On h-Quasi-Hemi-Slant Riemannian Maps

Abstract

In the present article, we indroduce and study h-quasi-hemi-slant (in short, h-qhs) Riemannian maps and almost h-qhs Riemannian maps from almost quaternionic Hermitian manifolds to Riemannian manifolds. We investigate some fundamental results mainly on h-qhs Riemannian maps: the integrability of distributions, geometry of foliations, the condition for such maps to be totally geodesic, etc. At the end of this article, we give two non-trivial examples of this notion.

1. Introduction

In Riemannian geometry, there are few appropriate maps among Riemannian manifolds that compare their geometric properties. In this direction, as a generalization of the notions of isometric immersions and Riemannian submersions, Riemannian maps between Riemannian manifolds were initiated by Fischer [1], while isometric immersions and Riemannian submersions were widely studied in [2] and [3], respectively. However, the notion of Riemannian maps is a new research topic for geometers. More precisely, a differentiable map $\pi :(N_1,g_1)\to (N_2,g_2)$ between Riemannian manifolds $(N_1,g_1)$ and $(N_2,g_2)$ is called a Riemannian map ($0<rank{\pi }_{\ast }<min\{m,n\},$ where $dim{N}_1=m$ and $dim{N}_2=n$) if it satisfies the following equation:

$$g_2({\pi }_{\ast }{W}_1,{\pi }_{\ast }{W}_2)=g_1(W_1,W_2),\mathrm{for}{W}_1,W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp },$$

(1)

where ${\pi }_{\ast }$ is the differentiable map of $\pi .$

Consequently, isometric immersions and Riemannian submersions are particular cases of Riemannian maps with $ker{\pi }_{\ast }=0$ and ${\left(range{\pi }_{\ast }\right)}^{\perp }=0,$ respectively [1].

The other prominent basic map for comparing geometric structures between Riemannian manifolds is Riemannian submersion, and it was studied by O’Neill [4] and Gray [5]. In 1976, Watson [6] studied Riemannian submersion between Riemannian manifolds equipped with differentiable structures. After that, several kinds of Riemannian submersions were introduced and studied, including Riemannian submersion [3], H-anti-invariant submersion [7], H-semi-invariant submersion [8] and H-semi-slant submersion [9].

Currently, one of the most inventive topics in differential geometry is the theory of Riemannian maps between different Riemannian manifolds. It is well known that differentiable maps between Riemannian manifolds have wide applications in differential geometry as well as in physics, such as in Yang–Mills theory [10], Kaluza–Klein theory [11], and supergravity and superstring theories [12].

We also note that quarternionic manifolds have many applications, including for nonlinear $\sigma$ models with super symmetry [12], in the theory of harmonic differential forms [13] and obtaining estimates for the Betti numbers of the manifold [14,15]. In this paper, we have for the first time investigated h-qhs Riemannian maps from almost quarternionic manifolds to Riemannian manifolds. Here, we mainly focus on the most fundamental and interesting geometric properties on the fibers and distributions of these maps.

Nowadays, Riemannian maps and related topics have been actively studied by many authors, such as invariant and anti-invariant Riemannian maps [16], semi-invariant Riemannian maps [17], slant Riemannian maps [18], semi-slant Riemannian maps [19,20], hemi-slant Riemannian maps [21], quasi-hemi-slant Riemannian maps [22], almost h-semi-slant Riemannian maps [23], V-quasi-bi-slant Riemannian maps [24] and Clairaut semi-invariant Riemannian maps [25]. As a generalization of h-slant Riemannian maps [26], h-semi-slant Riemannian maps [9] and h-hemi-slant Riemannian maps, we define and study h-qhs Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. In the near future, we plan to work on conformal h-qhs submersions, conformal h-qhs submersions, h-qhs semi-Riemannian submersions, etc.

This paper is structured as follows. In Section 2, we recall basic facts about Riemannian maps and almost Hermitian manifolds. In Section 3, we define h-qhs Riemannian maps and study the geometry of leaves of distributions that are involved in the definition of such maps. We give necessary and sufficient conditions for h-qhs Riemannian maps to be totally geodesic. Finally, we provide two concrete examples of h-qhs Riemannian maps.

2. Preliminaries

Let $(N_1,g_1)$ and $(N_2,g_2)$ be Riemannian manifolds and $\pi :(N_1,g_1)\to (N_2,g_2)$ be a $C^{\infty }$-Riemannian map [1].

We define O’Neill’s tensors $\mathcal{T}$ and $\mathcal{A}$ [4] by

$${\mathcal{A}}_{F_1}{F}_2=\mathcal{H}{\nabla }_{\mathcal{H}{F}_1}\mathcal{V}{F}_2+\mathcal{V}{\nabla }_{\mathcal{H}{F}_1}\mathcal{H}{F}_2,$$

(2)

$${\mathcal{T}}_{F_1}{F}_2=\mathcal{H}{\nabla }_{\mathcal{V}{F}_1}\mathcal{V}{F}_2+\mathcal{V}{\nabla }_{\mathcal{V}{F}_1}\mathcal{H}{F}_2,$$

(3)

for any vector fields $F_1,F_2$ on $N_1,$ where ∇ is the Levi-Civita connection of $g_1.$

From Equations (2) and (3), we have

$${\nabla }_{Y_1}{Y}_2={\mathcal{T}}_{Y_1}{Y}_2+\mathcal{V}{\nabla }_{Y_1}{Y}_2,$$

(4)

$${\nabla }_{Y_1}{U}_1={\mathcal{T}}_{Y_1}{U}_1+\mathcal{H}{\nabla }_{Y_1}{U}_1,$$

(5)

$${\nabla }_{U_1}{Y}_1={\mathcal{A}}_{U_1}{Y}_1+\mathcal{V}{\nabla }_{U_1}{Y}_1,$$

(6)

$${\nabla }_{U_1}{U}_2=\mathcal{H}{\nabla }_{U_1}{U}_2+{\mathcal{A}}_{U_1}{U}_2,$$

(7)

for $Y_1,Y_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $U_1,U_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp },$ where $\mathcal{H}{\nabla }_{Y_1}{U}_1={\mathcal{A}}_{U_1}{Y}_1$ and $U_1$ is basic.

Let $\pi :(N_1,E,g_1)\to (N_2,g_2)$ be a $C^{\infty }$ map. The second fundamental form of $\pi$ is given by

$$(\nabla {\pi }_{\ast })(V_1,V_2)={\nabla }_{V_1}^{\pi }{\pi }_{\ast }\left(V_2\right)-{\pi }_{\ast }\left({\nabla }_{V_1}^{N_1}{V}_2\right),$$

(8)

for $V_1,V_2\in \mathsf{\Gamma }\left(T{N}_1\right),$ where ${\nabla }^{\pi }$ is the pullback connection [27]. The map $\pi$ is said to be a total geodesic if $(\nabla {\pi }_{\ast })(V_1,V_2)=0$ for $V_1,V_2\in$ $\mathsf{\Gamma }\left(T{N}_1\right)$.

Let $(N_1,E,g_1)$ be an almost quaternionic Hermitian manifold, where $g_1$ is a Riemanian metric on the maniifold $N_1$ and E is a rank 3 subbundle of $End\left(T{N}_1\right)$ such that for any point $p\in N_1$ within some neighborhood U, there exists a local basis $\{J_1,J_2,J_3\}$ of sections of E on U satisfying all $\alpha \in \{1,2,3\}$ in which

$$J_{\alpha }^2=-id,J_{\alpha }{J}_{\alpha +1}=-J_{\alpha +1}{J}_{\alpha }=J_{\alpha +2},$$

(9)

$$g_1(J_{\alpha }{X}_2,J_{\alpha }{X}_1)=g_1(X_1,X_2),$$

(10)

for $X_1,X_2\in \mathsf{\Gamma }\left(T{N}_1\right),$ where the indices are taken from $\{1,2,3\}$ modulo 3 and $\{J_1,J_2,J_3\}$ is called the quaternionic Hermitian basis. The structure $(N_1,E,g_1)$ is called a quaternionic kähler manifold if there exist locally defined 1 forms ${\omega }_1,{\omega }_2,{\omega }_3$ such that for $\alpha \in \{1,2,3\}$, we have

$${\nabla }_{X_1}{J}_{\alpha }={\omega }_{\alpha +2}\left(X_1\right)J_{\alpha +1}-{\omega }_{\alpha +1}\left(X_1\right)J_{\alpha +2},$$

(11)

for $X_1\in \mathsf{\Gamma }\left(T{N}_1\right),$ where the indices are taken from $\{1,2,3\}$ modulo $3.$ If there exists a global parallel quaternionic Hermitian basis $\{J_1,J_2,J_3\}$ of sections of E on $N_1$, then $(N_1,E,g_1)$ is called a hyperkähler. The structure $\{J_1,J_2,J_3,g_1\},$ where $g_1$, a hyperkähler metric, is called a hyperkähler structure on $N_1$.

