On the Geometry of the Kähler Golden Manifold

Abstract

The main objective of this paper is to investigate the properties related to the sectional curvatures of a Kähler golden manifold, an almost Hermitian golden manifold whose almost complex golden structure is parallel with respect to the Levi–Civita connection. Under certain conditions, we prove that a Kähler golden manifold with constant sectional curvature is flat. We introduce the concepts of $\Phi$-holomorphic sectional curvature and $\Phi$-holomorphic bi-sectional curvature on a Kähler golden manifold, and compare them respectively with the holomorphic sectional curvature and holomorphic bi-sectional curvature on a Kähler manifold.

1. Introduction

The almost complex golden structure was first introduced in [1]. Since then, the geometry of metric manifolds equipped with special second-order polynomial structures has been extensively studied. Notable contributions in this area include [2,3,4].

The concept of a complex golden number arises by extending the classical golden ratio $\varphi ={\displaystyle \frac{1+\sqrt{5}}{2}}$ to the complex golden ratio ${\phi }_c={\displaystyle \frac{1+i\sqrt{5}}{2}}$, where i denotes the imaginary unit (i.e., $i^2=-1$).

The equation $x^2-x+{\displaystyle \frac{3}{2}}=0$ has two complex conjugate solutions: ${\phi }_c$ and its complex conjugate $\overline{{\phi }_c}={\displaystyle \frac{1-i\sqrt{5}}{2}}$.

We consider a polynomial structure on a differentiable manifold M, given by

$$P(\Phi ):={\Phi }^2-\Phi +{\displaystyle \frac{3}{2}}I,$$

as defined in the general case in [5], where $\Phi$ is a smooth $(1,1)$-tensor field and I denotes the identity transformation on the Lie algebra of vector fields on M such that ${\Phi }^2\left(x\right),\Phi \left(x\right)$ and I are linearly independent, for every $x\in M$. Moreover, $P(\Phi )$ is called the structure polynomial.

The almost complex golden structure was generalized in [6], under the name of the almost $(\alpha ,p)$-golden structure. If $(\alpha ,p)=(-1,1)$, then one obtains an almost complex golden structure.

Let $\tilde{M}$ be a differentiable manifold and $\Phi$ an almost complex golden structure. Then the pair $(\tilde{M},\Phi )$ is called almost complex golden manifold. Several properties of such manifolds are discussed in [7,8].

Sectional curvature properties in various types of golden or metallic manifolds have been studied in [9,10,11,12]. The main distinction between these works and the present study lies in the use of a Hermitian metric on a Riemannian manifold, which leads to the emergence of different properties related to the sectional curvature.

The structure of the present work is organized as follows:

Following the Introduction, Section 2 provides a brief overview of almost Hermitian golden manifolds, intended for later reference. It is shown that any almost complex structure J on an even-dimensional manifold $\tilde{M}$ gives rise to two almost complex golden structures, denoted by ${\Phi }_{ϵ}$, where $ϵ\in \{-1,1\}$. We also consider a Hermitian metric $\tilde{g}$ that satisfies certain compatibility conditions with the structure ${\Phi }_{ϵ}$. Several examples of almost Hermitian golden manifolds are also presented.

In Section 3, we give the definition of Kähler golden manifold and establish several necessary and sufficient conditions for an almost Hermitian golden manifold to be Kähler golden manifold.

In Section 4, we investigate the curvature tensors of a Kähler golden manifold. After presenting some preliminary results concerning the Riemannian curvature transformation and the Riemannian curvature tensor, we focus on identifying specific properties of the sectional curvature in this geometric setting. Under suitable conditions, it is proved that a Kähler golden manifold with constant sectional curvature is flat. Next, we define the $\Phi$-holomorphic sectional curvature of a Kähler golden manifold, where the almost complex golden structure ${\Phi }_{ϵ}$ is $ϵ$-associated with the almost complex structure J. We then compare this curvature with the holomorphic sectional curvature of the underlying Kähler manifold. Finally, we define the $\Phi$-holomorphic bi-sectional curvature of a Kähler golden manifold and establish several relations between this curvature and another special curvature known as the golden sectional curvature as introduced in [10].

2. Almost Hermitian Golden Manifold

Let $\tilde{M}$ be an even-dimensional manifold, and let J be an almost complex structure on $\tilde{M}$ (that is, $J^2=-I$).

The triple $(\tilde{M},\tilde{g},J)$ is an almost Hermitian manifold if $\tilde{g}$ is a Hermitian metric that satisfies the equality [13]

$$\tilde{g}(JX,JY)=\tilde{g}(X,Y),$$

(1)

which is equivalent to

$$\tilde{g}(JX,Y)=-\tilde{g}(X,JY),$$

(2)

for any vector fields $X,Y\in \chi \left(\tilde{M}\right)$. The compatibility of the metric $\tilde{g}$ with the almost complex structure J is understood in accordance with relations (1) or (2).

An almost complex golden manifold $(\tilde{M},\Phi )$ is a Riemannian manifold endowed with an almost complex golden structure $\Phi$, which is an endomorphism of the tangent bundle $T\tilde{M}$, which satisfies the equation [1]

$${\Phi }^2=\Phi -{\displaystyle \frac{3}{2}}·I.$$

(3)

Using a particular case of the ${\Phi }_{\alpha ,p}$-golden structure (studied in [6]) obtained for $\alpha =-1$ and $p=1$, one obtains the compatibility between an almost complex golden structure and an almost complex structure as follows.

Proposition 1.

If $\overline{M}$ is an even-dimensional manifold, then every almost complex structure J on $\tilde{M}$ defines two almost complex golden structures, given by the equality

$${\Phi }_{ϵ}=ϵ{\displaystyle \frac{\sqrt{5}}{2}}J+{\displaystyle \frac{1}{2}}I,$$

(4)

where $ϵ\in \{-1,1\}$.

Conversely, two almost complex structures J can be obtained from a given almost complex golden structure Φ as follows:

$$J={\displaystyle \frac{2ϵ\sqrt{5}}{5}}\Phi -{\displaystyle \frac{ϵ\sqrt{5}}{5}}I,$$

(5)

for $ϵ\in \{-1,1\}$.

Proof. 

Let J be an almost complex structure on $\overline{M}$. We construct an almost complex golden structure $\Phi$ on $\overline{M}$, such that $\Phi =aJ+bI$, where a and b are two real numbers. Using Equation (3), one obtains $a=\pm {\displaystyle \frac{\sqrt{5}}{2}}$ and $b={\displaystyle \frac{1}{2}}$. Thus, for $ϵ\in \{-1,1\}$, one gets (4). The equality (5) can be obtained in the same manner.   □

The last proposition is a generalization of Proposition 2.2 from [7].

Definition 1.

The almost complex golden structure ${\Phi }_{ϵ}$, which verifies the equality (4), is called the almost complex golden structure ϵ-associated with the almost complex structure J.

Taking into account the equalities (2) and (4), one obtains the compatibility of the almost complex golden structure and the Hermitian metric $\tilde{g}$ as follows [7].

Definition 2.

A triple $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is an almost Hermitian golden manifold if the Hermitian metric $\tilde{g}$ verifies the equality

$$\tilde{g}({\Phi }_{ϵ}X,Y)+\tilde{g}(X,{\Phi }_{ϵ}Y)=\tilde{g}(X,Y),$$

(6)

for any $X,Y\in \chi \left(\tilde{M}\right)$, where ${\Phi }_{ϵ}$ is an almost complex golden structure ϵ-associated with J on $\tilde{M}$.

Remark 1.

If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is an almost Hermitian golden manifold, the equality (6) is equivalent to

$$\tilde{g}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)={\displaystyle \frac{3}{2}}\tilde{g}(X,Y),$$

(7)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

The compatibility of the metric $\tilde{g}$ with the almost complex golden structure ${\Phi }_{ϵ}$ is understood in accordance with relations (6) or (7).

Remark 2.

If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is an almost Hermitian golden manifold, then from (6), one gets

$$\tilde{g}({\Phi }_{ϵ}X,X)={\displaystyle \frac{1}{2}}\tilde{g}(X,X),$$

(8)

for any $X\in \Gamma \left(TM\right)$. Moreover, if $X,Y\in \Gamma \left(TM\right)$ are orthogonal vector fields, then

$$\tilde{g}({\Phi }_{ϵ}X,Y)=-\tilde{g}(X,{\Phi }_{ϵ}Y).$$

(9)

Proposition 2.

