},\phi ,g)$ was described in [5]. The curvature tensor of $\overline{N}$ with constant golden sectional curvature c can be written as

$$\begin{array}{ccc}\hfill \tilde{R}\left(X_1,Y_1\right)Z_1& & =\frac{c}{3}\left\{g\left(Y_1,Z_1\right)X_1-g\left(X_1,Z_1\right)Y_1\right.\hfill \\ & & -g\left(Y_1,\phi Z_1\right)X_1-g\left(Y_1,Z_1\right)\phi X_1+2g\left(Y_1,\phi Z_1\right)\phi X_1\hfill \\ & & \left.+g\left(X_1,\phi Z_1\right)Y_1+g\left(X_1,Z_1\right)\phi Y_1-2g\left(X_1,\phi Z_1\right)\phi Y_1\right\},\hfill \end{array}$$

(3)

for any vector fields $X_1,Y_1$ and $Z_1$ on $\overline{N}$.

Using (3), we obtain the expressions of the Ricci tensor field as [5]

$$S\left(X_1,Y_1\right)=\frac{c}{3}\left\{g\left(X_1,Y_1\right)((m-3)-\parallel \phi \parallel )+g\left(X_1,\phi Y_1\right)(2\parallel \phi \parallel -m)\right\},$$

and the scalar curvature by

$$\begin{array}{c}\hfill r=\frac{c}{3}\left\{m^2-3m+{2\parallel \phi \parallel }^2-2m\parallel \phi \parallel \right\},\end{array}$$

(4)

where $\parallel \phi \parallel ={\sum }_{i=1}^{m}g\left(\phi e_i,e_i\right)$, and $\{e_1,e_2,\dots ,e_m\}$ is an orthonormal basis of $T_p\overline{N}$.

3. Geometric Inequalities

In the following, let $(\overline{N},\phi ,g)$ be a locally decomposable golden Riemannian manifold with constant golden sectional curvature, or in short, a golden space form.

On a submanifold N in $(\overline{N},\phi ,g)$, we obtain optimal inequalities for ${\delta }_c(r;n-1)$ and ${\widehat{\delta }}_c(r;n-1)$.

Theorem 1.

Let N be an n-dimensional Riemannian manifold isometrically immersed in $(\overline{N},\phi ,g)$. Then, for $A_2=n(n-1)$:

(i) 

${\delta }_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\delta }_c(r;n-1)}{A_2}+\frac{c}{3A_2}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\},\hfill \end{array}$$

(5)

if $0<r<A_2.$

(ii) 

${\widehat{\delta }}_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\widehat{\delta }}_c(r;n-1)}{A_2}+\frac{c}{3A_2}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\},\hfill \end{array}$$

(6)

if $r>A_2.$

In addition, the equality holds identically in (5) or (6) iff N is invariantly quasi-umbilical, the normal connection of N in $\overline{N}$ is trivial and there exist an orthonormal tangent frame $\{E_1,\dots ,E_n\}$ and an orthonormal normal frame $\{E_{n+1},\dots ,E_m\}$ such that $S_r$, $r\in \{n+1,\dots ,m\}$, have the following forms:

$$\begin{array}{c}\hfill S_{n+1}=\left(\begin{array}{cccccc}b& 0& 0& \dots & 0& 0\\ 0& b& 0& \dots & 0& 0\\ 0& 0& b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & b& 0\\ 0& 0& 0& \dots & 0& \frac{1}{r}.A_2.b\end{array}\right),S_{n+2}=\cdots =S_m=0.\end{array}$$

(7)

Proof. 

(i) Let N be a submanifold isometrically immersed in $(\overline{N},\phi ,g)$.

The Equations (1) and (4) imply

$$\begin{array}{ccc}\hfill 2\tau \left(p\right)& =& \frac{c}{3}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}+n^2{\left|\right|H\left|\right|}^2-n\mathcal{C}.\hfill \end{array}$$

(8)

A quadratic polynomial Q in the components of h can be stated by the following formula:

$$\begin{array}{c}\hfill \mathcal{Q}=r\mathcal{C}+\frac{1}{rn}{A}_1.\mathcal{C}\left(\mathcal{L}\right)+\frac{c}{3}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}-2\tau \left(p\right).\end{array}$$

