Norden Golden Manifolds with Constant Sectional Curvature and Their Submanifolds

Abstract

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if a Norden golden manifold has a constant real sectional curvature, the manifold is flat. For this reason, the notions of holomorphic-like sectional curvature and holomorphic-like bisectional curvature on the Norden golden manifold are investigated, but it is seen that these notions do not work on the Norden golden manifold. This shows the need for a new concept of sectional curvature. In this direction, a new notion of sectional curvature (Norden golden sectional curvature) is proposed, an example is given, and if this new sectional curvature is constant, the curvature tensor field of the Norden golden manifold is expressed in terms of the metric tensor field. Since the geometry of the submanifolds of manifolds with constant sectional curvature has nice properties, the last section of this paper examines the semi-invariant submanifolds of the Norden golden space form.

1. Introduction

Manifolds with constant curvature are the most important class of manifolds in differential geometry. Such manifolds are suitable as prototypes for many physical phenomena in practice. In addition, the curvature tensor fields of such manifolds have a special form, allowing one to obtain sharp results about both manifolds themselves and their submanifolds.

The best-known examples of manifolds with endomorphism satisfying a certain condition are complex manifolds and their special case, Kaehler manifolds. If a Kaehler manifold has a real constant curvature, the sectional curvature of the manifold is obtained as zero (Proposition 4.3 in [1]). For this reason, the notion of holomorphic sectional curvature on the holomorphic plane in Kaehler manifolds is defined. Kaehler manifolds with constant holomorphic sectional curvature are called complex space forms, and the expression of the curvature tensor fields of such manifolds is also well known [2]. Similarly, for nearly Kaehler manifolds and locally conformal Kaehler manifolds, which are samples of complex manifolds, the curvature tensor field of the manifold takes a special form if the holomorphic sectional curvature is constant.

Golden manifolds were defined by Crasmareanu and Hretcanu in [3], and the geometries of such manifolds have been studied by many authors. In [4], the present authors showed that when the real sectional curvature and holomorphic-like sectional curvature of a locally decomposable golden manifold are constant, the sectional curvatures are zero under certain conditions. The authors also defined a new concept of sectional curvature and obtained the expression of the curvature tensor field when this new sectional curvature is constant.

Crasmareanu and Hretcanu also defined the almost complex golden manifold in the same paper [3]. Later, these manifolds were studied by many authors [5,6,7,8,9,10,11]. However, as far as we know, there has been no study in the literature about the sectional curvature of such manifolds.

The first aim of this study is to investigate the existence of real sectional curvature and holomorphic sectional curvature on almost complex golden manifolds endowed with a semi-Riemannian metric (almost Norden golden manifold). For this purpose, it has been shown that if the real sectional curvature of the almost Norden golden manifold and the holomorphic-like sectional curvature are constant, the sectional curvatures are zero. For this reason, a new sectional curvature is introduced for almost Norden golden manifolds, and examples are given. In addition, if this new sectional curvature is constant, the expression of the curvature tensor field is obtained and it is shown that the manifold is not always an Einstein manifold, in contrast to the real space form and complex space form. The second aim of this paper is to study the submanifolds of almost Norden golden manifolds with constant c sectional curvature. For this purpose, a characterization of the existence of a semi-invariant submanifold of an almost Norden golden manifold with constant c sectional curvature is obtained and the geometric properties of semi-invariant submanifolds are investigated.

2. Curvature Tensor Field of Norden Golden Semi-Riemannian Manifolds

We first recall the notion of almost Norden golden manifold from [3,7]. Let $\tilde{M}$ be a manifold and $\varphi$ an endomorphism on $\tilde{M}$ such that

$${\varphi }^2=\varphi -\frac{3}{2}I$$

where I denotes the identity map. Then, $(\tilde{M},\varphi )$ is called an almost complex golden manifold. Let g be a semi-Riemannian metric on $\tilde{M}$ such that

$$\begin{array}{c}\hfill g(\varphi {\xi }_1,{\xi }_2)=g({\xi }_1,\varphi {\xi }_2),\end{array}$$

(1)

then, $(\tilde{M},\varphi ,g)$ is called an almost Norden golden manifold. We note that (1) is equivalent to

$$g(\varphi {\xi }_1,\varphi {\xi }_2)=g(\varphi {\xi }_1,{\xi }_2)-\frac{3}{2}g({\xi }_1,{\xi }_2)$$

(2)

Moreover, if $\varphi$ is parallel with respect to a vector field X on $\tilde{M}$, $({\nabla }_X\varphi =0)$, then $(\tilde{M},\varphi ,g)$ is called a locally decomposable almost Norden golden semi-Riemannian manifold (in short Norden golden semi-Riemannian manifold).

We now denote the curvature tensor field of $\tilde{M}$ by $\mathfrak{R}$. Then, we have the following result.

Lemma 1.

Let $(\tilde{M},\varphi ,g)$ be a locally decomposable almost Norden golden semi-Riemannian manifold. Then, we have

$$\mathfrak{R}(\varphi {\xi }_1,\varphi {\xi }_2)=\mathfrak{R}({\xi }_1,\varphi {\xi }_2)-\frac{3}{2}\mathfrak{R}({\xi }_1,{\xi }_2)$$

(3)

for any ${\xi }_1$ and ${\xi }_2$ on $\tilde{M}$.