The triple $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is an almost Hermitian golden manifold if and only if the triple $(\tilde{M},\tilde{g},J)$ is an almost Hermitian manifold, where ${\Phi }_{ϵ}$ is the almost complex golden structure ϵ-associated with the almost complex structure J on $\tilde{M}$.

Example 1.

Let $({\mathbb{E}}^{2n},\tilde{g}:=<·,·>,J_1)$ be an almost Hermitian manifold of the real dimension $2n$ ($n\in {\mathbb{N}}^{*}$), and let $J_1$ be the canonical almost complex structure

$$J_1(X^1,\dots ,X^n,Y^1,\dots ,Y^n)=(-Y^1,\dots ,-Y^n,X^1,\dots ,X^n),$$

(10)

for any $(X^1,\dots ,X^n,Y^1,\dots ,Y^n)\in \chi \left({\mathbb{E}}^{2n}\right)$.

Let ${\Phi }_{ϵ}^1$ be the almost complex golden structure ϵ-associated with the almost complex structure $J_1$, given by the equality

$${\Phi }_{ϵ}^1(X^1,\dots ,X^n,Y^1,\dots ,Y^n)={\displaystyle \frac{1}{2}}(X^1-ϵ\sqrt{5}{Y}^1,\dots ,X^n-ϵ\sqrt{5}{Y}^n,Y^1+ϵ\sqrt{5}{X}^1,\dots ,Y^n+ϵ\sqrt{5}{X}^n).$$

By direct computations, one can obtain the equality

$$\tilde{g}({\Phi }_{ϵ}^1(X^i,Y^i),{\Phi }_{ϵ}^1({\tilde{X}}^i,{\tilde{Y}}^i))={\displaystyle \frac{1}{2}}\sum _{i=1}^n((X^i-ϵ\sqrt{5}{Y}^i)({\tilde{X}}^i-ϵ\sqrt{5}{\tilde{Y}}^i)+(Y^i+ϵ\sqrt{5}{X}^i)({\tilde{Y}}^i+ϵ\sqrt{5}{\tilde{X}}^i)),$$

for any $(X^i,Y^i),({\tilde{X}}^i,{\tilde{Y}}^i)\in \chi \left({\mathbb{E}}^{2n}\right)$, where $(X^i,Y^i):=(X^1,\dots ,X^n,Y^1,\dots ,Y^n)$ and $({\tilde{X}}^i,{\tilde{Y}}^i):=({\tilde{X}}^1,\dots ,{\tilde{X}}^n,{\tilde{Y}}^1,\dots ,{\tilde{Y}}^n)$. Thus, we have

$$\tilde{g}({\Phi }_{ϵ}^1(X^i,Y^i),{\Phi }_{ϵ}^1({\tilde{X}}^i,{\tilde{Y}}^i)={\displaystyle \frac{3}{2}}\tilde{g}((X^i,Y^i),({\tilde{X}}^i,{\tilde{Y}}^i))$$

and the Hermitian metric $\tilde{g}:=<·,·>$ verifies the equality (7). Therefore, the triple $({\mathbb{E}}^{2n},\tilde{g},{\Phi }_{ϵ}^1)$ is an almost Hermitian golden manifold.

Example 2.

Let $({\mathbb{E}}^{4m},\tilde{g}:=<·,·>,J_2)$ be an almost Hermitian manifold of the dimension $4m$, ($m\in {\mathbb{N}}^{*}$), and let $J_2$ be an almost complex structure, given by [14]

$$J_2(X^1,\dots ,X^{2m},Y^1,\dots ,Y^{2m})=(-X^2,X^1,\dots ,-X^{2m},X^{2m-1},Y^2,-Y^1,\dots ,Y^{2m},-Y^{2m-1}),$$

(11)

for any $(X^1,\dots ,X^{2m},Y^1,\dots ,Y^{2m})\in \chi \left({\mathbb{E}}^{4m}\right)$.

Let ${\Phi }_{ϵ}^2$ be the almost complex golden structure, ϵ-associated with the almost complex structures $J_2$, which satisfies the equality

$${\Phi }_{ϵ}^2(X^1,\dots ,X^{2m},Y^1,\dots ,Y^{2m})=(Z^1,\dots ,Z^{2m},W^1,\dots ,W^{2m})$$

(12)

where

$$Z^{2j-1}={\displaystyle \frac{1}{2}}{X}^{2j-1}-ϵ{\displaystyle \frac{\sqrt{5}}{2}}{X}^{2j},Z^{2j}={\displaystyle \frac{1}{2}}{X}^{2j}+ϵ{\displaystyle \frac{\sqrt{5}}{2}}{X}^{2j-1}$$

and

$$W^{2j-1}={\displaystyle \frac{1}{2}}{Y}^{2j-1}+ϵ{\displaystyle \frac{\sqrt{5}}{2}}{Y}^{2j},W^{2j}={\displaystyle \frac{1}{2}}{Y}^{2j}-ϵ{\displaystyle \frac{\sqrt{5}}{2}}{Y}^{2j-1},$$

for any $j\in \{1,2,\dots ,m\}$. By direct computations, one proves that the Hermitian metric $\tilde{g}:=<·,·>$ satisfies the equality (7). Thus, $({\mathbb{E}}^{4m},\tilde{g},{\Phi }_{ϵ}^2)$ is an almost Hermitian golden manifold.

Proposition 3.

If $(\tilde{g},J_1)$ and $(\tilde{g},J_2)$ are almost Hermitian structures on an even-dimensional manifold $\tilde{M}$ such that $J_1$ and $J_2$ are anti-commutative (i.e., $J_1{J}_2+J_2{J}_1=0$), then the endomorphism ${\Phi }_{ϵ,\theta }$ is defined as follows:

$${\Phi }_{ϵ,\theta }=ϵ{\displaystyle \frac{\sqrt{5}}{2}}cos\left(\theta \right)J_1+ϵ{\displaystyle \frac{\sqrt{5}}{2}}sin\left(\theta \right)J_2+{\displaystyle \frac{1}{2}}I,$$

(13)

is an almost complex golden structure, where θ is a smooth function on $\tilde{M}$, and $(\tilde{M},\tilde{g},{\Phi }_{ϵ,\theta })$ is an almost Hermitian golden manifold.

Proof. 

If $J_1$, $J_2$ are two almost complex structures on the metric manifold $(\tilde{M},\tilde{g})$, which verify $J_1{J}_2+J_2{J}_1=0$, then, from ([14], Example 3) the endomorphism given by the equality

$$J_{\theta }:=cos\left(\theta \right)J_1+sin\left(\theta \right)J_2,$$

(14)

is an almost complex structure on $\tilde{M}$, where $\theta$ is a smooth function on $\tilde{M}$. Hence, the triple $(\tilde{M},\tilde{g},J_{\theta })$ is an almost Hermitian manifold. Taking into account the equality (4) in (14), we obtain the almost complex golden structure ${\Phi }_{ϵ,\theta }$ given in the equality (13), which is $ϵ$-associated with $J_{\theta }$. Moreover, $\tilde{g}$ is compatible with the almost complex structure $J_{\theta }$ according to the relation (1), which implies that $\tilde{g}$ is compatible with the almost complex golden structure ${\Phi }_{ϵ,\theta }$ and this implies that $(\tilde{M},\tilde{g},{\Phi }_{ϵ,\theta })$ is an almost Hermitian golden manifold.    □

Remark 3.

In virtue of Proposition 3, an infinite number of complex golden structures can be constructed on a metric manifold $(\tilde{M},\tilde{g})$, using two anti-commutative complex structures $J_1$ and $J_2$ defined on $\tilde{M}$.

Example 3.