Let us rely on the fact that $\mathcal{L}$ is spanned by $\{E_1,\dots ,E_{n-1}\}$. Then, by standard computations, we have

$$\begin{array}{ccc}\hfill Q& =& \frac{r}{n}\sum _{\alpha =n+1}^m\sum _{i,j=1}^n{\left(h_{ij}^{\alpha }\right)}^2+\frac{1}{r.A_2}.A_1.\sum _{\alpha =n+1}^m\sum _{i,j=1}^{n-1}{\left(h_{ij}^{\alpha }\right)}^2-2\tau \left(p\right)\hfill \\ & & +\frac{c}{3}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}.\hfill \end{array}$$

(9)

By using Equations (8) and (9) we obtain

$$\begin{array}{ccc}\hfill Q& =& \frac{n+r}{n}\sum _{\alpha =n+1}^m\sum _{i,j=1}^n{\left(h_{ij}^{\alpha }\right)}^2+\frac{1}{r.A_2}.A_1.\sum _{\alpha =n+1}^m\sum _{i,j=1}^{n-1}{\left(h_{ij}^{\alpha }\right)}^2\hfill \\ & & -\sum _{\alpha =n+1}^m{\left(\sum _{i=1}^n{h}_{ii}^{\alpha }\right)}^2,\hfill \end{array}$$

or equivalently,

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \sum _{\alpha =n+1}^m\sum _{i=1}^{n-1}\left[\frac{n^2+n(r-1)-2r}{r}{\left(h_{ii}^{\alpha }\right)}^2+\frac{2(n+r)}{n}{\left(h_{in}^{\alpha }\right)}^2\right]\hfill \\ & & +\sum _{\alpha =n+1}^m\left[\frac{2(n+r)(n-1)}{r}\sum _{i<j=1}^{n-1}{\left(h_{ij}^{\alpha }\right)}^2-2\sum _{i<j=1}^n{h}_{ii}^{\alpha }{h}_{jj}^{\alpha }+\frac{r}{n}{\left(h_{nn}^{\alpha }\right)}^2\right].\hfill \end{array}$$

(10)

Then its critical points

$$\begin{array}{c}\hfill h^c=(h_{11}^{n+1},h_{12}^{n+1},\dots ,h_{nn}^{n+1},\dots ,h_{11}^m,\dots ,h_{nn}^m)\end{array}$$

are the solutions of:

$$\begin{array}{ccc}\hfill \frac{\partial Q}{\partial h_{ii}^{\alpha }}& =& \frac{2(n+r)(n-1)}{r}{h}_{ii}^{\alpha }-2\sum _{l=1}^n{h}_{ll}^{\alpha }=0,\hfill \\ \hfill \frac{\partial Q}{\partial h_{nn}^{\alpha }}& =& \frac{2r}{n}{h}_{nn}^{\alpha }-2\sum _{l=1}^{n-1}{h}_{ll}^{\alpha }=0,\hfill \\ \hfill \frac{\partial Q}{\partial h_{ij}^{\alpha }}& =& \frac{4(n+r)(n-1)}{r}{h}_{ij}^{\alpha }=0,\hfill \\ \hfill \frac{\partial Q}{\partial h_{in}^{\alpha }}& =& \frac{4(n+r)}{n}{h}_{in}^{\alpha }=0,\hfill \end{array}$$

(11)

$i,j\in \{1,2,\dots ,n-1\}$, $i\ne j$, $\alpha \in \{n+1,\dots ,m\}$.

Following that, each outcome $h^c$ has

$$h_{ij}^{\alpha }=0,i\ne j.$$

The associated determinant to the first two equations of (11) is 0. As a result, we have

$$\begin{array}{c}\hfill H\left(\mathcal{Q}\right)=\left(\begin{array}{ccc}{H}_1& 0& 0\\ 0& H_2& 0\\ 0& 0& H_3\end{array}\right),\end{array}$$

where

$$\begin{array}{c}\hfill H_1=\left(\begin{array}{ccccc}{A}_3-2& -2& \dots & -2& -2\\ -2& A_3-2& \dots & -2& -2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -2& -2& \dots & A_3-2& -2\\ -2& -2& \dots & -2& A_4\end{array}\right),\end{array}$$

$A_3=\frac{2(n+r)(n-1)}{r}$, $A_4=\frac{2r}{n}$, 0 display the null matrices of corresponding sizes.