Proof. 

By direct computations and using the well-known properties of $\mathfrak{R}$, we obtain

$$\begin{array}{ccc}\hfill g(\mathfrak{R}(\varphi {\xi }_1,\varphi {\xi }_2){\xi }_3,{\xi }_4)& =& g(\mathfrak{R}({\xi }_3,{\xi }_4)\varphi {\xi }_1,\varphi {\xi }_2)\hfill \\ & =& g(\varphi \mathfrak{R}({\xi }_3,{\xi }_4){\xi }_1,\varphi {\xi }_2)\hfill \\ & =& g(\mathfrak{R}({\xi }_3,{\xi }_4){\xi }_1,\varphi {\xi }_2)-\frac{3}{2}g(\mathfrak{R}({\xi }_3,{\xi }_4){\xi }_1,{\xi }_2)\hfill \\ & =& g(\mathfrak{R}({\xi }_1,\varphi {\xi }_2){\xi }_3,{\xi }_4)-\frac{3}{2}g(\mathfrak{R}({\xi }_1,{\xi }_2){\xi }_3,{\xi }_4).\hfill \end{array}$$

□

The following theorem shows that the notion of constant curvature for a locally decomposable almost Norden golden semi-Riemannian manifold is not essential.

Theorem 1.

Let $\tilde{M}$ be a locally decomposable almost Norden golden semi-Riemannian manifold. If $\tilde{M}$ is of constant curvature, then $\tilde{M}$ is either flat or $g(\varphi {\xi }_1,{\xi }_1)=g(\varphi {\xi }_2,{\xi }_2)$ for vector fields ${\xi }_1,{\xi }_2$ on $\tilde{M}$.

Proof. 

Suppose that $\tilde{M}$ is a locally decomposable almost Norden golden semi-Riemannian manifold with constant sectional curvature

$$K({\xi }_1\wedge {\xi }_2)=\frac{g(\mathfrak{R}({\xi }_1,{\xi }_2){\xi }_2,{\xi }_1)}{g({\xi }_1,{\xi }_1)g({\xi }_2,{\xi }_2)-g{({\xi }_1,{\xi }_2)}^2}=c.$$

Then, we have

$$\mathfrak{R}({\xi }_1,{\xi }_2){\xi }_3=c\{g({\xi }_2,{\xi }_3){\xi }_1-g({\xi }_1,{\xi }_3){\xi }_2\}.$$

(4)

Using (3), we obtain

$$\begin{array}{ccc}\hfill c\{g(\varphi {\xi }_2,{\xi }_3)\varphi {\xi }_1-g(\varphi {\xi }_1,{\xi }_3)\varphi {\xi }_2\}& =& c\{g(\varphi {\xi }_2,{\xi }_3){\xi }_1-g({\xi }_1,{\xi }_3)\varphi {\xi }_2\}\hfill \\ & -& \frac{3}{2}c\{g({\xi }_2,{\xi }_3){\xi }_1-g({\xi }_1,{\xi }_3){\xi }_2\}.\hfill \end{array}$$

Applying $\varphi$ to both sides, we have

$$\begin{array}{ccc}\hfill c\{-\frac{3}{2}g(\varphi {\xi }_2,{\xi }_3){\xi }_1-g(\varphi {\xi }_1,{\xi }_3)\varphi {\xi }_2+\frac{3}{2}g(\varphi {\xi }_1,{\xi }_3){\xi }_2\}& =& c\{-g({\xi }_1,{\xi }_3)\varphi {\xi }_2+\frac{3}{2}g({\xi }_1,{\xi }_3){\xi }_2\}\hfill \\ & -& \frac{3}{2}c\{g({\xi }_2,{\xi }_3)\varphi {\xi }_1-g({\xi }_1,{\xi }_3)\varphi {\xi }_2\}.\hfill \end{array}$$

Substituting ${\xi }_3$ by $\varphi {\xi }_1$, we obtain

$$\begin{array}{ccc}\hfill c\{-\frac{3}{2}g({\xi }_2,\varphi {\xi }_1){\xi }_1+\frac{3}{2}g({\xi }_1,{\xi }_1)\varphi {\xi }_2-\frac{9}{4}g({\xi }_1,{\xi }_1){\xi }_2\}& =& c\{-\frac{3}{2}g({\xi }_2,\varphi {\xi }_1)\varphi {\xi }_1\hfill \\ & +& \frac{3}{2}g({\xi }_1,\varphi {\xi }_1)\varphi {\xi }_2\}.\hfill \end{array}$$

for ${\xi }_1$ and ${\xi }_2$ orthogonal. Taking the inner product on both side of the above equation with ${\xi }_2$, we obtain

$$\begin{array}{ccc}\hfill \frac{3c}{2}\{\parallel {\xi }_1{\parallel }^{2}g(\varphi {\xi }_2,\varphi {\xi }_2)\}& =& \frac{3c}{2}\{g({\xi }_1,\varphi {\xi }_1)g({\xi }_2,\varphi {\xi }_2)-g{({\xi }_2,\varphi {\xi }_1)}^2\}.\hfill \end{array}$$

(5)