A map $\pi :(N_1,E_1,g_1)\to (N_2,E_2,g_2)$ is called an $(E_{{}_1},E_{{}_2})$-holomorphic map if for any point $p\in N_1$ and $J\in {\left(E_1\right)}_p$, there exists $J^{\prime }\in {\left(E_2\right)}_{\pi \left(p\right)}$ such that

$${\pi }_{\ast }\circ J=J^{\prime }\circ {\pi }_{\ast }.$$

A Riemannian submersion between quaternionic kähler manifolds $\pi :(N_1,E_1,g_1)\to (N_2,E_2,g_2),$ which is an $(E_1,E_2)$-holomorphic map, is known as a quaternionic kähler submersion (or a hyperkähler submersion) [9]:

Definition 1

([23]). A Riemannian map π from the almost quaternionic Hermitian manifold $(N_1,E,g_1)$ to the Riemannian manifold $(N_2,g_2)$ is called an h-semi-slant Riemannian map if, given a point $p\in N_1$ with a neighborhood U, there exists a quaternionic Hermitian basis $\{I,J,K\}$ of sections of E on U such that for any $R\in \{I,J,K\}$, the following is true:

$$ker{\pi }_{\ast }=D_1\oplus D_2,R\left(D_1\right)=D_1,$$

in which the angle ${\theta }_R={\theta }_R\left(Z_1\right)$ between $R{Z}_1$ and the space ${\left(D_2\right)}_q$ is constant for a non-zero $Z_1\in {\left(D_2\right)}_q$ and $q\in U,$ where $D_2$ is an orthogonal complement of $D_1$ in $ker{\pi }_{\ast }.$

Furthermore, assume we have

$$\theta ={\theta }_I={\theta }_J={\theta }_K,$$

Then, we call the map $\pi :(N_1,E,g_1)\to (N_2,g_2)$ a strictly h-semi-slant Riemannian map, the basis $\{I,J,K\}$ a strictly h-semi-slant basis and the angle $\theta$ a strictly h-semi-slant angle.

3. h-Quasi-Hemi-Slant Riemannian Maps

Motivated by the studies given in Section 2, we give the definition of the h-qhs Riemannian map as follows:

Definition 2.

A Riemannian map π from the almost quaternionic Hermitian manifold $(N_1,E,g_1)$ to the Riemannian manifold $(N_2,g_2)$ is called an h-qhs Riemannian map if, given a point $p\in N_1$ with a neighborhood U, there exists a quaternionic Hermitian basis $\{I,J,K\}$ of sections of E on U such that for any $R\in \{I,J,K\}$, there is a distribution $D\subset (ker{\pi }_{\ast })$ on U such that

$$ker{\pi }_{\ast }=D\oplus D_1\oplus D_2,R\left(D\right)=D,R\left(D_2\right)\subset {(ker{\pi }_{\ast })}^{\perp },$$

and the angle ${\theta }_R={\theta }_R\left(Z_1\right)$ between $R{Z}_1$ and the space ${\left(D_1\right)}_q$ is constant for a non-zero $Z_1\in {\left(D_1\right)}_q$ and $q\in U,$ where $ker{\pi }_{\ast }$ admits three orthogonal complementary distributions $D,D_1$ and $D_2$ such that D is invariant, $D_1$ is a slant with an angle ${\theta }_R$ and $D_2$ is anti-invariant.

We call the basis $\{I,J,K\}$ an h-qhs basis and the angles $\{{\theta }_I,{\theta }_J,{\theta }_K\}$h-qhs angles.

Furthermore, let us say we have

$$\theta ={\theta }_I={\theta }_J={\theta }_K,$$

Then, we call the map $\pi :(N_1,E,g_1)\to (N_2,g_2)$ a strictly h-qhs Riemannian map, the basis $\{I,J,K\}$ a strictly quasi-hemi-slant basis and the angle $\theta$ a strictly quasi-hemi-slant angle:

Definition 3.

A Riemannian map π from the almost quaternionic Hermitian manifold $(N_1,E,g_1)$ to the Riemannian manifold $(N_2,g_2)$ is called an almost h-qhs Riemannian map if, given a point $p\in N_1$ with a neighborhood U, there exists a quaternionic Hermitian basis $\{I,J,K\}$ of sections of E on U such that for any $R\in \{I,J,K\},$ there is a distribution $D^R\subset (ker{\pi }_{\ast })$ on U such that

$$ker{\pi }_{\ast }=D^R\oplus D_1^R\oplus D_2^R,R\left(D^R\right)=D^R,R\left(D_2^R\right)\subset {(ker{\pi }_{\ast })}^{\perp },$$

and the angle ${\theta }_R={\theta }_R\left(Z_1\right)$ between $R{Z}_1$ and the space ${\left(D_1^R\right)}_q$ is constant for a non-zero $Z_1\in {\left(D_1^R\right)}_q$ and $q\in U,$ where the vertical distribution $ker{\pi }_{\ast }$ admits three orthogonal complementary distributions $D^R,D_1^R$ and $D_2^R$ such that $D^R$ is invariant, $D_1^R$ is a slant with an angle ${\theta }_R$ and $D_2^R$ is anti-invariant.

We call the basis $\{I,J,K\}$ an almost h-qhs basis and the angles $\{{\theta }_I,{\theta }_J,{\theta }_K\}$ almost h-qhs angles.

Let $\pi :(N_1,E,g_1)\to (N_2,g_2)$ be an almost h-qhs Riemannian map. We can easily observe the following:

(a)

If $dim{D}^R\ne 0$, $dim{D}_1^R\ne 0,$ $0<{\theta }_R<\frac{\pi }{2}$ and $dim{D}_2^R=0,$ then $\pi$ is an almost proper h-semi-slant Riemannian map with a semi-slant angle ${\theta }_R$;

(b)

If $dim{D}^R=0$, $dim{D}_1^R\ne 0,$ $0<{\theta }_R<\frac{\pi }{2}$ and $dim{D}_2^R\ne 0,$ then $\pi$ is an almost h-hemi-slant Riemannian map.

We say that the almost h-qhs Riemannian map $\pi :(N_1,E,g_1)\to (N_2,g_2)$ is proper if $D^R\ne \left\{0\right\}$, $D_2^R\ne \left\{0\right\}$ and ${\theta }_R\ne 0,\frac{\pi }{2}.$ Thus, one can easily see that the h-hemi-slant Riemannian map, h-semi-invariant Riemannian map and h-semi-slant Riemannian map are examples of h-qhs Riemannian maps.

Thus, we have

$${(ker{\pi }_{\ast })}^{\perp }={\omega }_R\left(D_1^R\right)\oplus R\left(D_2^R\right)\oplus {\mu }_R.$$

Obviously, ${\mu }_R$ is an invariant sub-bundle of ${(ker{\pi }_{\ast })}^{\perp }$ with respect to the complex structure R.

For $V_1\in \mathsf{\Gamma }(ker{\pi }_{\ast }),$ we have

$$V_1=P_R{V}_1+Q_R{V}_1+S_R{V}_1,$$

(12)

where $P_R{V}_1\in \mathsf{\Gamma }\left(D^R\right)$, $Q_R{V}_1\in \mathsf{\Gamma }\left(D_1^R\right)$, $S_R{V}_1\in \mathsf{\Gamma }\left(D_2^R\right)$ and $R\in \{I,J,K\}.$

For $Z_1\in \mathsf{\Gamma }(ker{\pi }_{\ast }),$ we obtain

$$R{Z}_1={\varphi }_R{Z}_1+{\omega }_R{Z}_1,$$

(13)

where ${\varphi }_R{Z}_1\in \mathsf{\Gamma }(ker{\pi }_{\ast })$, ${\omega }_R{Z}_1\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}.$

For $X_1\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp },$ we have

$$R{X}_1=B_R{X}_1+C_R{X}_1,$$

(14)

where $B_R{X}_1\in \mathsf{\Gamma }(ker{\pi }_{\ast })$, $C_R{X}_1\in \mathsf{\Gamma }\left({\mu }_R\right)$ and $R\in \{I,J,K\}.$

We will denote an almost h-qhs Riemannian map from a hyperkähler manifold $(N_1,I,J,K,g_1)$ onto a Riemannian manifold $(N_2,g_2)$ such that $(I,J,K)$ is an almost h-qhs basis by $\pi$.

The following lemmas can be easily obtained:

Lemma 1.

For $\pi :(N_1,g_1,E_1)\to (N_2,g_2,E_2)$, we get

$${\varphi }_R{D}^R=D^R,{\omega }_R{D}^R=0,{\varphi }_R{D}_2^R=0,{\omega }_R{D}_2^R\subset {(ker{\pi }_{\ast })}^{\perp },$$

where $R\in \{I,J,K\}.$

Lemma 2.