The diagonal matrices $H_2$, $H_3$ are

$$\begin{array}{c}\hfill H_2=\mathrm{diag}(A_3,A_3,\dots ,A_3),\end{array}$$

$$\begin{array}{c}\hfill H_3=\mathrm{diag}(\frac{2(n+r)}{n},\frac{2(n+r)}{n},\dots ,\frac{2(n+r)}{n}).\end{array}$$

The eigenvalues of $H\left(Q\right)$ are

$$\begin{array}{c}\hfill {\lambda }_{11}=0,{\lambda }_{22}=\frac{2(n^3-n^2+r^2)}{rn},{\lambda }_{33}=\cdots ={\lambda }_{nn}=\frac{2(n+r)\left(1\right)}{r},\end{array}$$

$$\begin{array}{c}\hfill {\lambda }_{ij}=\frac{2(n+r)(n-1)}{r},{\lambda }_{in}=\frac{2(n+r)}{n},\end{array}$$

for all $i,j\in \{1,2,\dots ,n-1\}$, $i\ne j$.

As a result, we draw the conclusion that Q is parabolic and achieves a minimum $Q\left(h^c\right)$ at any $h^c$ of (11). By (10) and (11), we obtain $Q\left(h^c\right)=0$, which implies

$$Q\ge 0.$$

Consequently

$$\begin{array}{c}\hfill 2\tau \left(p\right)\le r\mathcal{C}+\frac{1}{rn}.A_1.\mathcal{C}\left(\mathcal{L}\right)+\frac{c}{3}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}.\end{array}$$

Then, we obtain

$$\begin{array}{c}\hfill \rho \le \frac{r}{A_2}\mathcal{C}+\frac{1}{rn.a_2}.A_1\mathcal{C}\left(\mathcal{L}\right)+\frac{c}{3A_2}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\},\end{array}$$

for any hyperplane L of $T_{p}N$.

The above inequality implies (5).

In addition, the equality holds in (5) if

$$\begin{array}{c}\hfill h_{ij}^{\alpha }=0,\forall i,j\in \{1,\dots ,n\},i\ne j,\end{array}$$

(12)

$$\begin{array}{c}\hfill h_{nn}^{\alpha }=\frac{1}{r}{A}_2{h}_{11}^{\alpha }=\frac{1}{r}{A}_2{h}_{22}^{\alpha }\cdots =\frac{1}{r}{A}_2{h}_{n-1n-1}^{\alpha },\end{array}$$

(13)

$\forall \alpha \in \{n+1,\dots ,m\}.$

By (12) and (13), we have equality in (5) iff N is invariantly quasi-umbilical, the normal connection is trivial and $S_r$ meet (7).

(ii) The inequality (6) can be proven in a similar way. □

Next, assume that $\overline{N}$ is a golden space form. Then the golden Ricci tensor on $\overline{N}$ was determined in [5]:

$$\begin{array}{ccc}\hfill S^G\left(X_1,Y_1\right)& & =S\left(X_1,\phi Y_1\right)\hfill \\ & & =\frac{c}{3}\left\{g\left(X_1,\phi Y_1\right)[\parallel \phi \parallel -3]+g\left(X_1,Y_1\right)[2\parallel \phi \parallel -n]\right\}.\hfill \end{array}$$

Furthermore, the golden scalar curvature of $\overline{N}$ is defined by [5]

$$\begin{array}{c}\hfill r^G=\frac{c}{3}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}.\end{array}$$

Using the golden scalar curvature and with the aid of the following result, we prove some relations.

Lemma 1

([26]). Let N be a Riemannian submanifold of a Riemannian manifold $(\overline{N},g)$ and define a differentiable function by

$$f:N\to \mathbb{R}.$$

Assuming that the constrained extremum problem $mi{n}_{x\in N}f\left(x\right)$ has a solution $y\in N$ and h indicates the second fundamental form of N, we have

(i) 

$\left(gradf\right)\left(y\right)\in T_y^{\perp }N;$

(ii) 

the bilinear form $L:T_{y}N\times T_{y}N\to \mathbb{R};$

$L(X,Y)=g(h(X,Y),\left(grad\left(f\right)\right)\left(y\right))+Hes{s}_f(X,Y)$

is positive semi-definite,

where $grad\left(f\right)$ denotes the gradient of f.