Interchanging the role of ${\xi }_1$ and ${\xi }_2$ and subtracting the resulting equation from (5), we obtain

$$\begin{array}{c}\hfill \frac{3c}{2}\{\parallel {\xi }_2{\parallel }^{2}g(\varphi {\xi }_1,\varphi {\xi }_1)\}-\parallel {\xi }_1{\parallel }^2\left\{g(\varphi {\xi }_2,\varphi {\xi }_2)\right\}=0.\end{array}$$

Then, from (1), we derive

$$\begin{array}{c}\hfill \frac{3c}{2}\{g(\varphi {\xi }_1,{\xi }_1)-g(\varphi {\xi }_2,{\xi }_2)\}=0,\end{array}$$

(6)

for unit vector fields ${\xi }_1$ and ${\xi }_2$. Since g is a semi-Riemannian metric, (6) implies that $c=0$ or $g(\varphi {\xi }_1,{\xi }_1)-g(\varphi {\xi }_2,{\xi }_2)=0$, which completes proof. □

We also have the following result for the curvature tensor field of $\tilde{M}$.

Lemma 2.

Let $\tilde{M}$ be a locally decomposable almost Norden golden semi-Riemannian manifold and $\mathfrak{R}$ a curvature tensor of $\tilde{M}$. Then, we have the following relations

$$\begin{array}{ccc}\hfill g(\mathfrak{R}({\xi }_1,\varphi {\xi }_1){\xi }_2,{\xi }_3)& =& 0\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill g(\mathfrak{R}({\xi }_1,\varphi {\xi }_1){\xi }_2,\varphi {\xi }_2)& =& 0\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill g(\mathfrak{R}({\xi }_1,\varphi {\xi }_1)\varphi {\xi }_1,{\xi }_1)& =& 0\hfill \end{array}$$

(9)

for any ${\xi }_1$, ${\xi }_2$ and ${\xi }_3$ on $\tilde{M}$.

Proof. 

Equation (7) is obvious. On the other hand, from (1) and (3), we obtain (8). Equation (9) is similar.

□

3. A New Sectional Curvature Proposal for Norden Golden Semi-Riemannian Manifolds

Lemma 2 above shows that the notion of holomorphic-like sectional curvature and holomorphic bisectional curvature do not work for Norden golden semi-Riemannian manifold. Thus, a new notion of sectional curvature is needed for Norden golden semi-Riemannian manifolds. In the following definition, a new notion is proposed in this direction.

Definition 1.

Let $(\tilde{M},\varphi ,g)$ be a locally decomposable almost Norden golden Semi-Riemannian manifold. Let $\mathbb{P}$ be a nondegenerate plane in $T_x\tilde{M}$ spanned by ${\xi }_{1x}$ and $\varphi {\xi }_{2x}$ for ${\xi }_1,{\xi }_2\in \mathrm{\Gamma }\left(T\tilde{M}\right)$, i.e., a real two-dimensional subspace of $T_x\tilde{M}$ spanned by ${\xi }_{1x}$ and $\varphi {\xi }_{2x}$ for ${\xi }_1,{\xi }_2\in \mathrm{\Gamma }\left(T\tilde{M}\right)$. We define

$$K({\xi }_1\wedge \varphi {\xi }_2)=\frac{g(\mathfrak{R}({\xi }_1,\varphi {\xi }_2)\varphi {\xi }_2,{\xi }_1)}{g({\xi }_1,{\xi }_1)g(\varphi {\xi }_2,\varphi {\xi }_2)-g{({\xi }_1,\varphi {\xi }_2)}^2}$$

for any plane $\mathbb{P}$ spanned by non-null vector fields ${\xi }_1$ and $\varphi {\xi }_2$. The expression $K({\xi }_1\wedge \varphi {\xi }_2)$ is called the Norden golden sectional curvature of $\tilde{M}$ with respect to the section spanned by ${\xi }_1$ and $\varphi {\xi }_2$.

If the Norden golden sectional curvature is a constant c independent of $\mathbb{P}$ at each point x, then $\tilde{M}$ is called a space of constant Norden golden sectional curvature or simply a Norden golden space form, in short NGSF(c). The following example shows that there are many examples for the notion given above.

Example 1.

Let $S_1^{n-1}\left(r\right)$ be a pseudosphere with radius $r=\frac{1}{\sigma }$, where σ is the Norden golden ratio. Then, the shape operator $\mathfrak{W}$ of the pseudosphere is $\mathfrak{W}{\xi }_1=\sigma {\xi }_1$ for any vector field ${\xi }_1$. Since $E_1^n$ is flat, considering $S_1^{n-1}\left(r\right)$ as a hypersurface in $E_1^n$, we have

$$\begin{array}{ccc}\hfill g(\mathfrak{R}({\xi }_1,{\xi }_2){\xi }_3,{\xi }_4)& =& g(\mathfrak{W}{\xi }_2,{\xi }_3)g(\mathfrak{W}{\xi }_1,{\xi }_4)-g(\mathfrak{W}{\xi }_1,{\xi }_3)g(\mathfrak{W}{\xi }_2,{\xi }_4)\hfill \end{array}$$

for any ${\xi }_1,{\xi }_2,{\xi }_3$ and ${\xi }_4$ on $S^{n-1}\left(r\right)$. Thus, using (1), we obtain

$$K({\xi }_1\wedge \mathfrak{W}{\xi }_2)=\frac{1}{{\sigma }^2}.$$

Hence, $S_1^{n-1}\left(r\right)$ with radius $r=\frac{1}{\sigma }$ is a Norden golden space form.