For $\pi :(N_1,g_1,E_1)\to (N_2,g_2,E_2)$, we have

$$\begin{array}{ccc}\hfill {\varphi }_R^2{Z}_1+B_R{\omega }_R{Z}_1& =& -Z_1,{\omega }_R{\varphi }_R{Z}_1+C_R{\omega }_R{Z}_1=0,\hfill \\ \hfill {\varphi }_R{B}_R{Z}_2+B_R{C}_R{Z}_2& =& 0,{\omega }_R{B}_R{Z}_2+C_R^2{Z}_2=-Z_2,\hfill \end{array}$$

for any $Z_1\in \mathsf{\Gamma }(ker{\pi }_{\ast })$, $Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}.$

Proof. 

Using Equations (9), (13) and (14), we can find all equations of Lemma 2: □

Lemma 3.

With $\pi :(N_1,I,J,K,g_1)\to (N_2,g_2)$ being an almost h-qhs Riemannian map, we then obtain

$$\mathcal{V}{\nabla }_{X_1}{\varphi }_R{X}_2+{\mathcal{T}}_{X_1}{\omega }_R{X}_2=B_R{\mathcal{T}}_{X_1}{X}_2+{\varphi }_R\mathcal{V}{\nabla }_{X_1}{X}_2,$$

(15)

$${\mathcal{T}}_{X_1}{\varphi }_R{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_R{X}_2=C_R{\mathcal{T}}_{X_1}{X}_2+{\omega }_R\mathcal{V}{\nabla }_{X_1}{X}_2,$$

(16)

$${\mathcal{T}}_{X_1}{B}_R{Z}_1+\mathcal{H}{\nabla }_{X_1}{C}_R{Z}_1=C_R\mathcal{H}{\nabla }_{X_1}{Z}_1+{\omega }_R{\mathcal{T}}_{X_1}{Z}_1,$$

(17)

$$\mathcal{V}{\nabla }_{X_1}{B}_R{Z}_1+{\mathcal{T}}_{X_1}{C}_R{Z}_1=B_R\mathcal{H}{\nabla }_{X_1}{Z}_1+\varphi {\mathcal{T}}_{X_1}{Z}_1,$$

(18)

$$\mathcal{V}{\nabla }_{Z_1}{\varphi }_R{X}_1+{\mathcal{A}}_{Z_1}{\omega }_R{X}_1=B_R{\mathcal{A}}_{Z_1}{X}_1+{\varphi }_R\mathcal{V}{\nabla }_{Z_1}{X}_1,$$

(19)

$${\mathcal{A}}_{Z_1}{\varphi }_R{X}_1+\mathcal{H}{\nabla }_{Z_1}{\omega }_R{X}_1=C_R{\mathcal{A}}_{Z_1}{X}_1+{\omega }_R\mathcal{V}{\nabla }_{Z_1}{X}_1,$$

(20)

$${\mathcal{A}}_{Z_1}{B}_R{Z}_2+\mathcal{H}{\nabla }_{Z_1}{C}_R{Z}_2=C_R\mathcal{H}{\nabla }_{Z_1}{Z}_2+{\omega }_R{\mathcal{A}}_{Z_1}{Z}_2,$$

(21)

$$\mathcal{V}{\nabla }_{Z_1}{B}_R{Z}_2+{\mathcal{A}}_{Z_1}{C}_R{Z}_2=B_R\mathcal{H}{\nabla }_{Z_1}{Z}_2+{\varphi }_R{\mathcal{A}}_{Z_1}{Z}_2,$$

(22)

for $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$, $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}.$

Proof. 

Using Equations (4)–(7), (13) and (14), we can easily obtain Equations (15)–(22). □

Now, we define

$$\left({\nabla }_{X_1}{\varphi }_R\right)X_2=\mathcal{V}{\nabla }_{X_1}{\varphi }_R{X}_2-{\varphi }_R\mathcal{V}{\nabla }_{X_1}{X}_2,$$

(23)

$$\left({\nabla }_{X_1}{\omega }_R\right)X_2=\mathcal{H}{\nabla }_{X_1}{\omega }_R{X}_2-{\omega }_R\mathcal{V}{\nabla }_{X_1}{X}_2,$$

(24)

$$\left({\nabla }_{Z_1}{B}_R\right)Z_2=\mathcal{V}{\nabla }_{Z_1}{B}_R{Z}_2-B_R\mathcal{H}{\nabla }_{Z_1}{Z}_2,$$

(25)

$$\left({\nabla }_{Z_1}{C}_R\right)Z_2=\mathcal{H}{\nabla }_{Z_1}{C}_R{Z}_2-C_R\mathcal{H}{\nabla }_{Z_1}{Z}_2,$$

(26)

for $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$, $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}.$

Lemma 4.

For $\pi :(N_1,I,J,K,g_1)\to (N_2,g_2)$, we find

$$\left({\nabla }_{X_1}{\varphi }_R\right)X_2=B_R{\mathcal{T}}_{X_1}{X}_2-{\mathcal{T}}_{X_1}{\omega }_R{X}_2,\left({\nabla }_{X_1}{\omega }_R\right)X_2=C_R{\mathcal{T}}_{X_1}{X}_2-{\mathcal{T}}_{X_1}{\varphi }_R{X}_2,$$

$$\left({\nabla }_{Z_1}{C}_R\right)Z_2={\omega }_R{\mathcal{A}}_{Z_1}{Z}_2-{\mathcal{A}}_{Z_1}{B}_R{Z}_2,\left({\nabla }_{Z_1}{B}_R\right)Z_2={\varphi }_R{\mathcal{A}}_{Z_1}{Z}_2-{\mathcal{A}}_{Z_1}{C}_R{Z}_2,$$

for all $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast }),$ $Z_1$, $Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}.$

Proof. 

Using Equations (15) and (16) as well as Equations (21)–(26), Lemma 4 follows. □

If the tensors ${\varphi }_R$ and ${\omega }_R$ are parallel with respect to the linear connection ∇ on $N_1$, then

$$B_R{\mathcal{T}}_{X_1}{X}_2={\mathcal{T}}_{X_1}{\omega }_R{X}_2,C_R{\mathcal{T}}_{X_1}{X}_2={\mathcal{T}}_{X_1}{\varphi }_R{X}_2,$$

for all $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $R\in \{I,J,K\}$:

Lemma 5.

Let $\pi :(N_1,E,g_1)\to (N_2,g_2)$, be an almost h-qhs Riemannian map. Then, we obtain

$${\varphi }_R^2{V}_1=-{cos}^2{\theta }_R{V}_1,$$

(27)

for any non-zero vector field $V_1\in \mathsf{\Gamma }\left(D_1^R\right)$ and $R\in \{I,J,K\}$, where $\{I,J,K$} is an almost h-qhs basis with the almost h-qhs angles $\{{\theta }_I,{\theta }_J,{\theta }_K\}$.

Proof. 

For any non-zero vector field $V_1\in \mathsf{\Gamma }\left(D_1^R\right)$ and $R\in \{I,J,K\}$, we have

$$cos{\theta }_R=\frac{\Vert {\varphi }_R{V}_1\Vert }{\Vert R{V}_1\Vert },$$

(28)

and

$$cos{\theta }_R=\frac{g_1(R{V}_1,{\varphi }_R{V}_1)}{\Vert {\varphi }_R{V}_1\Vert \Vert R{V}_1\Vert },$$

(29)

where ${\theta }_R\left(V_1\right)$ is the h-qhs angle.

Using Equations (9) and (13), we obtain

$$cos{\theta }_R=-\frac{g_1(V_1,{{\varphi }_R}^2{V}_1)}{\Vert {\varphi }_R{V}_1\Vert \Vert R{V}_1\Vert }.$$

(30)

From Equations (29) and (30), Equation (27) follows. □

Theorem 1.

Let π be an h-qhs Riemannian map from an almost hyperkahler manifold $(N_1,I,J,K,g_1)$ to a Riemannian manifold $(N_2,g_2)$. Then, the following cases are equivalent:

(a)

$D^R$is integrable;

(b)

$$g_1({\mathcal{T}}_{Z_2}I{Z}_1-{\mathcal{T}}_{Z_1}I{Z}_2,{\omega }_I{Q}_I{U}_1+I{S}_I{U}_1)=g_1(\mathcal{V}{\nabla }_{Z_1}I{Z}_2-\mathcal{V}{\nabla }_{Z_2}I{Z}_1,{\varphi }_I{Q}_I{U}_1)$$

for $Z_1,Z_2\in \mathsf{\Gamma }\left(D^I\right)$ and $U_1\in \mathsf{\Gamma }(D_1^I\oplus D_2^I)$;

(c)

$$g_1({\mathcal{T}}_{Z_2}J{Z}_1-{\mathcal{T}}_{Z_1}J{Z}_2,{\omega }_J{Q}_J{U}_1+J{S}_J{U}_1)=g_1(\mathcal{V}{\nabla }_{Z_1}J{Z}_2-\mathcal{V}{\nabla }_{Z_2}J{Z}_1,{\varphi }_J{Q}_J{U}_1)$$

for $Z_1,Z_2\in \mathsf{\Gamma }\left(D^J\right)$ and $U_1\in \mathsf{\Gamma }(D_1^J\oplus D_2^J)$;

(d)

$$g_1({\mathcal{T}}_{Z_2}K{Z}_1-{\mathcal{T}}_{Z_1}K{Z}_2,{\omega }_K{Q}_K{U}_1+K{S}_K{U}_1)=g_1(\mathcal{V}{\nabla }_{Z_1}K{Z}_2-\mathcal{V}{\nabla }_{Z_2}K{Z}_1,{\varphi }_K{Q}_K{U}_1)$$

for $Z_1,Z_2\in \mathsf{\Gamma }\left(D^K\right)$ and $U_1\in \mathsf{\Gamma }(D_1^K\oplus D_2^K).$

Proof. 