Theorem 2.

Let N be an n-dimensional submanifold isometrically immersed in $(\overline{N},\phi ,g)$.

Then:

(i) 

${\delta }_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill {\rho }^G& \le & \frac{{\delta }_c(r;n-1)}{A_2}+\frac{c}{3A_2}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\},\hfill \end{array}$$

(14)

$0<r<A_2$.

(ii) 

${\widehat{\delta }}_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill {\rho }^G& \le & \frac{{\widehat{\delta }}_c(r;n-1)}{A_2}+\frac{c}{3A_2}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\},\hfill \end{array}$$

(15)

$r>A_2$.

In addition, the equality holds in (14) or (15) iff N satisfies the conditions discussed in Theorem 1 for the equality, and $S_r$ achieve the following forms:

$$\begin{array}{c}\hfill S_{n+1}=\left(\begin{array}{cccccc}b& 0& 0& \dots & 0& 0\\ 0& b& 0& \dots & 0& 0\\ 0& 0& b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & b& 0\\ 0& 0& 0& \dots & 0& \frac{1}{r}{A}_{2}b\end{array}\right),S_{n+2}=\cdots =S_m=0.\end{array}$$

(16)

Proof. 

(i) From (1) and (14), we have

$$\begin{array}{ccc}\hfill 2\tau \left(p\right)& =& \frac{c}{3}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}+n^2{\left|\right|H\left|\right|}^2-n\mathcal{C}.\hfill \end{array}$$

(17)

Consider a quadratic polynomial Q:

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& r\mathcal{C}+\frac{1}{rn}{A}_1\mathcal{C}\left(\mathcal{L}\right)-2\tau \left(p\right)+\frac{c}{3}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}.\hfill \end{array}$$

Let $\{E_1,\dots ,E_{n-1}\}$ be an orthonormal basis of $\mathcal{L}$. One can write

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \frac{r}{n}\sum _{\alpha =n+1}^m\sum _{i,j=1}^n{\left(h_{ij}^{\alpha }\right)}^2+\frac{1}{r{A}_2}{A}_1\sum _{\alpha =n+1}^m\sum _{i,j=1}^{n-1}{\left(h_{ij}^{\alpha }\right)}^2-2\tau \left(p\right)\hfill \\ & & +\frac{c}{3}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}.\hfill \end{array}$$

(18)

Equations (17) and (18) produce

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \frac{n+r}{n}\sum _{\alpha =n+1}^m\sum _{i,j=1}^n{\left(h_{ij}^{\alpha }\right)}^2+\frac{1}{r{A}_2}{A}_1\sum _{\alpha =n+1}^m\sum _{i,j=1}^{n-1}{\left(h_{ij}^{\alpha }\right)}^2\hfill \\ & & -\sum _{\alpha =n+1}^m{\left(\sum _{i=1}^n{h}_{ii}^{\alpha }\right)}^2.\hfill \end{array}$$

One also obtains

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \sum _{\alpha =n+1}^m\sum _{i=1}^{n-1}\left[\frac{2(n+r)}{n}{\left(h_{in}^{\alpha }\right)}^2+\frac{n^2+n(r-1)-2r}{r}{\left(h_{ii}^{\alpha }\right)}^2\right]\hfill \\ & & +\sum _{\alpha =n+1}^m\left[\frac{2(n+r)(n-1)}{r}\sum _{i<j=1}^{n-1}{\left(h_{ij}^{\alpha }\right)}^2+\frac{r}{n}{\left(h_{nn}^{\alpha }\right)}^2-2\sum _{i<j=1}^n{h}_{ii}^{\alpha }{h}_{jj}^{\alpha }\right]\hfill \\ & & \ge \sum _{\alpha =n+1}^m\sum _{i=1}^{n-1}\left[\frac{n^2+n(r-1)-2r}{r}{\left(h_{ii}^{\alpha }\right)}^2+\frac{r}{n}{\left(h_{nn}^{\alpha }\right)}^2-2\sum _{i<j=1}^n{h}_{ii}^{\alpha }{h}_{jj}^{\alpha }\right]\hfill \end{array}$$

(19)