Consider the almost complex structures $J_1$ defined in (10) (for $n=2m$) and $J_2$ defined in (11), in a $4m$-dimensional Riemannian manifold ${\mathbb{E}}^{4m}$, and let $\tilde{g}:=<·,·>$ be a Hermitian metric compatible with $J_1$ and $J_2$, respectively. We can verify, by direct computation, that $J_1$ and $J_2$ verify $J_1{J}_2+J_2{J}_1=0$. Thus, using Proposition 3, one obtains some new almost complex golden structures as follows:

1. 

if $\theta ={\displaystyle \frac{\pi }{6}}+2k\pi$, $k\in \mathbb{Z}$, then ${\Phi }_{ϵ,{\displaystyle \frac{\pi }{6}}}=ϵ{\displaystyle \frac{\sqrt{15}}{4}}{J}_1+ϵ{\displaystyle \frac{\sqrt{5}}{4}}{J}_2+{\displaystyle \frac{1}{2}}I$;

2. 

if $\theta ={\displaystyle \frac{\pi }{4}}+2k\pi$, $k\in \mathbb{Z}$, then ${\Phi }_{ϵ,{\displaystyle \frac{\pi }{4}}}=ϵ{\displaystyle \frac{\sqrt{10}}{4}}{J}_1+ϵ{\displaystyle \frac{\sqrt{10}}{4}}{J}_2+{\displaystyle \frac{1}{2}}I$;

3. 

if $\theta ={\displaystyle \frac{\pi }{3}}+2k\pi$, $k\in \mathbb{Z}$, then ${\Phi }_{ϵ,{\displaystyle \frac{\pi }{3}}}=ϵ{\displaystyle \frac{\sqrt{5}}{4}}{J}_1+ϵ{\displaystyle \frac{\sqrt{15}}{4}}{J}_2+{\displaystyle \frac{1}{2}}I$.

3. Kähler Golden Manifolds

Consider an almost Hermitian golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$, where ${\Phi }_{ϵ}$ is the almost complex golden structure, which is $ϵ$-associated with an almost complex structure J on $\tilde{M}$. If $\tilde{\nabla }$ is the Levi–Civita connection on $\tilde{M}$, then we consider

$$\left({\tilde{\nabla }}_X{\Phi }_{ϵ}\right)Y:={\tilde{\nabla }}_X\left({\Phi }_{ϵ}Y\right)-{\Phi }_{ϵ}\left({\tilde{\nabla }}_{X}Y\right),$$

(15)

If $N_J(X,Y):=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]$ is the Nijenhuis tensor field of the almost complex structure J, then the Nijenhuis tensor field of the golden complex structure ${\Phi }_{ϵ}$ can be defined as

$$N_{{\Phi }_{ϵ}}(X,Y):=[{\Phi }_{ϵ}X,{\Phi }_{ϵ}Y]-{\Phi }_{ϵ}[{\Phi }_{ϵ}X,Y]-{\Phi }_{ϵ}[X,{\Phi }_{ϵ}Y]+{\Phi }_{ϵ}^2[X,Y].$$

for any vector fields $X,Y\in \Gamma \left(T\tilde{M}\right)$.

From equalities (4) and (15) and using a similar method as in ([7], Proposition 2.3), we show that

$$\left({\tilde{\nabla }}_X{\Phi }_{ϵ}\right)Y=ϵ{\displaystyle \frac{\sqrt{5}}{2}}\left({\tilde{\nabla }}_{X}J\right)Y$$

(16)

and

$$N_{{\Phi }_{ϵ}}\left(X,Y\right)={\displaystyle \frac{5}{4}}{N}_J(X,Y),$$

(17)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Definition 3.

An almost complex golden structure ${\Phi }_{ϵ}$ on a differentiable manifold is integrable if the Nijenhuis tensor field $N_{{\Phi }_{ϵ}}$ vanishes identically (i.e., $N_{{\Phi }_{ϵ}}=0$). An integrable almost complex golden structure is called a complex golden structure.

In virtue of equality (17), if ${\Phi }_{ϵ}$ is an almost complex golden structure $ϵ$-associated with the almost complex structure J, then ${\Phi }_{ϵ}$ is integrable if and only if J is integrable (i.e., $N_J=0$).

Let us now consider the second fundamental form $\Omega$ on the almost Hermitian manifold $(\tilde{M},\tilde{g},J)$, defined as follows:

$$\Omega (X,Y):=\tilde{g}(JX,Y),$$

(18)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

From (18), we find that $\Omega$ is the same as the second fundamental form defined and studied in [7]. Using (5) and (18), we observe that it satisfies the equality

$$\Omega (X,Y)={\displaystyle \frac{2ϵ\sqrt{5}}{5}}\tilde{g}({\Phi }_{ϵ}X,Y)-{\displaystyle \frac{ϵ\sqrt{5}}{5}}\tilde{g}(X,Y),$$

(19)

for any $X,Y\in \chi \left(\tilde{M}\right)$. Thus, $\Omega (X,Y)$ given in (19) may be regarded as the second fundamental form on the almost Hermitian golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$.

Proposition 4

([7]). If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is an almost Hermitian golden manifold, then the second fundamental form satisfies the following equalities:

$$\Omega (X,Y)=-\Omega (Y,X)$$

and

$$\Omega ({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)={\displaystyle \frac{3}{2}}\Omega (X,Y),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Proposition 5

([7], Theorem 3.4). If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is an almost Hermitian golden manifold, then the following conditions

$$\left(i\right)\tilde{\nabla }{\Phi }_{ϵ}=0;\left(ii\right)\tilde{\nabla }\Omega =0;\left(iii\right)N_{{\Phi }_{ϵ}}=0andd\Omega =0,$$

are equivalent, where $\tilde{\nabla }$ denotes the Levi–Civita connection on $\tilde{M}$.

Definition 4

([7], Definition 3.4). A Kähler golden manifold is an almost Hermitian golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ whose almost complex golden structure ${\Phi }_{ϵ}$ is parallel with respect to the Levi–Civita connection $\tilde{\nabla }$ (i.e., $\tilde{\nabla }{\Phi }_{ϵ}=0$).

From Definition 4 and Proposition 5, one obtains the following result.

Proposition 6.

An almost Hermitian golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is a Kähler golden manifold if and only if its almost complex golden structure ${\Phi }_{ϵ}$ is integrable (i.e., $N_{{\Phi }_{ϵ}}=0$) and $d\Omega =0$.

By virtue of (16), one can obtain a necessary and sufficient condition for an almost Hermitian golden manifold to be a Kähler golden manifold.

Proposition 7.

A necessary and sufficient condition for an almost Hermitian golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ to be a Kähler golden manifold is that the almost Hermitian manifold $(\tilde{M},\tilde{g},J)$ is a Kähler manifold, where ${\Phi }_{ϵ}$ is the almost complex golden structure ϵ-associated with the almost complex structure J.

4. Curvature Tensors of the Kähler Golden Manifold

In this section, we consider a Kähler golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$, of real dimension $2m$, and we establish several properties of some types of curvature tensors of $\tilde{M}$ and their correspondence with the curvature tensors of the Kähler manifold $(\tilde{M},\tilde{g},J)$, where ${\Phi }_{ϵ}$ is the almost complex golden structure $ϵ$-associated with the almost complex structure J.

4.1. Riemannian Curvature

Let $\tilde{R}(X,Y)$ be the Riemannian curvature transformation on $T_x\left(\tilde{M}\right)$, which is defined in [13] as follows:

$$\tilde{R}(X,Y):={\tilde{\nabla }}_X{\tilde{\nabla }}_Y-{\tilde{\nabla }}_Y{\tilde{\nabla }}_X-{\tilde{\nabla }}_{[X,Y]},$$

(20)

where $\tilde{\nabla }$ is the Levi–Civita connection on $\tilde{M}$. It is obvious that

$$\tilde{R}(X,Y)=-\tilde{R}(Y,X),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$.

On a Kähler manifold $(\tilde{M},\tilde{g},J)$, the Riemannian curvature tensor field of covariant degree 4 of $\tilde{M}$ is defined as follows:

$$\tilde{R}(X,Y,Z,W):=\tilde{g}(\tilde{R}(Z,W)Y,X)$$

(21)

for any $X,Y,Z,W\in \chi \left(\tilde{M}\right)$. The following equalities are satisfied:

$$\tilde{R}(X,Y,Z,W)=\tilde{R}(Z,W,X,Y)=-\tilde{R}(Y,X,Z,W)=-\tilde{R}(X,Y,W,Z)$$

(22)

and the first Bianchi identity implies that

$$\tilde{R}(X,Y,Z,W)+\tilde{R}(X,Z,W,Y)+\tilde{R}(X,W,Y,Z)=0,$$

(23)

for any $X,Y,Z,W\in \chi \left(\tilde{M}\right)$, [13], p. 132.

The Ricci tensor is defined as follows:

$$\tilde{S}(X,Y):=\sum _{i=1}^{2m}\tilde{g}(\tilde{R}(E_i,X)Y,E_i),$$

(24)

where $E_1,\dots ,E_{2m}$ are orthogonal vector fields on $\tilde{M}$.