The following theorem is the main theorem of this section, and it expresses the curvature tensor field $\mathfrak{R}$ of an NGSF(c).

Theorem 2.

Let $\tilde{M}\left(c\right)$ be a Norden golden space form. We have

$$\begin{array}{ccc}& & \mathfrak{R}({\xi }_1,{\xi }_2){\xi }_3=\frac{c}{9}\{-3g({\xi }_1,{\xi }_3){\xi }_2+3g({\xi }_2,{\xi }_3){\xi }_1-4g({\xi }_2,\varphi {\xi }_3)\varphi {\xi }_1+2g({\xi }_2,\varphi {\xi }_3){\xi }_1\hfill \\ & & +2g({\xi }_2,{\xi }_3)\varphi {\xi }_1+4g({\xi }_1,\varphi {\xi }_3)\varphi {\xi }_2-2g({\xi }_1,\varphi {\xi }_3){\xi }_2-2g({\xi }_1,{\xi }_3)\varphi {\xi }_2\}\hfill \end{array}$$

(10)

for any ${\xi }_1$, ${\xi }_2$ and ${\xi }_3$.

Proof. 

From the constancy of the expression $K({\xi }_1\wedge \varphi {\xi }_2)$ for ${\xi }_1$ and ${\xi }_2$, we derive

$$\begin{array}{c}\hfill g(\mathfrak{R}({\xi }_1,\varphi {\xi }_2)\varphi {\xi }_2,{\xi }_1)=c\{g({\xi }_1,{\xi }_1)g(\varphi {\xi }_2,\varphi {\xi }_2)-g\left({({\xi }_1,\varphi {\xi }_2)}^2\right\}.\end{array}$$

(11)

From Bianchi’s identity, (8), (1) and (2) we have

$$\begin{array}{ccc}\hfill -g(\mathfrak{R}({\xi }_1,{\xi }_2)\varphi {\xi }_1,{\xi }_2)+\frac{3}{2}g(\mathfrak{R}({\xi }_1,{\xi }_2){\xi }_1,{\xi }_2)& =& c\{g({\xi }_1,{\xi }_1)g(\varphi {\xi }_2,\varphi {\xi }_2)\hfill \\ & -& g{({\xi }_1,\varphi {\xi }_2)}^2\}.\hfill \end{array}$$

(12)

Substituting ${\xi }_2$ by $\varphi {\xi }_2$ in (12), we derive

$$\begin{array}{ccc}\hfill \frac{3}{2}g(\mathfrak{R}({\xi }_1,{\xi }_2)\varphi {\xi }_1,{\xi }_2)& =& c\{\frac{3}{2}g({\xi }_1,{\xi }_1)g(\varphi {\xi }_2,\varphi {\xi }_2)-\frac{3}{2}g{({\xi }_1,\varphi {\xi }_2)}^2\hfill \\ & -& 3g({\xi }_1,{\xi }_1)g(\varphi {\xi }_2,{\xi }_2)+\frac{9}{4}g({\xi }_1,{\xi }_1)g({\xi }_2,{\xi }_2)\hfill \\ & +& 3g({\xi }_1,\varphi {\xi }_2)g({\xi }_1,{\xi }_2)+\frac{9}{4}g{({\xi }_1,{\xi }_2)}^2\}.\hfill \end{array}$$

(13)

Putting (13) in (12), we obtain

$$\begin{array}{ccc}\hfill g(\mathfrak{R}({\xi }_1,{\xi }_2){\xi }_1,{\xi }_2)& =& \frac{2}{3}c\{-\frac{3}{2}g({\xi }_1,{\xi }_1)g({\xi }_2,{\xi }_2)-2g{({\xi }_1,\varphi {\xi }_2)}^2\hfill \\ & +& 2g({\xi }_1,\varphi {\xi }_2)g({\xi }_1,{\xi }_2)+\frac{3}{2}g{({\xi }_1,{\xi }_2)}^2\}.\hfill \end{array}$$

(14)

Substituting ${\xi }_1$ by ${\xi }_1+{\xi }_3$ in (14), we obtain

$$\begin{array}{ccc}\hfill \mathfrak{R}({\xi }_1,{\xi }_2){\xi }_2& =& \frac{c}{3}\{3g({\xi }_2,{\xi }_2){\xi }_1+4g({\xi }_1,\varphi {\xi }_2)\varphi {\xi }_2-2g({\xi }_1,\varphi {\xi }_2){\xi }_2-2g({\xi }_1,{\xi }_2)\varphi {\xi }_2\hfill \\ & -& 3g({\xi }_1,{\xi }_2){\xi }_2\}.\hfill \end{array}$$

(15)