For $Z_1,Z_2\in \mathsf{\Gamma }\left(D^R\right)$, $U_1\in \mathsf{\Gamma }(D_1^R\oplus D_2^R)$, $U_2\in {(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, since $[Z_1,Z_2]\in (ker{\pi }_{\ast }),$ we have $g_1([Z_1,Z_2],U_2)=0.$ Thus, $D^R$ is integrable ⇔ $g_1([Z_1,Z_2],U_1)=0.$ Now, using Equations (4) and (12)–(14), we have

$$\begin{array}{ccc}& & g_1([Z_1,Z_2],U_1)\hfill \\ & =& g_1(R{\nabla }_{Z_1}{Z}_2,R{U}_1)-g_1(R{\nabla }_{Z_2}{Z}_1,R{U}_1),\hfill \\ & =& g_1({\nabla }_{Z_1}R{Z}_2,R{U}_1)-g_1({\nabla }_{Z_2}R{Z}_1,R{U}_1),\hfill \\ & =& g_1({\mathcal{T}}_{Z_1}R{Z}_2-{\mathcal{T}}_{Z_2}R{Z}_1,{\omega }_R{Q}_R{U}_1+JR{U}_1)\hfill \\ & & -g_1(\mathcal{V}{\nabla }_{Z_1}R{Z}_2-\mathcal{V}{\nabla }_{Z_2}R{Z}_1,{\varphi }_R{Q}_R{U}_1).\hfill \end{array}$$

Since $D^R$ is R-invariant, we have

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, we obtain the result. □

Theorem 2.

The following cases are equivalent for the map π defined in Theorem 1:

(a)

$D_1^R$is integrable;

(b)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Y_1}{\omega }_I{\varphi }_I{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_I{\varphi }_I{Y}_1,V_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_I{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_I{Y}_1,{\varphi }_I{P}_I{V}_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{Y}_2-\mathcal{H}{\nabla }_{Y_2}{\omega }_I{Y}_1,{\omega }_I{S}_I{V}_1)\hfill \end{array}$$

for all $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^I\right)$ and $V_1\in \mathsf{\Gamma }(D^I\oplus D_2^I)$;

(c)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Y_1}{\omega }_J{\varphi }_J{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_J{\varphi }_J{Y}_1,V_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_J{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_J{Y}_1,{\varphi }_J{P}_J{V}_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_J{Y}_2-\mathcal{H}{\nabla }_{Y_2}{\omega }_J{Y}_1,{\omega }_J{S}_J{V}_1)\hfill \end{array}$$

for all $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^J\right)$ and $V_1\in \mathsf{\Gamma }(D^J\oplus D_2^J)$;

(d)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Y_1}{\omega }_K{\varphi }_K{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_K{\varphi }_K{Y}_1,V_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_K{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_K{Y}_1,{\varphi }_K{P}_K{V}_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_K{Y}_2-\mathcal{H}{\nabla }_{Y_2}{\omega }_K{Y}_1,{\omega }_K{S}_K{V}_1)\hfill \end{array}$$

for all $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^K\right)$ and $V_1\in \mathsf{\Gamma }(D^K\oplus D_2^K).$

Proof. 

For $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^R\right)$, $V_1\in \mathsf{\Gamma }(D^R\oplus D_2^R)$, $V_2\in {(ker{F}_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, since $[Y_1,Y_2]\in (ker{\pi }_{\ast })$, we have $g_1([Y_1,Y_2],V_2)=0.$ Thus, $D_1^R$ is integrable ⇔$g_1([Y_1,Y_2],V_1)=0.$ Using Equations (4), (5), (12) and (13) as well as Lemma $5,$ we have

$$\begin{array}{ccc}& & g_1([Y_1,Y_2],V_1)\hfill \\ & =& g_1({\nabla }_{Y_1}R{Y}_2,R{V}_1)-g_1({\nabla }_{Y_2}R{Y}_1,R{V}_1),\hfill \\ & =& g_1({\nabla }_{Y_1}{\varphi }_R{Y}_2,R{V}_1)+g_1({\nabla }_{Y_1}{\omega }_R{Y}_2,R{V}_1)-g_1({\nabla }_{Y_2}{\varphi }_R{Y}_1,R{V}_1)-g_1({\nabla }_{Y_2}{\omega }_R{Y}_1,R{V}_1),\hfill \\ & =& {cos}^2{\theta }_R{g}_1({\nabla }_{Y_1}{Y}_2,V_1)-{cos}^2{\theta }_R{g}_1({\nabla }_{Y_2}{Y}_1,V_1)-g_1({\mathcal{T}}_{Y_1}{\omega }_R{\varphi }_R{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_R{\varphi }_R{Y}_1,V_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{Y}_2+{\mathcal{T}}_{Y_1}{\omega }_R{Y}_2,R{P}_R{V}_1+{\omega }_R{S}_R{V}_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{Y_2}{\omega }_R{Y}_1+{\mathcal{T}}_{Y_2}{\omega }_R{Y}_1,R{P}_R{V}_1+{\omega }_R{S}_R{V}_1),\hfill \end{array}$$

which gives

$$\begin{array}{ccc}\hfill {sin}^2{\theta }_1{g}_1([Y_1,Y_2],V_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_R{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_R{Y}_1,R{P}_R{V}_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{Y}_2-\mathcal{H}{\nabla }_{Y_2}{\omega }_R{Y}_1,{\omega }_R{S}_R{V}_1)\hfill \\ & & -g_1({\mathcal{T}}_{Y_1}{\omega }_R{\varphi }_R{Y}_2-{\mathcal{T}}_{Y_2}{\omega }_R{\varphi }_R{Y}_1,V_1).\hfill \end{array}$$

Since $D_1^R$ is an R-slant distribution, therefore, we obtain

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, we find the result. □

Theorem 3.

For the h-qhs Riemannian map π defined in Theorem 1, $D_2^R$ is always integrable.

Proof. 

We can easily prove the Theorem as hemi-slant case given in [21]. □

Theorem 4.

For the h-qhs Riemannian map π defined in Theorem 1, any one of the following assertions implies the others:

(a)

${(ker{\pi }_{\ast })}^{\perp }$defines a totally geodesic foliation on $N_1$;

(b)

$$\begin{array}{ccc}\hfill g_1({\mathcal{A}}_{Z_1}{Z}_2,P_I{W}_1+{cos}^2{\theta }_I{Q}_I{W}_1)& =& g_1(\mathcal{H}{\nabla }_{Z_1}{Z}_2,{\omega }_I{\varphi }_I{P}_I{W}_1+{\omega }_I{\varphi }_I{Q}_I{W}_1)\hfill \\ & & -g_1({\mathcal{A}}_{Z_1}{B}_I{Z}_2+\mathcal{H}{\nabla }_{Z_1}{C}_I{Z}_2,{\omega }_I{W}_1)\hfill \end{array}$$

for $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $W_1\in \mathsf{\Gamma }(ker{\pi }_{\ast })$;

(c)

$$\begin{array}{ccc}\hfill g_1({\mathcal{A}}_{Z_1}{Z}_2,P_J{W}_1+{cos}^2{\theta }_J{Q}_J{W}_1)& =& g_1(\mathcal{H}{\nabla }_{Z_1}{Z}_2,{\omega }_J{\varphi }_J{P}_J{W}_1+{\omega }_J{\varphi }_J{Q}_J{W}_1)\hfill \\ \hfill & & -g_1({\mathcal{A}}_{Z_1}{B}_J{Z}_2+\mathcal{H}{\nabla }_{Z_1}{C}_J{Z}_2,{\omega }_J{W}_1)\hfill \end{array}$$

for $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $W_1\in \mathsf{\Gamma }(ker{\pi }_{\ast })$;

(d)

$$\begin{array}{ccc}\hfill g_1({\mathcal{A}}_{Z_1}{Z}_2,P_K{W}_1+{cos}^2{\theta }_K{Q}_K{W}_1)& =& g_1(\mathcal{H}{\nabla }_{Z_1}{Z}_2,{\omega }_K{\varphi }_K{P}_K{W}_1+{\omega }_K{\varphi }_K{Q}_K{W}_1)\hfill \\ & & -g_1({\mathcal{A}}_{Z_1}{B}_K{Z}_2+\mathcal{H}{\nabla }_{Z_1}{C}_K{Z}_2,{\omega }_K{W}_1)\hfill \end{array}$$

for $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $W_1\in \mathsf{\Gamma }(ker{\pi }_{\ast }).$

Proof. 