Let

$$f_{\alpha }:{\mathbb{R}}^n\to \mathbb{R}$$

be a function given by

$$\begin{array}{c}\hfill f_{\alpha }(h_{11}^{\alpha },\dots ,h_{nn}^{\alpha })=\sum _{i=1}^{n-1}\frac{n^2+n(r-1)-2r}{r}{\left(h_{ii}^{\alpha }\right)}^2+\frac{r}{n}{\left(h_{nn}^{\alpha }\right)}^2-2\sum _{i<j=1}^n{h}_{ii}^{\alpha }{h}_{jj}^{\alpha },\end{array}$$

and the constrained extremum problem $\min f_{\alpha }$ such that

$$F:h_{11}^{\alpha }+\cdots +h_{nn}^{\alpha }={\beta }^{\alpha },\forall \alpha \in \{n+1,\dots ,m\},$$

${\beta }^{\alpha }$ being a real constant, for any $\alpha \in \{n+1,...,m\}$.

In addition, we find

$$\begin{array}{ccc}\hfill \frac{\partial f_{\alpha }}{\partial h_{ii}^{\alpha }}& =& \frac{2(n+r)(n-1)}{r}{h}_{ii}^{\alpha }-2\sum _{l=1}^n{h}_{ll}^{\alpha }=0,\hfill \\ \hfill \frac{\partial f_{\alpha }}{\partial h_{nn}^{\alpha }}& =& \frac{2r}{n}{h}_{nn}^{\alpha }-2\sum _{l=1}^{n-1}{h}_{ll}^{\alpha }=0,\hfill \end{array}$$

(20)

$i,j=\{1,2,\dots ,n-1\},i\ne j$.

To find the solution of the above problem, $grad\left(f_{\alpha }\right)$ is normal to F. By using (20), we have:

$$\begin{array}{c}\hfill h_{11}^{\alpha }=h_{22}^{\alpha }=\dots h_{n-1n-1}^{\alpha }=\frac{r}{(n+r)(n-1)}{\beta }^{\alpha },h_{nn}^{\alpha }=\frac{n}{n+r}{\beta }^{\alpha }\end{array}$$

(21)

Fix $y\in F.$ Then, in view of Lemma 1, we consider the bilinear form

$$L:T_{y}F\times T_{y}F\to \mathbb{R},$$

such that

$$L(X,Y)=g(h(X,Y),\left(grad{f}_{\alpha }\right)\left(y\right))+Hes{s}_{f_{\alpha }}(X,Y),$$

h is used for the second fundamental form of F in ${\mathbb{R}}^n$.

Furthermore, one can observe

$$\begin{array}{c}\hfill Hess\left(f_{\alpha }\right)=\left(\begin{array}{ccccc}{A}_3-2& -2& \dots & -2& -2\\ -2& A_3-2& \dots & -2& -2\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -2& -2& \dots & A_3-2& -2\\ -2& -2& \dots & -2& A_4\end{array}\right).\end{array}$$

Let y be an arbitrary point on F and $X=(X_1,X_2,\dots ,X_n)$ be any vector tangent to F at y. Then using the fact that F is totally geodesic in ${\mathbb{R}}^n$, one observes

$$\begin{array}{ccc}\hfill L(X,X)& =& \frac{2(n^2-n+rn-2r)}{r}\sum _{l=1}^{n-1}{X}_i^2-2{\left(\sum _{l=1}^n{X}_i\right)}^2+\frac{2r}{n}{X}_n^2\hfill \\ & =& \frac{2(n^2-n+rn-2r)}{r}\sum _{l=1}^{n-1}{X}_i^2+\frac{2r}{n}{X}_n^2\hfill \\ & \ge & 0.\hfill \end{array}$$

Using Lemma 1, one obtains that $(h_{11}^{\alpha },\dots ,h_{nn}^{\alpha })$ (see (21)) is a global minimum point. Furthermore,

$$f_{\alpha }(h_{11}^{\alpha },\dots ,h_{nn}^{\alpha })=0.$$

This implies

$$\begin{array}{c}\hfill \mathcal{Q}\ge 0,\end{array}$$

(22)

which shows

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& r\mathcal{C}+\frac{1}{rn}{A}_1\mathcal{C}\left(\mathcal{L}\right)+\frac{c}{3}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}.\hfill \end{array}$$

One can easily write

$$\begin{array}{ccc}\hfill \rho & \le & \frac{r}{A_2}\mathcal{C}+\frac{1}{rn{A}_2}{A}_1\mathcal{C}\left(\mathcal{L}\right)+\frac{c}{3A_2}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\};\hfill \end{array}$$

(23)

the inequality (14) or (15) obviously follows from (23).