Let us consider an almost Hermitian manifold $(\tilde{M},\tilde{g},J)$ of real dimension $2m$ ($m\in {\mathbb{N}}^{*}$). Let $E_1$ be a unit vector in the tangent space $T_x\tilde{M}$ (for any $x\in \tilde{M}$) and let $J{E}_1$ be a unit vector in $T_x\tilde{M}$, which is orthogonal to $E_1$. If we take a unit vector $E_2$ in $T_x\tilde{M}$, which is orthogonal on $span\{E_1,J{E}_1\}$, then the unit vector $J{E}_2$ is orthogonal on $span\{E_1,J{E}_1,E_2\}$. Following these steps, one obtains an orthonormal basis in $T_{x}M$, $\{E_1,\dots ,E_m,J{E}_1,\dots ,J{E}_m\}$, which is named J-basis on $(\tilde{M},\tilde{g},J)$ [15].

Lemma 1.

If $(\tilde{M},\tilde{g},J)$ is an almost Hermitian manifold, of real dimension $2m$, having an orthonormal basis $\{E_1,\dots ,E_m,J{E}_1,\dots ,J{E}_m\}$ in $T_x\tilde{M}$, then we can construct another basis on $(\tilde{M},\tilde{g})$, considering the almost complex golden structure ${\Phi }_{ϵ}$ (which is ϵ-associated with the almost complex structure J) given by

$$\{E_1,\dots ,E_m,{\Phi }_{ϵ}{E}_1,\dots ,{\Phi }_{ϵ}{E}_m\},$$

named the ${\Phi }_{ϵ}$-basis.

Proof. 

Indeed, if $a_i$ and $b_i$ are real numbers, for $i\in \{1,2,\dots ,m\}$, such that

$$\sum _{i=1}^m{a}_i{E}_i+\sum _{i=1}^m{b}_i{\Phi }_{ϵ}{E}_i=0,$$

then from (4), we obtain

$$\sum _{i=1}^m(a_i+{\displaystyle \frac{1}{2}}{b}_i)E_i+{\displaystyle \frac{ϵ\sqrt{5}}{2}}\sum _{i=1}^m{b}_{i}J{E}_i=0.$$

Taking into account that $\{E_1,\dots ,E_m,J{E}_1,\dots ,J{E}_m\}$ is an orthonormal basis in $(\tilde{M},\tilde{g})$, it follows that $\{E_1,\dots ,E_m,{\Phi }_{ϵ}{E}_1,\dots ,{\Phi }_{ϵ}{E}_m\}$ is also a basis in $(\tilde{M},\tilde{g})$.    □

We observe that the ${\Phi }_{ϵ}$-basis $\{E_1,\dots ,E_m,{\Phi }_{ϵ}{E}_1,\dots ,{\Phi }_{ϵ}{E}_m\}$ is not orthogonal because, by virtue of (8), we have $\tilde{g}({\Phi }_{ϵ}{E}_i,E_i)={\displaystyle \frac{1}{2}}\tilde{g}(E_i,E_i)={\displaystyle \frac{1}{2}}$. On the other hand, if $\tilde{g}(E_i,E_j)=0$ for $i\ne j$, then from the equality (7) one gets

$$\tilde{g}({\Phi }_{ϵ}{E}_i,{\Phi }_{ϵ}{E}_j)={\displaystyle \frac{3}{2}}\tilde{g}(E_i,E_j)=0.$$

Proposition 8.

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a $2m$-dimensional Kähler golden manifold. If $\tilde{R}$ is the curvature tensor field and $\tilde{S}$ is the Ricci tensor of $\tilde{M}$, then the following equalities hold:

$$\tilde{R}(X,Y){\Phi }_{ϵ}W={\Phi }_{ϵ}\tilde{R}(X,Y)W,$$

(25)

$$\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)W={\displaystyle \frac{3}{2}}\tilde{R}(X,Y)W,$$

(26)

$$\tilde{R}({\Phi }_{ϵ}X,Y)W+\tilde{R}(X,{\Phi }_{ϵ}Y)W=\tilde{R}(X,Y)W,$$

(27)

$$\tilde{S}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)={\displaystyle \frac{3}{2}}\tilde{S}(X,Y)$$

(28)

and

$$\tilde{S}({\Phi }_{ϵ}X,Y)+\tilde{S}(X,{\Phi }_{ϵ}Y)=\tilde{S}(X,Y),$$

(29)

for any $X,Y,W\in \chi \left(\tilde{M}\right)$.

Proof. 

Using $\tilde{\nabla }{\Phi }_{ϵ}=0$ in the equality (15), one gets ${\tilde{\nabla }}_X{\Phi }_{ϵ}Y={\Phi }_{ϵ}{\tilde{\nabla }}_{X}Y$, for any $X,Y\in \chi \left(\tilde{M}\right)$. Applying this in the equality (20), we have (25).

From the equalities (21) and (22), we obtain

$$\tilde{g}(\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)W,Z)=\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y,Z,W)=\tilde{g}(\tilde{R}(Z,W){\Phi }_{ϵ}Y,{\Phi }_{ϵ}X)$$

and using (25), one gets

$$\tilde{g}(\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)W,Z)=\tilde{g}({\Phi }_{ϵ}\tilde{R}(Z,W)Y,{\Phi }_{ϵ}X).$$

Thus, by virtue of (4), we have

$$\tilde{g}({\Phi }_{ϵ}\tilde{R}(Z,W)Y,{\Phi }_{ϵ}X)={\displaystyle \frac{3}{2}}\tilde{g}(\tilde{R}(Z,W)Y,X)={\displaystyle \frac{3}{2}}\tilde{g}(\tilde{R}(X,Y)W,Z),$$

for any $X,Y,Z,W\in \chi \left(\tilde{M}\right)$, which implies (26).

From (22) one gets

$$\tilde{g}(\tilde{R}({\Phi }_{ϵ}X,Y)W,Z)=\tilde{g}(\tilde{R}(Z,W)Y,{\Phi }_{ϵ}X)$$

and using (6) and (25), it follows that

$$\tilde{g}(\tilde{R}(Z,W)Y,{\Phi }_{ϵ}X)=-\tilde{g}({\Phi }_{ϵ}\tilde{R}(Z,W)Y,X)+\tilde{g}(\tilde{R}(Z,W)Y,X)$$

$$=-\tilde{g}(\tilde{R}(Z,W){\Phi }_{ϵ}Y,X)+\tilde{g}(\tilde{R}(Z,W)Y,X)=\tilde{g}(-\tilde{R}(X,{\Phi }_{ϵ}Y)W+\tilde{R}(X,Y)W,Z),$$

for any $X,Y,Z,W\in \chi \left(\tilde{M}\right)$, which implies (27).

From (24) we have

$$\tilde{S}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)=\sum _{i=1}^{2m}\tilde{g}(\tilde{R}(E_i,{\Phi }_{ϵ}X){\Phi }_{ϵ}Y,E_i),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$, where $E_1,\dots ,E_{2m}$ are orthogonal vector fields on $\tilde{M}$.

Thus, by virtue of the compatibility relation (4) between the almost complex golden structure ${\Phi }_{ϵ}$ and the almost complex structure J, one obtains

$$\tilde{S}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y)={\displaystyle \frac{1}{4}}\tilde{S}(X,Y)+{\displaystyle \frac{5}{4}}\tilde{S}(JX,JY)+{\displaystyle \frac{ϵ\sqrt{5}}{4}}(\tilde{S}(X,JY)+\tilde{S}(JX,Y)),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Using the equality $\tilde{S}(JX,JY)=\tilde{S}(X,Y)$ from [13] (Proposition 4.1, p. 130) we have

$$\tilde{S}(X,JY)=\tilde{S}(JX,J^{2}Y)=-\tilde{S}(JX,Y),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$. Hence, the equality (28) is satisfied.

By virtue of the Equation (28), one gets

$$\tilde{S}({\Phi }_{ϵ}X,Y)={\displaystyle \frac{2}{3}}\tilde{S}({\Phi }_{ϵ}^{2}X,{\Phi }_{ϵ}Y)={\displaystyle \frac{2}{3}}\tilde{S}({\Phi }_{ϵ}X-{\displaystyle \frac{3}{2}}X,{\Phi }_{ϵ}Y)=\tilde{S}(X,Y)-\tilde{S}(X,{\Phi }_{ϵ}Y),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$, which implies (29).    □

Corollary 1.