For $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp },W_1\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $R\in \{I,J,K\}$, using Equations (6), (7) and (12)–(14) as well as Lemma 5, we have

$$\begin{array}{ccc}& & g_1({\nabla }_{Z_1}{Z}_2,W_1)\hfill \\ & =& g_1(R{\nabla }_{Z_1}{Z}_2,R{W}_1),\hfill \\ & =& g_1(R{\nabla }_{Z_1}{Z}_2,{\varphi }_R{P}_R{W}_1+{\varphi }_R{Q}_R{W}_1+{\omega }_R{Q}_R{W}_1+{\omega }_R{S}_R{W}_1),\hfill \\ & =& -g_1({\nabla }_{Z_1}{Z}_2,{\varphi }_R^2{P}_R{W}_1+{\omega }_R{\varphi }_R{P}_R{W}_1+{\omega }_R{\varphi }_R{Q}_R{W}_1)\hfill \\ & & +g_1({\nabla }_{Z_1}{B}_R{Z}_2,{\omega }_R{Q}_R{W}_1+{\omega }_R{S}_R{W}_1)+g_1({\nabla }_{Z_1}{C}_R{Z}_2,{\omega }_R{Q}_R{W}_1+{\omega }_R{S}_R{W}_1),\hfill \\ & =& g_1({\mathcal{A}}_{Z_1}{Z}_2,P_R{W}_1+{cos}^2{\theta }_R{Q}_R{W}_1)-g_1(\mathcal{H}{\nabla }_{Z_1}{Z}_2,{\omega }_R{\varphi }_R{P}_R{W}_1+{\omega }_R{\varphi }_R{Q}_R{W}_1)\hfill \\ & & +g_1({\mathcal{A}}_{Z_1}{B}_R{Z}_2,{\omega }_R{Q}_R{W}_1+{\omega }_R{S}_R{W}_1)+g_1(\mathcal{H}{\nabla }_{Z_1}{C}_R{Z}_2,{\omega }_R{Q}_R{W}_1+{\omega }_R{S}_R{W}_1).\hfill \end{array}$$

Thus, we obtain

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, the result follows. □

Theorem 5.

The following conditions are equivalent for the h-qhs Riemannian map π:

(a)

$(ker{\pi }_{\ast })$defines a totally geodesic foliation on $N_1$;

(b)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{X_1}{P}_I{X}_2+{cos}^2{\theta }_I{\mathcal{T}}_{X_1}{Q}_I{X}_2,Y_1)& =& g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_I{\varphi }_I{P}_I{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_I{\varphi }_I{Q}_I{X}_2,Y_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_I{Q}_I{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_I{S}_I{X}_2,C_I{Y}_1)\hfill \\ & & -g_1({\mathcal{T}}_{X_1}{\omega }_I{Q}_I{X}_2+{\mathcal{T}}_{X_1}{\omega }_I{S}_I{X}_2,B_I{Y}_1)\hfill \end{array}$$

for $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $Y_1\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(c)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{X_1}{P}_J{X}_2+{cos}^2{\theta }_J{\mathcal{T}}_{X_1}{Q}_J{X}_2,Y_1)& =& g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_J{\varphi }_J{P}_J{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_J{\varphi }_J{Q}_J{X}_2,Y_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_J{Q}_J{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_J{S}_J{X}_2,C_J{Y}_1)\hfill \\ & & -g_1({\mathcal{T}}_{X_1}{\omega }_J{Q}_J{X}_2+{\mathcal{T}}_{X_1}{\omega }_J{S}_J{X}_2,B_J{Y}_1)\hfill \end{array}$$

for $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $Y_1\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(d)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{X_1}{P}_K{X}_2+{cos}^2{\theta }_K{\mathcal{T}}_{X_1}{Q}_K{X}_2,Y_1)& =& g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_K{\varphi }_K{P}_K{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_K{\varphi }_K{Q}_K{X}_2,Y_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_K{Q}_K{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_K{S}_K{X}_2,C_K{Y}_1)\hfill \\ & & -g_1({\mathcal{T}}_{X_1}{\omega }_K{Q}_K{X}_2+{\mathcal{T}}_{X_1}{\omega }_K{S}_K{X}_2,B_K{Y}_1)\hfill \end{array}$$

for $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $Y_1\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

For $X_1,X_2\in \mathsf{\Gamma }(ker{\pi }_{\ast }),Y_1\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, using Equations (4), (5) and (12)–(14) as well as Lemma 5, we have

$$\begin{array}{ccc}& & g_1({\nabla }_{X_1}{X}_2,Y_1)\hfill \\ & =& g_1(R{\nabla }_{X_1}{X}_2,R{Y}_1),\hfill \\ & =& g_1({\nabla }_{X_1}{\varphi }_R{P}_R{X}_2,R{Y}_1)+g_1({\nabla }_{X_1}{\varphi }_R{Q}_R{X}_2,R{Y}_1)\hfill \\ & & +g_1({\nabla }_{X_1}{\omega }_R{Q}_R{X}_2,R{Y}_1)+g_1({\nabla }_{X_1}{\omega }_R{S}_R{X}_2,R{Y}_1),\hfill \\ & =& g_1({\mathcal{T}}_{X_1}{P}_R{X}_2,Y_1)+{cos}^2{\theta }_R{g}_1({\mathcal{T}}_{X_1}{Q}_R{X}_2,Y_1)-g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_R{\varphi }_R{P}_R{X}_2,Y_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_R{\varphi }_R{Q}_R{X}_2,Y_1)+g_1(\mathcal{H}{\nabla }_{X_1}{\omega }_R{Q}_R{X}_2+\mathcal{H}{\nabla }_{X_1}{\omega }_R{S}_R{X}_2,C_R{Y}_1)\hfill \\ & & +g_1({\mathcal{T}}_{X_1}{\omega }_R{Q}_R{X}_2+{\mathcal{T}}_{X_1}{\omega }_R{S}_R{X}_2,B_R{Y}_1).\hfill \end{array}$$

Thus, we obtain

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, the result follows. □

Theorem 6.

Let π be an h-qhs Riemannian map from an almost hyperkahler manifold $(N_1,I,J,K,g_1)$ to a Riemannian manifold $(N_2,g_2)$. Then, any one of the following assertions implies the others:

(a)

$D^R$defines a totally geodesic foliation on $N_1$;

(b)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Z_1}I{P}_I{Z}_2,{\omega }_I{Q}_I{Y}_1+{\omega }_I{S}_I{Y}_1)& =& -g_1(\mathcal{V}{\nabla }_{Z_1}I{P}_I{Z}_2,{\varphi }_I{Y}_1),\hfill \\ \hfill g_1({\mathcal{T}}_{Z_1}I{P}_I{Z}_2,C_I{Y}_2)& =& -g_1(\mathcal{V}{\nabla }_{Z_1}I{P}_I{Z}_2,B_I{Y}_2)\hfill \end{array}$$

for $Z_1,Z_2\in \mathsf{\Gamma }\left(D^I\right),$ $Y_1\in \mathsf{\Gamma }(D_1^I\oplus D_2^I)$ and $Y_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(c)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Z_1}J{P}_J{Z}_2,{\omega }_J{Q}_J{Y}_1+{\omega }_J{S}_J{Y}_1)& =& -g_1(\mathcal{V}{\nabla }_{Z_1}J{P}_J{Z}_2,{\varphi }_J{Y}_1),\hfill \\ \hfill g_1({\mathcal{T}}_{Z_1}J{P}_J{Z}_2,C_J{Y}_2)& =& -g_1(\mathcal{V}{\nabla }_{Z_1}J{P}_J{Z}_2,B_J{Y}_2)\hfill \end{array}$$

for $Z_1,Z_2\in \mathsf{\Gamma }\left(D^J\right),$ $Y_1\in \mathsf{\Gamma }(D_1^J\oplus D_2^J)$ and $Y_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(d)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Z_1}K{P}_K{Z}_2,{\omega }_K{Q}_K{Y}_1+{\omega }_K{S}_K{Y}_1)& =& -g_1(\mathcal{V}{\nabla }_{Z_1}K{P}_K{Z}_2,{\varphi }_K{Y}_1),\hfill \\ \hfill g_1({\mathcal{T}}_{Z_1}K{P}_K{Z}_2,C_K{Y}_2)& =& -g_1(\mathcal{V}{\nabla }_{Z_1}K{P}_K{Z}_2,B_K{Y}_2)\hfill \end{array}$$

for $Z_1,Z_2\in \mathsf{\Gamma }\left(D^K\right),$ $Y_1\in \mathsf{\Gamma }(D_1^K\oplus D_2^K)$ and $Y_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