Substituting ${\xi }_2$ by ${\xi }_2+{\xi }_3$ in (15) and using (15) again, we have

$$\begin{array}{ccc}\hfill \mathfrak{R}({\xi }_1,{\xi }_2){\xi }_3+\mathfrak{R}({\xi }_1,{\xi }_3){\xi }_2& =& \frac{c}{3}\{6g({\xi }_2,{\xi }_3){\xi }_1+4g({\xi }_1,\varphi {\xi }_2)\varphi {\xi }_3+4g({\xi }_1,\varphi {\xi }_3)\varphi {\xi }_2\hfill \\ & -& 2g({\xi }_1,\varphi {\xi }_2){\xi }_3-2g({\xi }_1,\varphi {\xi }_3){\xi }_2-2g({\xi }_1,{\xi }_2)\varphi {\xi }_3\hfill \\ & -& 2g({\xi }_1,{\xi }_3)\varphi {\xi }_2-3g({\xi }_1,{\xi }_2){\xi }_3-3g({\xi }_1,{\xi }_3){\xi }_2\}.\hfill \end{array}$$

(16)

Substituting ${\xi }_1$ and ${\xi }_2$ by ${\xi }_2$ and $-{\xi }_1$ in (16), respectively, we obtain

$$\begin{array}{ccc}\hfill -\mathfrak{R}({\xi }_2,{\xi }_1){\xi }_3-\mathfrak{R}({\xi }_2,{\xi }_3){\xi }_1& =& \frac{c}{3}\{-6g({\xi }_1,{\xi }_3){\xi }_2-4g({\xi }_2,\varphi {\xi }_1)\varphi {\xi }_3\hfill \\ & -& 4g({\xi }_2,\varphi {\xi }_3)\varphi {\xi }_1+2g({\xi }_2,\varphi {\xi }_1){\xi }_3\hfill \\ & +& 2g({\xi }_2,\varphi {\xi }_3){\xi }_1+2g({\xi }_2,{\xi }_1)\varphi {\xi }_3\hfill \\ & +& 2g({\xi }_2,{\xi }_3)\varphi {\xi }_1+3g({\xi }_2,{\xi }_1){\xi }_3\hfill \\ & +& 3g({\xi }_2,{\xi }_3){\xi }_1\}.\hfill \end{array}$$

(17)

Then, from (16) and (17), we obtain (10). □

It is well known that spaces of constant curvature are Einstein manifolds. Our current goal is to examine whether the Norden golden space form is an Einstein manifold. We now investigate the Ricci tensor field of a Norden golden space form. The Norden golden Ricci tensor field is defined as follows.

Definition 2.

Let $\tilde{M}\left(c\right)$ be an NGSF(c). The Norden golden Ricci tensor field is defined as

$$r=S({\xi }_1,{\xi }_2)=\sum _{i=1}^{n}g(\mathfrak{R}(e_i,{\xi }_1){\xi }_2,e_i)$$

(18)

$\forall {\xi }_1,{\xi }_2\in \chi \left(\tilde{M}\right)$.

From (18), (10) and (2), we have

$$\begin{array}{c}\hfill S({\xi }_1,{\xi }_2)=\frac{c}{9}\{3n^2-9n+4n\parallel \varphi \parallel -{4\parallel \varphi \parallel }^2\}\end{array}$$

(19)

where $\{e_1,e_2,\dots ,e_n\}$ is an orthonormal basis on $\tilde{M}$, and $\parallel \varphi \parallel ={\sum }_{i=1}^{n}g(\varphi e_i,e_i)$.

An Einstein manifold is defined by the following equation between the Ricci tensor field and the metric of the manifold [1].

$$S(\mathfrak{U},\mathfrak{V})=\lambda g(\mathfrak{U},\mathfrak{V}),\lambda \in \mathbb{R},\forall \mathfrak{U},\mathfrak{V}\in \chi \left(\tilde{M}\right).$$

Unlike a Riemannian space form, (19) implies that a Norden golden space form is not an Einstein manifold, in general.

Thus, from (19), we have the following result.

Theorem 3.

Let $\tilde{M}\left(c\right)$ be a Norden golden space form. NGSF(c) is an Einstein manifold if and only if $\parallel \varphi \parallel =\frac{n}{2}$, where $dim\tilde{M}=n$, ${\sum }_{i=1}^{n}g(\varphi e_i,e_i)=\parallel \varphi \parallel$.

4. Applications of Constant Norden Golden Sectional Curvature to Submanifolds

Because of the relationship of the curvature tensors of constant curvature spaces with the metric tensor, submanifolds of such manifolds can be studied from this point of view and nice results can be obtained. This section examines the geometry of the submanifolds of a Norden golden space form.

We first recall that M is called a semi-invariant submanifold of Norden golden manifold if there exist two orthogonal complementary distributions $\mathfrak{D}$ and ${\mathfrak{D}}^{\perp }$ on M satisfying the following conditions

• $\mathfrak{J}\left({\mathfrak{D}}_p\right)={\mathfrak{D}}_p\subseteq T_{p}M$,
• $\mathfrak{J}\left({\mathfrak{D}}_p^{\perp }\right)\subseteq T_p{M}^{\perp }$

for each point $p\in M$, where $\mathfrak{D}$ and ${\mathfrak{D}}^{\perp }$ are said to be a $\mathfrak{J}$-invariant distribution and $\mathfrak{J}$-anti-invariant distribution, respectively [12].

Our first result in this section shows that $c=0$ under certain conditions.

Theorem 4.