If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is a Kähler golden manifold, then the Riemannian curvature tensor field $\tilde{R}$ verifies the equalities

$$\tilde{R}(X,Y,Z,{\Phi }_{ϵ}W)+\tilde{R}(X,Y,{\Phi }_{ϵ}Z,W)=\tilde{R}(X,Y,Z,W),$$

(30)

$$\tilde{R}(X,{\Phi }_{ϵ}Y,Z,W)+\tilde{R}({\Phi }_{ϵ}X,Y,Z,W)=\tilde{R}(X,Y,Z,W),$$

(31)

$$\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y,Z,W)=\tilde{R}(X,Y,{\Phi }_{ϵ}Z,{\Phi }_{ϵ}W)={\displaystyle \frac{3}{2}}\tilde{R}(X,Y,Z,W),$$

(32)

and

$$\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y,{\Phi }_{ϵ}Z,{\Phi }_{ϵ}W)={\displaystyle \frac{9}{4}}\tilde{R}(X,Y,Z,W),$$

for any $X,Y,Z,W\in \chi \left(\tilde{M}\right)$.

Proof. 

Using the definition of the Riemannian curvature tensor given in (21) and the equality (27), we obtain (30).

Interchanging X with Z and Y with W in the equality (30) and using the first equality from (22), one gets (31).

Taking into account the equality (26), one gets (32).

The last equality results from (32) as follows:

$$\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y,{\Phi }_{ϵ}Z,{\Phi }_{ϵ}W)={\displaystyle \frac{3}{2}}\tilde{R}({\Phi }_{ϵ}X,{\Phi }_{ϵ}Y,Z,W)={\displaystyle \frac{9}{4}}\tilde{R}(X,Y,Z,W),$$

for any $X,Y,Z,W\in \chi \left(\tilde{M}\right)$.    □

By virtue of the equality (4), we obtain the following result.

Lemma 2.

If $(\tilde{M},\tilde{g},J)$ is a Kähler manifold and ${\Phi }_{ϵ}$ is the almost complex golden structure ϵ-associated with the almost complex structure J, then we obtain

$$\tilde{R}(X,{\Phi }_{ϵ}Y)Z={\displaystyle \frac{ϵ\sqrt{5}}{2}}\tilde{R}(X,JY)Z+{\displaystyle \frac{1}{2}}\tilde{R}(X,Y)Z,$$

(33)

for any $X,Y,Z\in \chi \left(\tilde{M}\right)$.

Lemma 3.

If $(\tilde{M},\tilde{g},J)$ is a Kähler manifold and ${\Phi }_{ϵ}$ is the almost complex golden structure ϵ-associated with the almost complex structure J, then it follows that

$$\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)={\displaystyle \frac{5}{4}}\tilde{R}(X,JX,Y,JY)$$

(34)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Proof. 

Taking into account (4) and (33), we have

$$\tilde{R}(X,{\Phi }_{ϵ}X){\Phi }_{ϵ}Y={\displaystyle \frac{5}{4}}\tilde{R}(X,JX)JY+{\displaystyle \frac{ϵ\sqrt{5}}{4}}\tilde{R}(X,JX)Y,$$

(35)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Applying $\tilde{g}(·,Y)$ in (35) and using the equalities (21) and (22), we deduce that

$$\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)=\tilde{g}(\tilde{R}(X,{\Phi }_{ϵ}X){\Phi }_{ϵ}Y,Y)={\displaystyle \frac{5}{4}}\tilde{R}(X,JX,Y,JY)+{\displaystyle \frac{ϵ\sqrt{5}}{4}}\tilde{R}(X,JX,Y,Y).$$

From the first equality of (22), one gets

$$\tilde{R}(X,JX,Y,Y)=\tilde{R}(Y,Y,X,JX)=\tilde{g}(\tilde{R}(Y,Y)JX,X)=0.$$

Hence, the equality (34) is satisfied.    □

4.2. Sectional Curvature

Let $\pi$ be a plane in the tangent space $T_x\tilde{M}$, where $x\in \tilde{M}$. For any $X,Y\in T_x\tilde{M}$, we use the notation

$$\tilde{\Delta }(X,Y):=\tilde{g}(X,X)\tilde{g}(Y,Y)-\tilde{g}{(X,Y)}^2.$$

(36)

The plane $\pi$ is non-degenerate if $\tilde{\Delta }(X,Y)\ne 0,$ for a basis $\{X,Y\}$ in ${\pi }_x$ and any $x\in \tilde{M}$, [16].

For each non-degenerate plane $\pi$ of $T_x\tilde{M}$, the sectional curvature is given by [13]:

$$\tilde{K}(X,Y):={\displaystyle \frac{\tilde{R}(X,Y,X,Y)}{\tilde{g}(X,X)\tilde{g}(Y,Y)-\tilde{g}{(X,Y)}^2}},$$

(37)

where $\{X,Y\}$ is an arbitrary basis for $\pi$.

If $\{X,Y\}$ is an orthonormal basis for $\pi$, then one obtains

$$\tilde{K}(X,Y)=\tilde{R}(X,Y,X,Y).$$

The right-hand side of the sectional curvature $\tilde{K}(X,Y)$ depends only on $\pi$, not on the choice of the basis $\{X,Y\}$ for $\pi$ [17].

If the sectional curvature of the Riemannian manifold $(\tilde{M},\tilde{g})$ is a constant $\tilde{c}$, for any $x\in \tilde{M}$ and any plane $\pi$ spanned by arbitrary unit vector fields $X,Y\in \chi \left(\tilde{M}\right)$, then $(\tilde{M},\tilde{g})$ is called a space form. According to [13], the sectional curvature satisfies the relation

$$\tilde{R}(X,Y)Z=\tilde{c}\left(\tilde{g}(Y,Z)X-\tilde{g}(X,Z)Y\right),$$

(38)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

We can show that a Kähler golden manifold with constant sectional curvature is flat under certain conditions.

Theorem 1.

If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is a Kähler golden manifold of dimension $2m>2$, having constant sectional curvature $\tilde{c}$, then either $\tilde{M}$ is flat or, for any orthonormal vector fields $X,Y\in \chi \left(\tilde{M}\right)$, the equality $\tilde{g}{({\Phi }_{ϵ}X,Y)}^2={\displaystyle \frac{5}{4}}$ holds.

Proof. 

If we suppose that $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is a Kähler golden manifold with constant sectional curvature $\tilde{c}$, then from (27) and (38), we have

$$\tilde{c}\left(\tilde{g}(Y,W){\Phi }_{ϵ}X-\tilde{g}({\Phi }_{ϵ}X,W)Y+\tilde{g}({\Phi }_{ϵ}Y,W)X-\tilde{g}(X,W){\Phi }_{ϵ}Y\right)=\tilde{c}\left(\tilde{g}(Y,W)X-\tilde{g}(X,W)Y\right),$$

for any $X,Y,W\in \chi \left(\tilde{M}\right)$, which is equivalent to the equality

$$\tilde{c}\left(\tilde{g}(Y,W)({\Phi }_{ϵ}X-X)-\tilde{g}(X,W)({\Phi }_{ϵ}Y-Y)+\tilde{g}({\Phi }_{ϵ}Y,W)X-\tilde{g}({\Phi }_{ϵ}X,W)Y\right)=0.$$

Applying ${\Phi }_{ϵ}$ in the last equality and using the Equation (3) satisfied by ${\Phi }_{ϵ}$, we obtain

$$\tilde{c}\left({\displaystyle \frac{3}{2}}\tilde{g}(X,W)Y-{\displaystyle \frac{3}{2}}\tilde{g}(Y,W)X+\tilde{g}({\Phi }_{ϵ}Y,W){\Phi }_{ϵ}X-\tilde{g}({\Phi }_{ϵ}X,W){\Phi }_{ϵ}Y\right)=0.$$