For $Z_1,Z_2\in \mathsf{\Gamma }\left(D^R\right),Y_1\in \mathsf{\Gamma }(D_1^R\oplus D_2^R),Y_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, using Equations (4), (12) and (13), we have

$$\begin{array}{ccc}& & g_1({\nabla }_{Z_1}{Z}_2,Y_1)\hfill \\ & =& g_1({\nabla }_{Z_1}R{Z}_2,R{Y}_1),\hfill \\ & =& g_1({\nabla }_{Z_1}R{P}_R{Z}_2,R{Q}_R{Y}_1+R{S}_R{Y}_1),\hfill \\ & =& g_1({\mathcal{T}}_{Z_1}{\varphi }_R{P}_R{Z}_2,{\omega }_R{Q}_R{Y}_1+{\omega }_R{S}_R{Y}_1)+g_1(\mathcal{V}{\nabla }_{Z_1}{\varphi }_R{P}_R{Z}_2,{\varphi }_R{Q}_R{Y}_1).\hfill \end{array}$$

Moreover, using Equations (4), (12) and (14), we obtain

$$\begin{array}{ccc}& & g_1({\nabla }_{Z_1}{Z}_2,Y_2)\hfill \\ & =& g_1({\nabla }_{Z_1}R{Z}_2,R{Y}_2),\hfill \\ & =& g_1({\nabla }_{Z_1}R{P}_R{Z}_2,B_R{Y}_2+C_R{Y}_2),\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{Z_1}R{P}_R{Z}_2,B_R{Y}_2)+g_1({\mathcal{T}}_{Z_1}J{P}_R{Z}_2,C_R{Y}_2).\hfill \end{array}$$

Hence, we have

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, the result follows. □

Theorem 7.

With $\pi :(N_1,I,J,K,g_1)\to (N_2,g_2)$ being an h-qhs Riemannian map, the following conditions are equivalent:

(a)

$D_1^R$defines a totally geodesic foliation on $N_1$;

(b)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Y_1}{\omega }_I{\varphi }_I{Y}_2,Z_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_I{Y}_2,{\varphi }_I{P}_I{Z}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{Y}_2,{\omega }_I{S}_I{Z}_1),\hfill \\ \hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{\varphi }_I{Y}_2,Z_2)& =& g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{Y}_2,C_I{Z}_2)+g_1({\mathcal{T}}_{Y_1}{\omega }_I{Y}_2,B_I{Z}_2)\hfill \end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^I\right),$ $Z_1\in \mathsf{\Gamma }(D^I\oplus D_2^I)$ and $Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(c)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Y_1}{\omega }_J{\varphi }_J{Y}_2,Z_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_J{Y}_2,{\varphi }_J{P}_J{Z}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_J{Y}_2,{\omega }_J{S}_J{Z}_1),\hfill \\ \hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_J{\varphi }_J{Y}_2,Z_2)& =& g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_J{Y}_2,C_J{Z}_2)+g_1({\mathcal{T}}_{Y_1}{\omega }_J{Y}_2,B_J{Z}_2)\hfill \end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^J\right),$ $Z_1\in \mathsf{\Gamma }(D^J\oplus D_2^J)$ and $Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(d)

$$\begin{array}{ccc}\hfill g_1({\mathcal{T}}_{Y_1}{\omega }_K{\varphi }_K{Y}_2,Z_1)& =& g_1({\mathcal{T}}_{Y_1}{\omega }_K{Y}_2,{\varphi }_K{P}_K{Z}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_K{Y}_2,{\omega }_K{S}_K{Z}_1),\hfill \\ \hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_K{\varphi }_K{Y}_2,Z_2)& =& g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_K{Y}_2,C_K{Z}_2)+g_1({\mathcal{T}}_{Y_1}{\omega }_K{Y}_2,B_K{Z}_2)\hfill \end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^K\right),$ $Z_1\in \mathsf{\Gamma }(D^K\oplus D_2^K)$ and $Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

For $Y_1,Y_2\in \mathsf{\Gamma }\left(D_1^R\right),$ $Z_1\in \mathsf{\Gamma }(D^R\oplus D_2^R),$ $Z_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, using Equations (5), (12) and (13) as well as Lemma $5,$ we have

$$\begin{array}{ccc}& & g_1({\nabla }_{Y_1}{Y}_2,Z_1)\hfill \\ & =& g_1({\nabla }_{Y_1}R{Y}_2,R{Z}_1),\hfill \\ & =& g_1({\nabla }_{Y_1}{\varphi }_R{Y}_2,R{Z}_1)+g_1({\nabla }_{Y_1}{\omega }_R{Y}_2,R{Z}_1),\hfill \\ & =& {cos}^2{\theta }_R{g}_1({\nabla }_{Y_1}{Y}_2,Z_1)-g_1({\mathcal{T}}_{Y_1}{\omega }_R{\varphi }_R{Y}_2,Z_1)\hfill \\ & & +g_1({\mathcal{T}}_{Y_1}{\omega }_R{Y}_2,{\varphi }_R{P}_R{Z}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{Y}_2,{\omega }_R{S}_R{Z}_1),\hfill \end{array}$$

which gives

$$\begin{array}{ccc}& & {sin}^2{\theta }_R{g}_1({\nabla }_{Y_1}{Y}_2,Z_1)\hfill \\ & =& -g_1({\mathcal{T}}_{Y_1}{\omega }_R{\varphi }_R{Y}_2,Z_1)+g_1({\mathcal{T}}_{Y_1}{\omega }_R{Y}_2,R{P}_R{Z}_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{Y}_2,{\omega }_R{S}_R{Z}_1).\hfill \end{array}$$

Moreover, from Equations (5), (13) and (14) as well as Lemma 5, we have

$$\begin{array}{ccc}& & g_1({\nabla }_{Y_1}{Y}_2,Z_2)\hfill \\ & =& g_1({\nabla }_{Y_1}R{Y}_2,R{Z}_2),\hfill \\ & =& g_1({\nabla }_{Y_1}{\varphi }_R{Y}_2,R{Z}_2)+g_1({\nabla }_{Y_1}{\omega }_R{Y}_2,R{Z}_2),\hfill \\ & =& {cos}^2{\theta }_R{g}_1({\nabla }_{Y_1}{Y}_2,Z_2)-g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{\varphi }_R{Y}_2,Z_2)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{Y}_2,C_R{Z}_2)+g_1({\mathcal{T}}_{Y_1}{\omega }_R{Y}_2,B_R{Z}_2).\hfill \end{array}$$

Thus, we find that

$$\begin{array}{ccc}& & {sin}^2{\theta }_R{g}_1({\nabla }_{Y_1}{Y}_2,Z_2)\hfill \\ & =& -g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{\varphi }_R{Y}_2,Z_2)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{Y}_2,C_R{Z}_2)+g_1({\mathcal{T}}_{Y_1}{\omega }_R{Y}_2,B_R{Z}_2).\hfill \end{array}$$

Hence, we have

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, the result follows. □

Theorem 8.

For the h-qhs Riemannian map π defined in Theorem 1, any one of the following assertions implies the others:

(a)

$D_2^R$defines a totally geodesic foliation on $N_1$;

(b)

$$\begin{array}{ccc}\hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{Y}_2,{\omega }_I{Q}_I{W}_1)& =& -g_1({\mathcal{T}}_{Y_1}{\omega }_I{S}_I{Y}_2,{\varphi }_I{P}_I{W}_1+{\varphi }_I{Q}_I{W}_1),\hfill \\ \hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{S}_I{Y}_2,C_I{W}_2)& =& -g_1({\mathcal{T}}_{Y_1}{\omega }_I{S}_I{Y}_2,B_I{W}_2)\hfill \end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }\left(D_2^I\right),W_1\in \mathsf{\Gamma }(D^I\oplus D_1^I)$ and $W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(c)

$$\begin{array}{ccc}\hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_J{Y}_2,{\omega }_J{Q}_J{W}_1)& =& -g_1({\mathcal{T}}_{Y_1}{\omega }_J{S}_J{Y}_2,{\varphi }_J{P}_J{W}_1+{\varphi }_J{Q}_J{W}_1),\hfill \\ \hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_{J}S{Y}_2,C_J{W}_2)& =& -g_1({\mathcal{T}}_{Y_1}{\omega }_{J}S{Y}_2,B_J{W}_2)\hfill \end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }\left(D_2^J\right),W_1\in \mathsf{\Gamma }(D^J\oplus D_1^J)$ and $W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(d)

$$\begin{array}{ccc}\hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_K{Y}_2,{\omega }_K{Q}_K{W}_1)& =& -g_1({\mathcal{T}}_{Y_1}{\omega }_{K}S{Y}_2,{\varphi }_K{P}_K{W}_1+{\varphi }_K{Q}_K{W}_1),\hfill \\ \hfill g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_{K}S{Y}_2,C_K{W}_2)& =& -g_1({\mathcal{T}}_{Y_1}{\omega }_K{S}_K{Y}_2,B_K{W}_2)\hfill \end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }\left(D_2^K\right),W_1\in \mathsf{\Gamma }(D^K\oplus D_1^K)$ and $W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