Let $\tilde{M}\left(c\right)$ be a Norden golden space form and M a semi-invariant submanifold. If

$$\left({\nabla }_{\mathfrak{U}}h\right)(\xi ,\mathfrak{V})=\left({\nabla }_{\xi }h\right)(\mathfrak{U},\mathfrak{V}),\left(\mathrm{the}\mathrm{Codazzi}\mathrm{condition}\right)$$

for $\mathfrak{U},\mathfrak{V}\in \mathrm{\Gamma }\left(\mathfrak{D}\right)$ and $\xi \in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$, then $c=0$.

Proof. 

Let $\tilde{M}\left(c\right)$ be an NGSF(c) and M a submanifold of $\tilde{M}$. We first have

$$\begin{array}{ccc}\hfill \tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y}& =& \frac{c}{9}\{-3g(\mathcal{X},\mathcal{Z})\mathcal{Y}+3g(\mathcal{Y},\mathcal{Z})\mathcal{X}\hfill \\ & -& 4g(\mathcal{Y},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{X}+2g(\mathcal{Y},\mathfrak{J}\mathcal{Z})\mathcal{X}+2g(\mathcal{Y},\mathcal{Z})\mathfrak{J}\mathcal{X}\hfill \\ & +& 4g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{Y}-2g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathcal{Y}-2g(\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}\}\hfill \end{array}$$

for any $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$ on $\tilde{M}$, and $\tilde{\mathfrak{R}}$ is a curvature tensor field of $\tilde{M}$. Then, we have

$$\begin{array}{ccc}\hfill \tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y}& =& \frac{c}{9}\{-3g(\mathcal{X},\mathcal{Z})\mathcal{Y}+4g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{Y}-2g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathcal{Y}\hfill \\ & -& 2g(\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}\}\hfill \end{array}$$

(20)

where $\mathcal{X},\mathcal{Y}\in \mathrm{\Gamma }\left(\mathfrak{D}\right)$, $\mathcal{Z}\in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$. From the Gauss equation, we have

$$\begin{array}{ccc}\hfill \tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y}& =& \mathfrak{R}(\mathcal{X},\mathcal{Y})\mathcal{Z}-A_{h(\mathcal{Y},\mathcal{Z})}\mathcal{X}+A_{h(\mathcal{X},\mathcal{Z})}\mathcal{Y}\hfill \\ & +& \left({\nabla }_{\mathcal{X}}h\right)(\mathcal{Y},\mathcal{Z})-\left({\nabla }_{\mathcal{Y}}h\right)(\mathcal{X},\mathcal{Z}),\hfill \end{array}$$

(21)

where $\mathfrak{R}$ is the curvature tensor field of M. Thus, we find

$$\begin{array}{c}\hfill \left({\nabla }_{\mathcal{X}}h\right)(\mathcal{Y},\mathcal{Z})-\left({\nabla }_{\mathcal{Y}}h\right)(\mathcal{X},\mathcal{Z})=\frac{c}{9}\{4g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{Y}-2g(\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}\}=0.\end{array}$$

(22)

Suppose the Codazzi condition is satisfied, then replacing $\mathcal{X}$ by $\mathfrak{J}\mathcal{X}+\mathcal{X}$ in the above equation, we have

$$\begin{array}{c}\hfill \frac{c}{9}\{4g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{Y}+4g(\mathfrak{J}\mathcal{X},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{Y}-2g(\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}-2g(\mathfrak{J}\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}\}=0;\end{array}$$

here, if we use

$$g(\mathfrak{J}\mathcal{X},\mathfrak{J}\mathcal{Z})=g(\mathcal{X},\mathfrak{J}\mathcal{Z})-\frac{3}{2}g(\mathcal{X},\mathcal{Z}),$$

then we obtain

$$\begin{array}{c}\hfill \frac{c}{9}\{6g(\mathcal{X},\mathfrak{J}\mathcal{Z})\mathfrak{J}\mathcal{Y}-8g(\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}\}=0.\end{array}$$

(23)

From (22) and (23), we obtain

$$\begin{array}{c}\hfill \frac{10c}{9}g(\mathcal{X},\mathcal{Z})\mathfrak{J}\mathcal{Y}=0.\end{array}$$

Thus, $c=0$. □

From Theorem 4, we have the following nonexistence result for a semi-invariant submanifold.

Corollary 1.

There is no parallel proper semi-invariant submanifold of Norden golden space form $\tilde{M}$.

We now consider the totally umbilical semi-invariant submanifold of a Norden golden space form. Such submanifolds have been previously studied for Norden golden manifolds [3,4,13,14,15,16,17,18]. We recall that a submanifold M is called a totally umbilical submanifold if

$$h(\mathcal{X},\mathcal{Y})=g(\mathcal{X},\mathcal{Y})\mathcal{H}$$

(24)

for $\mathcal{X},\mathcal{Y}\in \mathrm{\Gamma }\left(TM\right)$, where $\mathcal{H}$ and h are the mean curvature vector field and second fundamental form of M, respectively.

Theorem 5.

Let $\tilde{M}\left(c\right)$ be a Norden golden space form and M a totally umbilical semi-invariant submanifold. For any $\mathcal{X}\in \mathrm{\Gamma }\left(\mathfrak{D}\right)$ and $\mathcal{Y}\in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$. If $c=0$, then we have

$$\mathfrak{K}(\mathcal{X}\wedge \mathcal{Y})=-{\parallel \mathcal{H}\parallel }^{2}g(\mathcal{X},\mathcal{X})g(\mathcal{Y},\mathcal{Y}),$$

where $\mathfrak{K}$ is the sectional curvature of M.