Replacing W by ${\Phi }_{ϵ}X$, one gets

$$\tilde{c}\left({\displaystyle \frac{3}{2}}\tilde{g}(X,{\Phi }_{ϵ}X)Y-{\displaystyle \frac{3}{2}}\tilde{g}(Y,{\Phi }_{ϵ}X)X+\tilde{g}({\Phi }_{ϵ}Y,{\Phi }_{ϵ}X){\Phi }_{ϵ}X-\tilde{g}({\Phi }_{ϵ}X,{\Phi }_{ϵ}X){\Phi }_{ϵ}Y\right)=0$$

and using (7) and (8), we have

$$\tilde{c}\left({\displaystyle \frac{3}{4}}{\Vert X\Vert }^{2}Y-{\displaystyle \frac{3}{2}}\tilde{g}(Y,{\Phi }_{ϵ}X)X+{\displaystyle \frac{3}{2}}\tilde{g}(Y,X){\Phi }_{ϵ}X-{\displaystyle \frac{3}{2}}{\Vert X\Vert }^2{\Phi }_{ϵ}Y\right)=0.$$

Suppose that X and Y are orthogonal vector fields; applying ${\displaystyle \frac{2}{3}}\tilde{g}(·,{\Phi }_{ϵ}Y)$ to both sides of the last equality, we obtain

$$\tilde{c}\left({\displaystyle \frac{1}{2}}{\Vert X\Vert }^2\tilde{g}(Y,{\Phi }_{ϵ}Y)-\tilde{g}(Y,{\Phi }_{ϵ}X)\tilde{g}(X,{\Phi }_{ϵ}Y)-{\Vert X\Vert }^2\tilde{g}({\Phi }_{ϵ}Y,{\Phi }_{ϵ}Y)\right)=0.$$

Based on relations (6), (8) and (9), the last equality can be rewritten as

$$\tilde{c}\left(\tilde{g}{({\Phi }_{ϵ}X,Y)}^2-{\displaystyle \frac{5}{4}}{\Vert X\Vert }^2{\Vert Y\Vert }^2\right)=0.$$

(39)

Therefore, the equality (39) leads to $\tilde{c}=0$, or $\tilde{g}{({\Phi }_{ϵ}X,Y)}^2={\displaystyle \frac{5}{4}}$ for any unitary vector fields $X,Y\in \chi \left(\tilde{M}\right)$.    □

Proposition 9.

The equality $\tilde{g}{({\Phi }_{ϵ}X,Y)}^2={\displaystyle \frac{5}{4}}$ from the Theorem 1 is equivalent to

$${sin}^2({\Phi }_{ϵ}X,Y)={sin}^2(X,{\Phi }_{ϵ}Y)={\displaystyle \frac{1}{6}},$$

for any orthonormal vector fields $X,Y\in \chi \left(\tilde{M}\right)$.

Proof. 

Indeed, using (7), which implies $\Vert {\Phi }_{ϵ}{X\Vert }^2={\displaystyle \frac{3}{2}}{\Vert X\Vert }^2$ and using Theorem 1, we obtain

$$\tilde{g}{({\Phi }_{ϵ}X,Y)}^2={\displaystyle \frac{3}{2}}{\Vert X\Vert }^2{\Vert Y\Vert }^2{cos}^2({\Phi }_{ϵ}X,Y)={\displaystyle \frac{5}{4}}.$$

Thus, for any orthonormal vector fields $X,Y\in \chi \left(\tilde{M}\right)$ we have ${cos}^2({\Phi }_{ϵ}X,Y)={\displaystyle \frac{5}{6}}$, which implies the conclusion.    □

4.3. $\Phi$-Holomorphic Sectional Curvature

Let $\pi$ be an invariant plane with respect to the almost complex structure J of the given Kähler manifold $(\tilde{M},\tilde{g},J)$. The holomorphic sectional curvature of $\tilde{M}$ is defined as the sectional curvature for the plane $\pi$, given as follows:

$${\tilde{K}}_{\pi }(X,JX):={\displaystyle \frac{\tilde{R}(X,JX,X,JX)}{\tilde{g}(X,X)\tilde{g}(JX,JX)-\tilde{g}{(X,JX)}^2}},$$

(40)

where $\pi$ is a plane spanned by $\{X,JX\}$.

In fact, from (36), we have

$$\tilde{\Delta }(X,JX)=\tilde{g}(X,X)\tilde{g}(JX,JX)-\tilde{g}{(X,JX)}^2=\tilde{g}{(X,X)}^2\ne 0,$$

for a plane $\pi$ spanned by an orthogonal basis $\{X,JX\}$. Thus, from (40), one gets

$${\tilde{K}}_{\pi }(X,JX)={\displaystyle \frac{\tilde{R}(X,JX,X,JX)}{\tilde{g}{(X,X)}^2}}.$$

The holomorphic sectional curvature $K_{\pi }$, for all J-invariant planes $\pi$, is determined by the Riemannian curvature tensor $\tilde{R}$ at x of $\tilde{M}$.

Moreover, if X is unitary, then ${\tilde{K}}_{\pi }(X,JX):=\tilde{R}(X,JX,X,JX)$.

If the holomorphic sectional curvature $K_{\pi }$ is constant for all J-invariant planes $\pi$ in $T_x\tilde{M}$ and for all points $x\in \tilde{M}$, then $\tilde{M}$ is called a space of constant holomorphic curvature or a complex space form [13], p. 134.

Based on the results presented above, we define a new type of sectional curvature on a Kähler golden manifold as follows.

Definition 5.

In a Kähler golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$, the Φ-holomorphic sectional curvature is defined by

$${\tilde{K}}_{\pi }(X,{\Phi }_{ϵ}X):={\displaystyle \frac{\tilde{R}(X,{\Phi }_{ϵ}X,X,{\Phi }_{ϵ}X)}{\tilde{g}(X,X)\tilde{g}({\Phi }_{ϵ}X,{\Phi }_{ϵ}X)-\tilde{g}{(X,{\Phi }_{ϵ}X)}^2}},$$

(41)

where π is a plane spanned by $\{X,{\Phi }_{ϵ}X\}$ which is invariant by the almost complex golden structure ${\Phi }_{ϵ}$.

Using the equalities (7) and (8), one gets

$$\tilde{\Delta }(X,{\Phi }_{ϵ}X)=\tilde{g}(X,X)\tilde{g}({\Phi }_{ϵ}X,{\Phi }_{ϵ}X)-\tilde{g}{(X,{\Phi }_{ϵ}X)}^2={\displaystyle \frac{5}{4}}\tilde{g}{(X,X)}^2,$$

(42)

for any $X\in \chi \left(\tilde{M}\right)$.

Hence, from (41) and (42), we derive the following property.

Proposition 10.

If $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is a Kähler golden manifold, then the Φ-holomorphic sectional curvature satisfies the equality

$${\tilde{K}}_{\pi }(X,{\Phi }_{ϵ}X)={\displaystyle \frac{4}{5}}\tilde{R}(X,{\Phi }_{ϵ}X,X,{\Phi }_{ϵ}X),$$

(43)

where $X\in \chi \left(\tilde{M}\right)$ is a unitary vector field, and π is a plane spanned by $\{X,{\Phi }_{ϵ}X\}$, and it is invariant by the almost complex golden structure ${\Phi }_{ϵ}$.

Based on (4) and Lemma 1, we obtain the following proposition.

Proposition 11.

Let $(\tilde{M},\tilde{g},J)$ be a Kähler manifold and let ${\Phi }_{ϵ}$ be the almost complex golden structure ϵ-associated with the almost complex structure J. The plane ${\pi }_J$, spanned by $\{X,JX\}$, is non-degenerate and is invariant by the almost complex structure J if and only if the plane ${\pi }_{\Phi }$, spanned by $\{X,{\Phi }_{ϵ}X\}$, is non-degenerate and is invariant by the almost complex golden structure ${\Phi }_{ϵ}$.

Definition 6.

If the Φ-holomorphic sectional curvature ${\tilde{K}}_{{\pi }_{\Phi }}(X,{\Phi }_{ϵ}X)$ is constant for all holomorphic planes ${\pi }_{\Phi }$ and all points $x\in \tilde{M}$, the Kähler golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is called a complex golden space form.

Using (34), (43) and Proposition 11, the following proposition holds.

Proposition 12.

Let $(\tilde{M},\tilde{g},J)$ be a Kähler manifold and let ${\Phi }_{ϵ}$ be the almost complex golden structure ϵ-associated with the almost complex structure J. Then the Φ-holomorphic sectional curvature ${\tilde{K}}_{{\pi }_{\Phi }}(X,{\Phi }_{ϵ}X)$ of the Kähler golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ and the holomorphic sectional curvature ${\tilde{K}}_{{\pi }_J}(X,JX)$ of $(\tilde{M},\tilde{g},J)$ satisfy

$${\tilde{K}}_{{\pi }_{\Phi }}(X,{\Phi }_{ϵ}X)={\tilde{K}}_{{\pi }_J}(X,JX),$$

for any unitary vector field $X\in \chi \left(\tilde{M}\right)$.