For $Y_1,Y_2\in \mathsf{\Gamma }\left(D_2^R\right),$ $W_1\in \mathsf{\Gamma }(D^R\oplus D_1^R),$ $W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, using Equations (5), (12) and (13), we have

$$\begin{array}{ccc}& & g_1({\nabla }_{Y_1}{Y}_2,W_1)\hfill \\ & =& g_1({\nabla }_{Y_1}R{Y}_2,R{W}_1)\hfill \\ & =& g_1({\nabla }_{Y_1}{\omega }_R{S}_R{Y}_2,{\varphi }_R{P}_R{W}_1+{\varphi }_R{Q}_R{W}_1+{\omega }_R{Q}_R{W}_1),\hfill \\ & =& g_1({\mathcal{T}}_{Y_1}{\omega }_R{S}_R{Y}_2,{\varphi }_R{P}_R{W}_1+{\varphi }_R{Q}_R{W}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{S}_R{Y}_2,{\omega }_R{Q}_R{W}_1).\hfill \end{array}$$

Again, using Equations (5), (13) and (14), we have

$$\begin{array}{ccc}\hfill g_1({\nabla }_{Y_1}{Y}_2,W_2)& =& g_1({\nabla }_{Y_1}R{Y}_2,R{W}_2)\hfill \\ & =& g_1({\nabla }_{Y_1}{\omega }_R{S}_R{Y}_2,B_R{W}_2+C_R{W}_2),\hfill \\ & =& g_1({\mathcal{T}}_{Y_1}{\omega }_R{S}_R{Y}_2,B_R{W}_2)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_{R}R{Y}_2,C_R{W}_2).\hfill \end{array}$$

Hence, we have

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Therefore, the result follows. □

Theorem 9.

Let π be an h-qhs Riemannian map from an almost hyperkahler manifold $(N_1,I,J,K,g_1)$ to a Riemannian manifold $(N_2,g_2)$. Then, the following conditions are equivalent:

(a)

π is a totally geodesic map;

(b)

$$\begin{array}{c}\hfill g_1({\mathcal{T}}_{Y_1}{P}_I{Y}_2+{cos}^2{\theta }_I{\mathcal{T}}_{Y_1}{Q}_I{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_I{\varphi }_I{P}_I{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_I{\varphi }_I{Q}_I{Y}_2,W_1)\\ \hfill =g_1({\mathcal{T}}_{Y_1}{\omega }_I{Q}_I{Y}_2+{\mathcal{T}}_{Y_1}{\omega }_I{S}_I{Y}_2,B_I{W}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_I{\varphi }_I{Q}_I{Y}_2+\mathcal{H}{\nabla }_{Y_1}{\omega }_I{\varphi }_I{S}_I{Y}_2,W_1),\end{array}$$

$$\begin{array}{c}\hfill g_1({\mathcal{A}}_{W_1}{P}_I{Y}_1+{cos}^2{\theta }_I{\mathcal{A}}_{W_1}{Q}_I{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_I{\varphi }_I{P}_I{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_I{\varphi }_I{Q}_I{Y}_1,W_2)\\ \hfill =g_1({\mathcal{A}}_{W_1}{\omega }_I{Q}_I{Y}_1+{\mathcal{A}}_{W_1}{\omega }_I{S}_I{Y}_1,B_I{W}_2)+g_1(\mathcal{H}{\nabla }_{W_1}{\omega }_I{Q}_I{Y}_1+\mathcal{H}{\nabla }_{W_1}{\omega }_I{S}_I{Y}_1,C_I{W}_2)\end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $W_1,W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(c)

$$\begin{array}{c}\hfill g_1({\mathcal{T}}_{Y_1}{P}_J{Y}_2+{cos}^2{\theta }_J{\mathcal{T}}_{Y_1}{Q}_J{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_J{\varphi }_J{P}_J{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_J{\varphi }_J{Q}_J{Y}_2,W_1)\\ \hfill =g_1({\mathcal{T}}_{Y_1}{\omega }_J{Q}_J{Y}_2+{\mathcal{T}}_{Y_1}{\omega }_J{S}_J{Y}_2,B_J{W}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_J{\varphi }_J{Q}_{J}J{Y}_2+\mathcal{H}{\nabla }_{Y_1}{\omega }_J{\varphi }_J{S}_J{Y}_2,W_1),\end{array}$$

$$\begin{array}{c}\hfill g_1({\mathcal{A}}_{W_1}{P}_J{Y}_1+{cos}^2{\theta }_J{\mathcal{A}}_{W_1}{Q}_J{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_J{\varphi }_J{P}_J{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_J{\varphi }_J{Q}_J{Y}_1,W_2)\\ \hfill =g_1({\mathcal{A}}_{W_1}{\omega }_J{Q}_J{Y}_1+{\mathcal{A}}_{W_1}{\omega }_J{S}_J{Y}_1,B_J{W}_2)+g_1(\mathcal{H}{\nabla }_{W_1}{\omega }_J{Q}_J{Y}_1+\mathcal{H}{\nabla }_{W_1}{\omega }_J{S}_J{Y}_1,C_J{W}_2)\end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $W_1,W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$;

(d)

$$\begin{array}{c}\hfill g_1({\mathcal{T}}_{Y_1}{P}_K{Y}_2+{cos}^2{\theta }_K{\mathcal{T}}_{Y_1}{Q}_K{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_K{\varphi }_K{P}_K{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_K{\varphi }_K{Q}_K{Y}_2,W_1)\\ \hfill =g_1({\mathcal{T}}_{Y_1}{\omega }_K{Q}_K{Y}_2+{\mathcal{T}}_{Y_1}{\omega }_K{S}_K{Y}_2,B_K{W}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_K{\varphi }_K{Q}_K{Y}_2+\mathcal{H}{\nabla }_{Y_1}{\omega }_K{\varphi }_K{S}_K{Y}_2,W_1),\end{array}$$

$$\begin{array}{c}\hfill g_1({\mathcal{A}}_{W_1}{P}_K{Y}_1+{cos}^2{\theta }_K{\mathcal{A}}_{W_1}{Q}_K{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_K{\varphi }_K{P}_K{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_K{\varphi }_K{Q}_K{Y}_1,W_2)\\ \hfill =g_1({\mathcal{A}}_{W_1}{\omega }_K{Q}_K{Y}_1+{\mathcal{A}}_{W_1}{\omega }_K{S}_K{Y}_1,B_K{W}_2)+g_1(\mathcal{H}{\nabla }_{W_1}{\omega }_K{Q}_K{Y}_1+\mathcal{H}{\nabla }_{W_1}{\omega }_K{S}_K{Y}_1,C_K{W}_2)\end{array}$$

for $Y_1,Y_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$ and $W_1,W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

Since π is a Riemannian map, therefore, we have

$$(\nabla {\pi }_{\ast })(W_1,W_2)=0,$$

for $W_1,W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }.$

For $Y_1,Y_2\in \mathsf{\Gamma }(ker{\pi }_{\ast })$, $W_1,W_2\in \mathsf{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ and $R\in \{I,J,K\}$, using Equations (4), (5) and (12)–(14) as well as Lemma 5, we have

$$\begin{array}{ccc}& & g_2((\nabla {\pi }_{\ast })(Y_1,Y_2),{\pi }_{\ast }\left(W_1\right))\hfill \\ & =& -g_1({\nabla }_{Y_1}{Y}_2,W_1)\hfill \\ & =& -g_1({\nabla }_{Y_1}R{Y}_2,R{W}_1)\hfill \\ & =& -g_1({\nabla }_{Y_1}R{P}_R{Y}_2,R{W}_1)-g_1({\nabla }_{Y_1}R{Q}_R{Y}_2,R{W}_1)-g_1({\nabla }_{Y_1}R{S}_R{Y}_2,R{W}_1),\hfill \\ & =& -g_1({\nabla }_{Y_1}{\varphi }_R{P}_R{Y}_2,R{W}_1)-g_1({\nabla }_{Y_1}{\varphi }_R{Q}_R{Y}_2,R{W}_1)\hfill \\ & & -g_1({\nabla }_{Y_1}{\omega }_R{Q}_R{Y}_2,R{W}_1)-g_1({\nabla }_{Y_1}{\omega }_R{S}_R{Y}_2,R{W}_1),\hfill \\ & =& -g_1({\mathcal{T}}_{Y_1}{P}_R{Y}_2+{cos}^2{\theta }_R{\mathcal{T}}_{Y_1}{Q}_R{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_R{\varphi }_R{P}_R{Y}_2-\mathcal{H}{\nabla }_{Y_1}{\omega }_R{\varphi }_R{Q}_R{Y}_2,W_1)\hfill \\ & & -g_1({\mathcal{T}}_{Y_1}{\omega }_R{Q}_R{Y}_2+{\mathcal{T}}_{Y_1}{\omega }_R{S}_R{Y}_2,B_R{W}_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{Y_1}{\omega }_R{\varphi }_R{Q}_R{Y}_2+\mathcal{H}{\nabla }_{Y_1}{\omega }_R{\varphi }_R{S}_R{Y}_2,W_1).\hfill \end{array}$$