Additionally, the equality sign holds in (14) or (15) iff

$$\begin{array}{c}\hfill h_{ij}^{\alpha }=0,\forall i,j\in \{1,\dots ,n\},i\ne j,\end{array}$$

(24)

and

$$\begin{array}{c}\hfill h_{nn}^{\alpha }=\frac{A_2}{r}{h}_{11}^{\alpha }=\frac{A_2}{r}{h}_{22}^{\alpha }\cdots =\frac{A_2}{r}{h}_{n-1n-1}^{\alpha },\end{array}$$

(25)

$\forall \alpha \in \{n+1,\dots ,m\}$.

Based on (24) and (25), we deduce that (14) (respectively, (15)) has the equality sign iff N is invariantly quasi-umbilical and the normal connection is trivial in $\overline{N}$, in which case $S_r$ have the form (16).

(ii) The second part follows in the same way. □

A submanifold N of $\overline{N}$ is said to be anti-invariant when $\phi$ maps any tangent space into a normal subspace, i.e.,

$$\phi \left(T_{p}N\right)\subset T_p^{\perp }N,\forall p\in N.$$

Obviously, in this case $\parallel \phi \parallel =0$.

Then one derives the following corollaries:

Corollary 1.

Let N be an n-dimensional anti-invariant submanifold isometrically immersed in $(\overline{N},\phi ,g)$. Then:

(i) 

${\delta }_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\delta }_c(r;n-1)}{A_2}+\frac{n-3}{3(n-1)}c\hfill \end{array}$$

(26)

$0<r<A_2$.

(ii) 

${\widehat{\delta }}_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & \frac{{\widehat{\delta }}_c(r;n-1)}{A_2}+\frac{n-3}{3(n-1)}c\hfill \end{array}$$

(27)

$r>A_2$.

In addition, the equality holds identically in (26) or (27) iff N satisfies the same conditions as in Theorem 1 for the equality, and $S_r$ can be represented as:

$$\begin{array}{c}{S}_{n+1}=\left(\begin{array}{cccccc}b& 0& 0& \dots & 0& 0\\ 0& b& 0& \dots & 0& 0\\ 0& 0& b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & b& 0\\ 0& 0& 0& \dots & 0& \frac{A_2}{r}b\end{array}\right),\hfill \\ \\ S_{n+2}=\cdots =S_m=0.\hfill \end{array}$$

Using Formula (14), we derive:

Corollary 2.

Let N be an n-dimensional anti-invariant submanifold of $(\overline{N},\phi ,g)$. Then:

(i) 

${\delta }_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill {\rho }^G& \le & \frac{{\delta }_c(r;n-1)}{A_2},0<r<A_2.\hfill \end{array}$$

(28)

(ii) 

${\widehat{\delta }}_c(r;n-1)$ satisfies

$$\begin{array}{ccc}\hfill {\rho }^G& \le & \frac{{\widehat{\delta }}_c(r;n-1)}{A_2},r>A_2.\hfill \end{array}$$

(29)

Along with that, the equality holds in (28) or (29) under the same conditions as in Theorem 1, and $S_r$ could be written as:

$$\begin{array}{c}{S}_{n+1}=\left(\begin{array}{cccccc}b& 0& 0& \dots & 0& 0\\ 0& b& 0& \dots & 0& 0\\ 0& 0& b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & b& 0\\ 0& 0& 0& \dots & 0& \frac{A_2}{r}b\end{array}\right),\hfill \\ \\ S_{n+2}=\cdots =S_m=0.\hfill \end{array}$$

Moreover, one can investigate other types of submanifolds, for instance invariant submanifolds, slant submanifolds, etc., in $(\overline{N},\phi ,g)$.

4. Consequences

As consequences of the Theorems 1 and 2, we write the relations using ${\delta }_c(n-1)$ and ${\widehat{\delta }}_c(n-1)$ for submanifolds of $(\overline{N},\phi ,g)$.

Corollary 3.