Proof. 

Let $\tilde{M}\left(c\right)$ be a Norden golden space form and M a totally umbilical semi-invariant submanifold. For $W\in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$, from (20), we have

$$\begin{array}{c}\hfill g(\tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Y})\mathcal{Z},W)=\frac{c}{9}\{-3g(\mathcal{X},\mathcal{Z})g(\mathcal{Y},W)-2g(\mathcal{X},\mathfrak{J}\mathcal{Z})g(\mathcal{Y},W)\}.\end{array}$$

and the following Gauss equation

$$\begin{array}{ccc}\hfill g(\tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Y})\mathcal{Z},W)& =& g\left(\mathfrak{R}\right(\mathcal{X},\mathcal{Y})\mathcal{Z},W)-g\left(h\right(\mathcal{Y},\mathcal{Z}),h(\mathcal{X},W\left)\right)\hfill \\ & +& g\left(h\right(\mathcal{X},\mathcal{Z}),h(\mathcal{Y},W\left)\right).\hfill \end{array}$$

The two equations above imply that

$$\begin{array}{ccc}\hfill -\frac{c}{9}\{3g(\mathcal{X},\mathcal{Z})g(\mathcal{Y},W)+2g(\mathcal{X},\mathfrak{J}\mathcal{Z})g(\mathcal{Y},W)\}& =& g\left(\mathfrak{R}\right(\mathcal{X},\mathcal{Y})\mathcal{Z},W)\hfill \\ & -& g\left(h\right(\mathcal{Y},\mathcal{Z}),h(\mathcal{X},W\left)\right)\hfill \\ & +& g\left(h\right(\mathcal{X},\mathcal{Z}),h(\mathcal{Y},W\left)\right).\hfill \end{array}$$

Putting $\mathcal{Z}=\mathcal{X}$ and $W=\mathcal{Y}$ in the above equation, we have

$$\begin{array}{ccc}\hfill \frac{c}{9}\{-3g(\mathcal{X},\mathcal{X})g(\mathcal{Y},\mathcal{Y})-2g(\mathcal{X},\mathfrak{J}\mathcal{X})g(\mathcal{Y},\mathcal{Y})\}& =& g\left(\mathfrak{R}\right(\mathcal{X},\mathcal{Y})\mathcal{X},\mathcal{Y})\hfill \\ & -& g\left(h\right(\mathcal{Y},X),h(\mathcal{X},\mathcal{Y}\left)\right)\hfill \\ & +& g\left(h\right(\mathcal{X},\mathcal{X}),h(\mathcal{Y},\mathcal{Y}\left)\right).\hfill \end{array}$$

Then, an umbilical M implies that

$$\begin{array}{ccc}\hfill \mathfrak{K}(\mathcal{X}\wedge \mathcal{Y})& =& \frac{c}{9}\{-3g(\mathcal{X},\mathcal{X})g(\mathcal{Y},\mathcal{Y})-2g(\mathcal{X},\mathfrak{J}\mathcal{X})g(\mathcal{Y},\mathcal{Y})\}\hfill \\ & -& {g(\mathcal{X},\mathcal{X})g(\mathcal{Y},\mathcal{Y})\parallel \mathcal{H}\parallel }^2.\hfill \end{array}$$

Hence, we derive

$$\begin{array}{c}\hfill \mathfrak{K}(\mathcal{X}\wedge \mathcal{Y})=g(\mathcal{Y},\mathcal{Y})\{\frac{-2c}{9}g(\mathcal{X},\mathfrak{J}\mathcal{X})-(\frac{c}{3}+{\parallel \mathcal{H}\parallel }^2\left)g(\mathcal{X},\mathcal{X})\right\}.\end{array}$$

This completes proof.

□

Theorem 6.

Let ($\tilde{M}\left(c\right),g,\mathfrak{J})$ be a Norden golden space form with $c\ne 0$ and M a submanifold of $\tilde{M}$. Assume that the endomorphism $\mathfrak{J}$ satisfies $\mathfrak{J}\mathfrak{D}={\varphi }_N\mathfrak{D}$ on the invariant distribution $\mathfrak{D}$, where ${\varphi }_N$ is an induced endomorphism on submanifold M by endomorphism $\mathfrak{J}$ in $\tilde{M}$. Then, M is a semi-invariant submanifold if and only if the maximal subspace

$${\mathfrak{D}}_x=T_{x}M\cap \mathfrak{D}{T}_{x}M,x\in M$$

define a non trivial distribution $\mathfrak{D}$ on M such that

$$\tilde{\mathfrak{R}}(\mathfrak{D},{\mathfrak{D}}^{\perp },\mathfrak{D},{\mathfrak{D}}^{\perp })=-\frac{c}{9}(6+2{\varphi }_N^2)g(\mathcal{X},\mathcal{Y})g(\mathcal{Z},W)$$

(25)

where ${\mathfrak{D}}^{\perp }$ is the orthogonal complementary distribution $\mathfrak{D}$ in $TM$.

Proof. 