Moreover, using Equations (4), (5) and (12)–(14) as well as Lemma 5, we have

$$\begin{array}{ccc}& & g_2((\nabla {\pi }_{\ast })(W_1,Y_1),{\pi }_{\ast }\left(W_2\right))\hfill \\ & =& -g_1({\nabla }_{W_1}{Y}_1,W_2),\hfill \\ & =& -g_1({\nabla }_{W_1}R{Y}_1,R{W}_2),\hfill \\ & =& -g_1({\nabla }_{W_1}R{P}_R{Y}_1,R{W}_2)-g_1({\nabla }_{W_1}R{Q}_R{Y}_1,R{W}_2)-g_1({\nabla }_{W_1}R{S}_R{Y}_1,R{W}_2),\hfill \\ & =& -g_1({\nabla }_{W_1}{\varphi }_R{P}_R{Y}_1,R{W}_2)-g_1({\nabla }_{W_1}{\varphi }_R{Q}_R{Y}_1,R{W}_2)\hfill \\ & & -g_1({\nabla }_{W_1}{\omega }_R{Q}_R{Y}_1,R{W}_2)-g_1({\nabla }_{W_1}{\omega }_R{S}_R{Y}_1,R{W}_2),\hfill \\ & =& -g_1({\mathcal{A}}_{W_1}{P}_R{Y}_1+{cos}^2{\theta }_R{\mathcal{A}}_{W_1}{Q}_R{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_R{\varphi }_R{P}_R{Y}_1-\mathcal{H}{\nabla }_{W_1}{\omega }_R{\varphi }_R{Q}_R{Y}_1,W_2)\hfill \\ & & -g_1({\mathcal{A}}_{W_1}{\omega }_R{Q}_R{Y}_1+{\mathcal{A}}_{W_1}{\omega }_R{S}_R{Y}_1,B_R{W}_2)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{W_1}{\omega }_R{Q}_R{Y}_1+\mathcal{H}{\nabla }_{W_1}{\omega }_R{S}_R{Y}_1,C_R{W}_2).\hfill \end{array}$$

Hence, we obtain

$$\left(a\right)\iff \left(b\right),\left(a\right)\iff \left(c\right),\left(a\right)\iff \left(d\right).$$

Thus, the theorem is proven. □

4. Example

Note that given a Euclidean space ${\mathbb{R}}^{4n}$ with coordinates $(x_1,x_2,.....,x_{4n}),$ we can canonically choose complex structures I, J and K on ${\mathbb{R}}^{4n}$ as follows:

$$\begin{array}{ccc}\hfill I(\frac{\partial }{\partial x_{4s+1}})& =& \frac{\partial }{\partial x_{4s+2}},I(\frac{\partial }{\partial x_{4s+2}})=-\frac{\partial }{\partial x_{4s+1}},I(\frac{\partial }{\partial x_{4s+3}})=\frac{\partial }{\partial x_{4s+4}},\hfill \\ \hfill I(\frac{\partial }{\partial x_{4s+4}})& =& -\frac{\partial }{\partial x_{4s+3}},J(\frac{\partial }{\partial x_{4s+1}})=\frac{\partial }{\partial x_{4s+3}},J(\frac{\partial }{\partial x_{4s+2}})=-\frac{\partial }{\partial x_{4s+4}},\hfill \\ \hfill J(\frac{\partial }{\partial x_{4s+3}})& =& -\frac{\partial }{\partial x_{4s+1}},J(\frac{\partial }{\partial x_{4s+4}})=\frac{\partial }{\partial x_{4s+2}},K(\frac{\partial }{\partial x_{4s+1}})=\frac{\partial }{\partial x_{4s+4}},\hfill \\ \hfill K(\frac{\partial }{\partial x_{4s+2}})& =& \frac{\partial }{\partial x_{4s+3}},K(\frac{\partial }{\partial x_{4s+3}})=-\frac{\partial }{\partial x_{4s+2}},K(\frac{\partial }{\partial x_{4s+4}})=-\frac{\partial }{\partial x_{4s+1}},\hfill \end{array}$$

for $s\in \{0,1,2,....,\dots ,n-1\}.$

Then, we can easily check that $(I,J,K,〈,〉)$ is a hyperkähler structure on ${\mathbb{R}}^{4n}$, where $〈,〉$ denotes the Euclidean metric on ${\mathbb{R}}^{4n}$. Throughout this section, we will use these notations.

Example 1.

Define a map $\pi :{\mathbb{R}}^{12}\to {\mathbb{R}}^6$ by

$$\pi (x_1,x_2,.........,x_{12})=(2020,x_2,x_6,\frac{x_8-x_9}{\sqrt{2}},x_{10},2022).$$

Then, the map $\pi$ is an almost h-qhs Riemannian map such that

$$ker{\pi }_{\ast }=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4},\frac{\partial }{\partial x_5},\frac{\partial }{\partial x_7},\frac{1}{\sqrt{2}}(\frac{\partial }{\partial x_8}+\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_{11}},\frac{\partial }{\partial x_{12}}〉,$$

$${(ker{\pi }_{\ast })}^{\perp }=〈\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_6},\frac{1}{\sqrt{2}}(\frac{\partial }{\partial x_8}-\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_{10}}〉,$$

$$D^I=〈\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4},\frac{\partial }{\partial x_{11}},\frac{\partial }{\partial x_{12}}〉,D_1^I=〈\frac{\partial }{\partial x_7},\frac{1}{\sqrt{2}}(\frac{\partial }{\partial x_8}+\frac{\partial }{\partial x_9})〉,D_2^I=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_5}〉,$$

$$D^J=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_5},\frac{\partial }{\partial x_7}〉,D_1^J=〈\frac{1}{\sqrt{2}}(\frac{\partial }{\partial x_8}+\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_{11}}〉,D_2^J=〈\frac{\partial }{\partial x_4},\frac{\partial }{\partial x_{12}}〉,$$

$$D^K=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_4}〉,D_1^K=〈\frac{\partial }{\partial x_5},\frac{1}{\sqrt{2}}(\frac{\partial }{\partial x_8}+\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_{12}}〉,D_2^K=〈\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_7},\frac{\partial }{\partial x_{11}}〉,$$

with the almost h-qhs angles $\{{\theta }_I={\theta }_J={\theta }_K=\frac{\pi }{4}\}.$

Example 2.

Define a map $\pi :{\mathbb{R}}^{16}\to {\mathbb{R}}^8$ by

$$\pi (x_1,x_2,.........,x_{16})=(101,\frac{\sqrt{3}{x}_5-x_9}{2},x_6,x_8,x_{11},x_{14},202,x_{15}).$$

Then, the map $\pi$ is an almost h-qhs Riemannian map such that

$$ker{\pi }_{\ast }=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4},\frac{1}{2}(\frac{\partial }{\partial x_5}+\sqrt{3}\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_7},\frac{\partial }{\partial x_{10}},\frac{\partial }{\partial x_{12}},\frac{\partial }{\partial x_{13}},\frac{\partial }{\partial x_{16}}〉,$$

$${(ker{\pi }_{\ast })}^{\perp }=〈\frac{1}{2}(\sqrt{3}\frac{\partial }{\partial x_5}-\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_6},\frac{\partial }{\partial x_8},\frac{\partial }{\partial x_{11}},\frac{\partial }{\partial x_{14}},\frac{\partial }{\partial x_{15}},〉,$$

$$D^I=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4}〉,D_1^I=〈\frac{1}{2}(\frac{\partial }{\partial x_5}+\sqrt{3}\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_{10}}〉,D_2^I=〈\frac{\partial }{\partial x_7},\frac{\partial }{\partial x_{12}},\frac{\partial }{\partial x_{13}},\frac{\partial }{\partial x_{16}}〉,$$

$$D^J=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4},\frac{\partial }{\partial x_{10}},\frac{\partial }{\partial x_{12}}〉,D_1^J=〈\frac{1}{2}(\frac{\partial }{\partial x_5}+\sqrt{3}\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_7}〉,D_2^J=〈\frac{\partial }{\partial x_{13}},\frac{\partial }{\partial x_{16}}〉,$$

$$D^K=〈\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3},\frac{\partial }{\partial x_4},\frac{\partial }{\partial x_{13}},\frac{\partial }{\partial x_{16}}〉,D_1^K=〈\frac{1}{2}(\frac{\partial }{\partial x_5}+\sqrt{3}\frac{\partial }{\partial x_9}),\frac{\partial }{\partial x_{12}}〉,D_2^K=〈\frac{\partial }{\partial x_7},\frac{\partial }{\partial x_{10}}〉,$$

with the almost h-qhs angles $\{{\theta }_I=\frac{\pi }{6},{\theta }_J=\frac{\pi }{3},{\theta }_K=\frac{\pi }{6}\}.$