Let N be an n-dimensional submanifold of $(\overline{N},\phi ,g)$. Then:

(i) 

${\delta }_c(n-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\delta }_c(n-1)+\frac{c}{3A_2}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}.\hfill \end{array}$$

(30)

Furthermore, the equality holds in (30) under the same conditions of the equality as in Theorem 1, and $S_r$ are written by

$$\begin{array}{c}{S}_{n+1}=\left(\begin{array}{cccccc}b& 0& 0& \dots & 0& 0\\ 0& b& 0& \dots & 0& 0\\ 0& 0& b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & b& 0\\ 0& 0& 0& \dots & 0& 2b\end{array}\right),\hfill \\ \\ S_{n+2}=\cdots =S_m=0.\hfill \end{array}$$

(ii) 

${\widehat{\delta }}_c(n-1)$ satisfies

$$\begin{array}{ccc}\hfill \rho & \le & {\widehat{\delta }}_c(n-1)+\frac{c}{3A_2}\left\{n^2-3n+{2\parallel \phi \parallel }^2-2n\parallel \phi \parallel \right\}.\hfill \end{array}$$

(31)

Moreover, the equality holds in (31) under the same equality conditions as in the Theorem 1, and $S_r$ satisfy

$$\begin{array}{c}{S}_{n+1}=\left(\begin{array}{cccccc}2b& 0& 0& \dots & 0& 0\\ 0& 2b& 0& \dots & 0& 0\\ 0& 0& 2b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & 2b& 0\\ 0& 0& 0& \dots & 0& b\end{array}\right),\hfill \\ \\ S_{n+2}=\cdots =S_m=0.\hfill \end{array}$$

Using Formula (14), we derive:

Corollary 4.

Let N be an n-dimensional submanifold of $(\overline{N},g,\phi )$. Then:

(i) 

${\delta }_c(n-1)$ satisfies

$$\begin{array}{ccc}\hfill {\rho }^G& \le & {\delta }_c(n-1)+\frac{c}{3A_2}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}.\hfill \end{array}$$

(32)

Additionally, the equality holds in (32) under the same equality conditions as in the Theorem 1, and $S_r$ are given by

$$\begin{array}{c}{S}_{n+1}=\left(\begin{array}{cccccc}b& 0& 0& \dots & 0& 0\\ 0& b& 0& \dots & 0& 0\\ 0& 0& b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & b& 0\\ 0& 0& 0& \dots & 0& 2b\end{array}\right),\hfill \\ \\ S_{n+2}=\cdots =S_m=0.\hfill \end{array}$$

(ii) 

${\widehat{\delta }}_c(n-1)$ satisfies

$$\begin{array}{ccc}\hfill {\rho }^G& \le & {\widehat{\delta }}_c(n-1)+\frac{c}{3A_2}\left\{\parallel \phi \parallel (3\parallel \phi \parallel -3-n)\right\}.\hfill \end{array}$$

(33)

Moreover, the equality holds in (33) under the same conditions for the equality as in the Theorem 1, and $S_r$ take the forms:

$$\begin{array}{c}{S}_{n+1}=\left(\begin{array}{cccccc}2b& 0& 0& \dots & 0& 0\\ 0& 2b& 0& \dots & 0& 0\\ 0& 0& 2b& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & 2b& 0\\ 0& 0& 0& \dots & 0& b\end{array}\right),\hfill \\ \\ S_{n+2}=\cdots =S_m=0.\hfill \end{array}$$

These results can be considered for the particular case of an anti-invariant submanifold by replacing $\parallel \phi \parallel =0$, as well as for other submanifolds, for example, invariant submanifolds, slant submanifolds, etc.

5. Conclusions

If the structure $\phi$ satisfies the relation

$${\phi }^2-p\phi -qI=0,$$

where $p,q$ are positive integers, we mention the following particular cases for different values of $p,q$:

• the golden structure: $\sigma =\frac{1+\sqrt{5}}{2}$;
• the silver structure: ${\sigma }_{2,1}=1+\sqrt{2}$;
• the bronze structure: ${\sigma }_{3,1}=\frac{3+\sqrt{13}}{2}$;
• the subtle structure: ${\sigma }_{4,1}=2+\sqrt{5}$;
• the copper structure: ${\sigma }_{1,2}=2$;
• the nickel structure: ${\sigma }_{1,3}=\frac{1+\sqrt{13}}{2}$, etc.

For the above mentioned circumstances, we can similarly establish relevant inequalities as in Theorem 1 and Corollary 3.