Suppose M is a semi-invariant submanifold of NGSF(c). Since NGSF(c) is a Norden golden space form, we have

$$\begin{array}{ccc}\hfill \tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y}& =& \frac{c}{9}\{-3g(\mathcal{X},\mathcal{Y})\mathcal{Z}\hfill \\ & +& 4g(\mathcal{X},\mathfrak{J}\mathcal{Y})\mathfrak{J}\mathcal{Z}-2g(\mathcal{X},\mathfrak{J}\mathcal{Y})\mathcal{Z}-2g(\mathcal{X},\mathcal{Y})\mathfrak{J}\mathcal{Z}\},\hfill \end{array}$$

(26)

for $\mathcal{X},\mathcal{Y}\in \mathrm{\Gamma }\left(\mathfrak{D}\right)$ and $\mathcal{Z}\in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$. For $W\in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$, we have

$$\begin{array}{ccc}\hfill g(\tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y},W)& =& \frac{c}{9}\{-6g(\mathcal{X},\mathcal{Y})g(\mathcal{Z},W)-2g(\mathfrak{J}\mathcal{X},\mathfrak{J}\mathcal{Y})g(\mathcal{Z},W)\}.\hfill \end{array}$$

Using $\mathfrak{J}\mathcal{X}={\varphi }_{N}X$ and $\mathfrak{J}\mathcal{Y}={\varphi }_{N}Y$, we have

$$\begin{array}{c}\hfill g(\tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y},W)=-\frac{c}{9}(6+2{\varphi }_N^2)g(\mathcal{X},\mathcal{Y})g(\mathcal{Z},W).\end{array}$$

Conversely, if

$$\mathfrak{J}\mathfrak{D}={\varphi }_N\mathfrak{D}$$

and

$$\tilde{\mathfrak{R}}(\mathfrak{D},{\mathfrak{D}}^{\perp },\mathfrak{D},{\mathfrak{D}}^{\perp })=-\frac{c}{9}(6+2{\varphi }_N^2)g(\mathcal{X},\mathcal{Y})g(\mathcal{Z},W)$$

for a maximal invariant distribution $\mathfrak{D}$, then the curvature tensor field of the Norden golden space form is

$$\begin{array}{ccc}\hfill g(\tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y},W)& =& \frac{c}{9}\{-3g(\mathcal{X},\mathcal{Y})g(\mathcal{Z},W)\hfill \\ & +& 4g(\mathcal{X},\mathfrak{J}\mathcal{Y})g(\mathfrak{J}\mathcal{Z},W)-2g(\mathcal{X},\mathfrak{J}\mathcal{Y})g(\mathcal{Z},W)\hfill \\ & -& 2g(\mathcal{X},\mathcal{Y})g(\mathfrak{J}\mathcal{Z},W)\}\hfill \end{array}$$

for $\mathcal{Z},W\in \mathrm{\Gamma }\left({\mathfrak{D}}^{\perp }\right)$. Since ${\mathfrak{D}}^{\perp }$ is an orthogonal complement distribution to $\mathfrak{D}$ and $\mathfrak{D}$ is invariant, it follows that $\mathfrak{J}{\mathfrak{D}}^{\perp }\perp \mathfrak{D}$. Thus, we have

$$\begin{array}{ccc}\hfill g(\tilde{\mathfrak{R}}(\mathcal{X},\mathcal{Z})\mathcal{Y},W)& =& \frac{c}{9}\{-(3+2{\varphi }_N)g(\mathcal{X},\mathcal{Y})g(\mathcal{Z},W)\hfill \\ & +& (4{\varphi }_N-2)g(\mathcal{X},\mathcal{Y})g(\mathfrak{J}\mathcal{Z},W)\}\hfill \end{array}$$

From assumption $(4{\varphi }_N-2)g(\mathcal{X},\mathcal{Y})g(\mathfrak{J}\mathcal{Z},W)=0$, $c=0$ and $4{\varphi }_N-2\ne 0$, thus $g(\mathfrak{J}\mathcal{Z},W)$$=0$. This shows that $\mathfrak{J}{\mathfrak{D}}^{\perp }$ is orthogonal to ${\mathfrak{D}}^{\perp }$. Since $\mathfrak{D}$ is invariant, $\mathfrak{J}{\mathfrak{D}}^{\perp }$ is also orthogonal to $\mathfrak{D}$. Thus, $\mathfrak{J}{\mathfrak{D}}^{\perp }\subset T{M}^{\perp }$. Therefore, M is a semi-invariant submanifold. □

5. Concluding Remarks

Manifolds with constant sectional curvature are very important spaces in differential geometry and mathematical physics. Obtaining the curvature tensor field of the manifold as an expression of the constant curvature and the metric tensor gives rise to geometrically rich properties. When we look at the applications of manifold theory, we see that manifolds with constant curvature are used as prototypes in many applications. The structure equations of the submanifolds of manifolds with constant sectional curvature are also reduced to a very useful form. This allows one to obtain very precise results about the geometry of the submanifolds. In this paper, the Norden golden manifold with constant sectional curvature was discussed, and a new sectional curvature was defined. When this sectional curvature is constant, the curvature tensor field of the manifold was obtained as an expression of the constant sectional curvature and metric tensor field. In this paper, only very basic results were obtained both about the manifold and its submanifolds. The geometry of submanifolds of constant curvature manifolds has interesting properties. Therefore, we invite the readers to explore new geometric results about Norden golden space forms and their submanifolds.