Corollary 2.

Let $(\tilde{M},\tilde{g},J)$ be a Kähler manifold and let ${\Phi }_{ϵ}$ be the almost complex golden structure ϵ-associated with the almost complex structure J. The Kähler golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ is a complex golden space form if and only if the Kähler manifold $(\tilde{M},\tilde{g},J)$ is a complex space form.

Using [13] (Theorem 4.2, p. 135), the holomorphic sectional curvature of a Kähler manifold $(\tilde{M},\tilde{g},J)$ is a constant $\tilde{c}$ if and only if the following equality holds:

$$\tilde{R}(X,Y)Z={\displaystyle \frac{\tilde{c}}{4}}\left(\tilde{g}(X,Z)Y-\tilde{g}(Y,Z)X+\tilde{g}(JX,Z)JY-\tilde{g}(JY,Z)JX+2\tilde{g}(JX,Y)JZ\right),$$

(44)

for any vector fields $X,Y,Z\in \chi \left(\tilde{M}\right)$.

The following theorem establishes a property of a Kähler golden manifold $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ with a constant holomorphic sectional curvature.

Theorem 2.

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a Kähler golden manifold. Then $\tilde{M}$ is a complex golden space form having the ${\Phi }_{ϵ}$-holomorphic sectional curvature, a constant $\tilde{c}$, if and only if the Riemannian curvature verifies the following equality:

$$\begin{gathered} \tilde{R}(X,Y)Z={\displaystyle \frac{\tilde{c}}{10}}(\tilde{g}(X,Z)({\Phi }_{ϵ}Y+2Y)-\tilde{g}(Y,Z)({\Phi }_{ϵ}X+2X)-\tilde{g}(X,Y)(2{\Phi }_{ϵ}Z-Z) \\ -\tilde{g}(X,{\Phi }_{ϵ}Z)(2{\Phi }_{ϵ}Y-Y)+\tilde{g}(Y,{\Phi }_{ϵ}Z)(2{\Phi }_{ϵ}X-X)+2\tilde{g}({\Phi }_{ϵ}X,Y)(2{\Phi }_{ϵ}Z-Z)), \end{gathered}$$

(45)

for any vector fields $X,Y,Z\in \chi \left(\tilde{M}\right)$.

Proof. 

If ${\Phi }_{ϵ}$ is the almost complex golden structure $ϵ$-associated with the almost complex structure J, then from (5), we obtain the equality

$$\tilde{g}(JX,Z)JY={\displaystyle \frac{4}{5}}\tilde{g}({\Phi }_{ϵ}X,Z){\Phi }_{ϵ}Y-{\displaystyle \frac{2}{5}}\tilde{g}({\Phi }_{ϵ}X,Z)Y-{\displaystyle \frac{2}{5}}\tilde{g}(X,Z){\Phi }_{ϵ}Y+{\displaystyle \frac{1}{5}}\tilde{g}(X,Z)Y,$$

(46)

for any vector fields $X,Y,Z\in \chi \left(\tilde{M}\right)$.

Then interchanging X with Y in (46), one gets

$$\tilde{g}(JY,Z)JX={\displaystyle \frac{4}{5}}\tilde{g}({\Phi }_{ϵ}Y,Z){\Phi }_{ϵ}X-{\displaystyle \frac{2}{5}}\tilde{g}({\Phi }_{ϵ}Y,Z)X-{\displaystyle \frac{2}{5}}\tilde{g}(Y,Z){\Phi }_{ϵ}X+{\displaystyle \frac{1}{5}}\tilde{g}(Y,Z)X.$$

(47)

On the other hand, interchanging Z with Y in (46), we have

$$\tilde{g}(JX,Y)JZ={\displaystyle \frac{4}{5}}\tilde{g}({\Phi }_{ϵ}X,Y){\Phi }_{ϵ}Z-{\displaystyle \frac{2}{5}}\tilde{g}({\Phi }_{ϵ}X,Y)Z-{\displaystyle \frac{2}{5}}\tilde{g}(X,Y){\Phi }_{ϵ}Z+{\displaystyle \frac{1}{5}}\tilde{g}(X,Y)Z.$$

(48)

Applying (46)–(48) and (6) to (44), and taking into account Proposition 12, we obtain equality (45).    □

4.4. $\Phi$-Holomorphic Bi-Sectional Curvature

Let $(\tilde{M},\tilde{g},J)$ be a Kähler manifold of real dimension $2m$ ($m\ge 1$). Let ${\pi }_1=span\{X,JX\}$ and ${\pi }_2=span\{Y,JY\}$ be two non-degenerate planes in $T_x\tilde{M}$, where X and Y are unitary vectors in $T_x\tilde{M}$, for any $x\in \tilde{M}$. The holomorphic bi-sectional curvature is given in the next definition.

Definition 7

([17]). The holomorphic bi-sectional curvature of the planes ${\pi }_1$ and ${\pi }_2$, which are invariant under the almost complex structure J, is defined by

$$H_{{\pi }_1,{\pi }_2}^J(X,Y):=\tilde{R}(X,JX,Y,JY),$$

(49)

where X is a unit vector in ${\pi }_1$ and Y is a unit vector in ${\pi }_2$.

For $X=Y$ in (49), ${\pi }_1={\pi }_2:=\pi$ and the holomorphic sectional curvature of $\pi$ is given as follows:

$$H_{\pi ,\pi }^J(X,X):=\tilde{R}(X,JX,X,JX).$$

We observe that, for any unit vector field $X\in \chi \left(\tilde{M}\right)$, the following equality holds:

$$H_{\pi ,\pi }^J(X,X)=K_{{\pi }_J}(X,JX),$$

where $K_{{\pi }_J}(X,JX)$ is the holomorphic sectional curvature given in (40).

From the first Bianchi identity, one obtains [18]

$$\tilde{R}(X,JX,Y,JY)=\tilde{R}(X,Y,X,Y)+\tilde{R}(X,JY,X,JY).$$

(50)

Based on the above results, we now introduce the following definition.

Definition 8.

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a Kähler golden manifold. If the planes ${\pi }_1$ and ${\pi }_2$ in $T_x\tilde{M}$ are invariant under the almost complex golden structure ${\Phi }_{ϵ}$, then the Φ-holomorphic bi-sectional curvature is defined as follows:

$$H_{{\pi }_1,{\pi }_2}^{{\Phi }_{ϵ}}(X,Y):=\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)$$

(51)

where X and Y are unit vectors in ${\pi }_1=span\{X,{\Phi }_{ϵ}X\}$ and, in ${\pi }_2=span\{Y,{\Phi }_{ϵ}Y\}$, respectively.

Taking into account the first equality in (22) as given in the preceding definition, it follows that

$$H_{{\pi }_1,{\pi }_2}^{{\Phi }_{ϵ}}(X,Y)=H_{{\pi }_1,{\pi }_2}^{{\Phi }_{ϵ}}(Y,X).$$

An identity similar to (50) for an almost complex gold structure ${\Phi }_{ϵ}$ can be deduced as follows.

Lemma 4.

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a Kähler golden manifold. Then

$$\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)={\displaystyle \frac{1}{2}}\left(\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)+\tilde{R}(Y,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}X)\right)+\tilde{R}(X,Y,X,Y),$$

(52)

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Proof. 

If we replace Z with X and W with ${\Phi }_{ϵ}Y$ in (31), one gets

$$\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)+\tilde{R}({\Phi }_{ϵ}X,Y,X,{\Phi }_{ϵ}Y)=\tilde{R}(X,Y,X,{\Phi }_{ϵ}Y),$$

for any $X,Y\in \chi \left(\tilde{M}\right)$.

Using the first equality from (22) in $\tilde{R}({\Phi }_{ϵ}X,Y,X,{\Phi }_{ϵ}Y)$, we have

$$\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)+\tilde{R}(X,{\Phi }_{ϵ}Y,{\Phi }_{ϵ}X,Y)=\tilde{R}(X,Y,X,{\Phi }_{ϵ}Y),$$

Applying the first Bianchi’s identity (23), we obtain

$$\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)-\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)-\tilde{R}(X,Y,{\Phi }_{ϵ}Y,{\Phi }_{ϵ}X)=\tilde{R}(X,Y,X,{\Phi }_{ϵ}Y).$$

Now, by taking into account (22) and (32), we deduce

$$\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)=\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)-{\displaystyle \frac{3}{2}}\tilde{R}(X,Y,X,Y)+\tilde{R}(X,Y,X,{\Phi }_{ϵ}Y).$$

(53)

Interchanging X with Y in (53) and using (22), we obtain

$$\tilde{R}(Y,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}X)=\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)-{\displaystyle \frac{3}{2}}\tilde{R}(X,Y,X,Y)+\tilde{R}(Y,X,Y,{\Phi }_{ϵ}X).$$

(54)

Adding the equalities (53) and (54), we have

$$\begin{gathered} \tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)+\tilde{R}(Y,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}X) \\ =2\tilde{R}(X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y)-3\tilde{R}(X,Y,X,Y)+\tilde{R}(X,Y,X,{\Phi }_{ϵ}Y)+\tilde{R}(X,Y,{\Phi }_{ϵ}X,Y). \end{gathered}$$

(55)

Using (31) in the last two terms of the equality (55), we obtain (52).    □

Let us consider, for each non-degenerate plane ${\pi }_3=span\{X,{\Phi }_{ϵ}Y\}$, the sectional curvature, given in (37), as follows:

$$\tilde{K}(X,{\Phi }_{ϵ}Y)={\displaystyle \frac{\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)}{\tilde{g}(X,X)\tilde{g}({\Phi }_{ϵ}Y,{\Phi }_{ϵ}Y)-\tilde{g}{(X,{\Phi }_{ϵ}Y)}^2}}.$$

(56)

The curvature given in (56) is named the golden sectional curvature in [10], and is noted by

$$K^G(X\wedge Y):=\tilde{K}(X,{\Phi }_{ϵ}Y).$$

(57)

Interchanging X with Y in (56), one obtains the sectional curvature

$$\tilde{K}(Y,{\Phi }_{ϵ}X)={\displaystyle \frac{\tilde{R}(Y,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}X)}{\tilde{g}(Y,Y)\tilde{g}({\Phi }_{ϵ}X,{\Phi }_{ϵ}X)-\tilde{g}{(Y,{\Phi }_{ϵ}X)}^2}},$$

for each non-degenerate plane ${\pi }_4=span\{Y,{\Phi }_{ϵ}X\}$.

Using the notation (36), we have

$$\tilde{\Delta }(X,{\Phi }_{ϵ}Y)=\tilde{g}(X,X)\tilde{g}({\Phi }_{ϵ}Y,{\Phi }_{ϵ}Y)-\tilde{g}{(X,{\Phi }_{ϵ}Y)}^2$$

and

$$\tilde{\Delta }(Y,{\Phi }_{ϵ}X)=\tilde{g}(Y,Y)\tilde{g}({\Phi }_{ϵ}X,{\Phi }_{ϵ}X)-\tilde{g}{(Y,{\Phi }_{ϵ}X)}^2$$

From the equalities (7) and (9), one obtains

$${\tilde{\Delta }}_{{\Phi }_{ϵ}}(X,Y):=\tilde{\Delta }(X,{\Phi }_{ϵ}Y)=\tilde{\Delta }(Y,{\Phi }_{ϵ}X)={\displaystyle \frac{3}{2}}\tilde{g}(X,X)\tilde{g}(Y,Y)-\tilde{g}{(X,{\Phi }_{ϵ}Y)}^2$$

(58)

for any normal vectors $X,Y\in T_x\tilde{M}$.

Theorem 3.

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a Kähler golden manifold of real dimension $2m$, where $m\ge 2$. The Φ-holomorphic bi-sectional curvature is given by the equality

$$H_{{\pi }_1,{\pi }_2}^{{\Phi }_{ϵ}}(X,Y)=({\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{2}}\tilde{g}{(X,{\Phi }_{ϵ}Y)}^2)(\tilde{K}(X,{\Phi }_{ϵ}Y)+\tilde{K}(Y,{\Phi }_{ϵ}X))+\tilde{K}(X,Y),$$

(59)

for any orthonormal vector fields $X,Y\in \chi \left(\tilde{M}\right)$ such that $\{X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y\}$ are linearly independent and ${\pi }_1=span\{X,{\Phi }_{ϵ}X\}$ and ${\pi }_2=span\{Y,{\Phi }_{ϵ}Y\}$ are invariant planes under the action of ${\Phi }_{ϵ}$.

Proof. 

By virtue of (51), (58), and Lemma 4, one gets

$$H_{{\pi }_1,{\pi }_2}^{{\Phi }_{ϵ}}(X,Y)={\displaystyle \frac{{\tilde{\Delta }}_{{\Phi }_{ϵ}}(X,Y)}{2}}\left({\displaystyle \frac{\tilde{R}(X,{\Phi }_{ϵ}Y,X,{\Phi }_{ϵ}Y)}{\tilde{\Delta }(X,{\Phi }_{ϵ}Y)}}+{\displaystyle \frac{\tilde{R}(Y,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}X)}{\tilde{\Delta }(Y,{\Phi }_{ϵ}X)}}\right)+\tilde{K}(X,Y),$$

(60)

for any orthonormal vector fields $X,Y\in \chi \left(\tilde{M}\right)$, where $\tilde{K}(X,Y)=\tilde{R}(X,Y,X,Y)$ is the sectional curvature. Using (51) and (56) in (60), the equality (59) holds.    □

If we consider $\tilde{g}(X,{\Phi }_{ϵ}Y)=0$ in (59), we obtain the following result.

Corollary 3.

In the conditions of Theorem 3, if X and ${\Phi }_{ϵ}Y$ are orthogonal vector fields in $\chi \left(\tilde{M}\right)$, then the Φ-holomorphic bi-sectional curvature has the form

$$H_{{\pi }_1,{\pi }_2}^{{\Phi }_{ϵ}}(X,Y)={\displaystyle \frac{3}{4}}(\tilde{K}(X,{\Phi }_{ϵ}Y)+\tilde{K}(Y,{\Phi }_{ϵ}X))+\tilde{K}(X,Y).$$

(61)

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a $2m$-dimensional Kähler golden manifold, where $m>1$, having a constant sectional curvature. If X and ${\Phi }_{ϵ}Y$ are orthogonal vector fields in $\chi \left(\tilde{M}\right)$, in the condition of Theorem 1, we have $\tilde{K}(X,Y)=0$. Thus, from the equality (61), in the conditions of Corollary 3, we obtain the following result.

Corollary 4.

Let $(\tilde{M},\tilde{g},{\Phi }_{ϵ})$ be a Kähler golden manifold of real dimension $2m$ and $m>1$, having a constant sectional curvature and let X and ${\Phi }_{ϵ}Y$ be orthogonal. If the golden sectional curvature is constant for any orthonormal vector fields $X,Y\in \chi \left(\tilde{M}\right)$ such that $\{X,{\Phi }_{ϵ}X,Y,{\Phi }_{ϵ}Y\}$ are linearly independent and ${\pi }_1=span\{X,{\Phi }_{ϵ}X\}$ and ${\pi }_2=span\{Y,{\Phi }_{ϵ}Y\}$ are invariant planes under the action of ${\Phi }_{ϵ}$, then the ${\Phi }_{ϵ}$-holomorphic bi-sectional curvature is constant and has the form

$$H_{{\pi }_1,{\pi }_2}^{\Phi }(X,Y)={\displaystyle \frac{3}{2}}{c}^G.$$

where $c^G$ is the constant golden sectional curvature of $\tilde{M}$, given in (57).

5. Conclusions

Kähler manifolds have significant applications at the intersection of differential geometry, complex analysis, and algebraic geometry. When equipped with additional structures, these manifolds exhibit special types of curvature, which possess intriguing properties.

In this work, initial steps are undertaken in the study of curvature properties of Kähler golden manifolds. We define the $\Phi$-holomorphic sectional curvature and $\Phi$-holomorphic bi-sectional curvature on such manifolds and investigate their fundamental properties. Moreover, several relations between the $\Phi$-holomorphic bi-sectional curvature and the golden sectional curvature were established.

However, this research can be further extended by analyzing the existence of certain types of subvarieties within Kähler golden manifolds possessing constant sectional curvature.