The Obata-Type Connection on a 3-Parameter Generalized Quaternion Structure

Abstract

This paper aims to define the Obata-type connection on a $4n$-dimensional smooth manifold equipped with a 3-parameter generalized quaternion structure. The paper demonstrates some general algebraic properties of this connection. The integrability of endomorphisms of the tangent bundle is interpreted as homomorphisms of certain skew-symmetric algebroids. The results confirm the uniqueness of a torsion-free connection compatible with generalized quaternion structures, assuming their integrability for non-zero parameters.

1. Introduction

An almost hypercomplex structure on a $4n$-dimensional smooth manifold M is a system of almost complex structures $(I, J, K)$ on M satisfying the the quaternionic conditions $IJ=-JI=K$. Additionally, if the almost complex structures I and J are integrable, then $(I, J, K)$ is called a hypercomplex structure, and a manifold M equipped with this structure is termed a hypercomplex manifold [1]. We denote such a manifold by $(M, I, J, K)$.

Recently, Şentürk and Ünal introduced generalized quaternions, designated as 3-parameter generalized quaternions (3PGQs) [2]. The set

$$\begin{array}{c}\mathbb{K}=\{a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3|a_0, a_1, a_2, a_3, {\lambda }_1, {\lambda }_2, {\lambda }_3\in \mathbb{R},\hfill \\ \hfill e_1^2=-{\lambda }_1{\lambda }_2, e_2^2=-{\lambda }_1{\lambda }_3, e_3^2=-{\lambda }_2{\lambda }_3, e_1{e}_2{e}_3=-{\lambda }_1{\lambda }_2{\lambda }_3\}\end{array}$$

is termed the set of 3-parameter generalized quaternions [2]. In [2], the previously known types of quaternions are generalized. For ${\lambda }_1={\lambda }_2={\lambda }_3=1$, the algebra of the Hamilton quaternions is obtained. For ${\lambda }_1={\lambda }_2=1$ and ${\lambda }_3=-1$, the algebra of split quaternions, introduced by Cockle [3], is archived. Furthermore, if ${\lambda }_1=1$, ${\lambda }_2, {\lambda }_3\in \mathbb{R}$, the algebra of 2-parameter generalized quaternions (2PGQs) is attained. Research on the 2PGQs was initiated by Dickson [4]. Subsequently, 2PGQs were studied by Griffits [5] and, more recently, by Jafari and Yayli [6]. In other cases, semi-quaternions, split semi-quaternions, and $1/4$-quaternions are specifically obtained if $\left({\lambda }_1, {\lambda }_2, {\lambda }_3\right)$ are equal to $(1, 1, 0)$, $(1, -1, 0)$ and $(1, 0, 0)$, respectively (cf. also [7]).

The Obata connection on the hypercomplex manifold $(M, I, J, K)$ is a unique torsion-free connection ∇ compatible with complex structures $I, J, K$ [8]. The holonomy group of the Obata connection lies in $GL(n, \mathbb{H})$, where $\mathbb{H}$ is the algebra of the Hamilton quaternions, $n={dim}_{\mathbb{H}}M$. The holonomy group is an important object of hypercomplex manifolds. In particular, it characterizes the existence of hyperkähler metrics. Recall that a hyperkähler manifold is a Riemannian manifold $(M, g)$ endowed with a hypercomplex structure. The manifold M admits a hyperkähler metric if and only if the holonomy of the Obata connection preserves positive definite metrics. For a hyperkähler manifold, the Obata connection is precisely the Levi–Civita connection with respect to the metric (cf. [9,10,11,12]).

As we mentioned, every hypercomplex manifold admits the Obata connection. Conversely, a manifold M equipped with three almost complex structures $I_1, I_2, I_3$, satisfying $I_1{I}_2=I_3$, together with a torsion-free connection ${\nabla }^0$ that preserves these structures (i.e., $\nabla I_i=0$ for $i=1, 2, 3$), is hypercomplex. This follows from the fact that $I_i$ are integrable. The integrability of these structures also follows from algebraic Lemma 1 of the presented paper, which shows an algebraic property of the Nijenhuis tensor of the endomorphism $\phi$ of the tangent bundle depending on the torsion of any linear connection ∇ and on $\nabla \phi$. Therefore, if we have two anticommutative complex structures, namely I and J, then $IJ$ is the third such structure that is anticommutative with both I and J. From Lemma 1, we immediately have that any endomorphism of the tangent bundle to a manifold M (not necessarily an almost complex structure) is integrable if it is preserved by some torsion-free connection on M.

This paper aims to consider an analogous connection equipped with endomorphisms of the tangent bundle satisfying some 3-parameter generalized quaternionic relations. For the hypercomplex manifold, Soldatenkov [10] derived the explicit formula for the Obata connection. The present paper generalizes this formula for 3-parameter generalized quaternion structures through non-zero parameters.

Let M be a manifold equipped with endomorphisms $I, J, K:TM\to TM$ satisfying the following properties of 3-parameter generalized quaternions:

$$\begin{array}{ccccc}\hfill I^2=-{\lambda }_1{\lambda }_2, & & IJ={\lambda }_{1}K, \hfill & & IK=-KI, \\ \hfill & & & & \\ \hfill JI=-IJ, & & J^2=-{\lambda }_1{\lambda }_3, \hfill & & JK={\lambda }_{3}I, \\ \hfill & & & & \\ \hfill KI={\lambda }_{2}J, & & KJ=-JK, \hfill & & K^2=-{\lambda }_2{\lambda }_3\end{array}$$

(1)

for some ${\lambda }_1, {\lambda }_2, {\lambda }_3\in \mathbb{R}$.

The $C^{\infty }\left(M\right)$-module of smooth vector fields on M is denoted by $\Gamma \left(TM\right)$. If ${\lambda }_1, {\lambda }_2$, and ${\lambda }_3$ are non-zero, we define a connection ∇ on M by

$${\nabla }_{X}Y={\displaystyle \frac{1}{2}}\left(\left[X, Y\right]+〈X, Y〉\right)$$

where

$$〈·, ·〉:\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(TM\right)$$

is the bilinear map fulfilling the conditions

$$〈X, f·Y〉=f·〈X, Y〉+X\left(f\right)·Y$$

and

$$〈f·X, Y〉=f·〈X, Y〉+Y\left(f\right)·X$$

for $X, Y\in \Gamma \left(TM\right)$, $f\in C^{\infty }\left(M\right)$. The connection ∇ is torsion-free if and only if $〈·, ·〉$ is a symmetric bracket [13]. In particular, if $T^{\nabla }=0$, then $〈X, Y〉={\nabla }_{X}Y+{\nabla }_{Y}X$ for $X, Y\in \Gamma \left(TM\right)$, i.e., $〈·, ·〉$ is the symmetric product defined by ∇, which was introduced by Crouch in [14].

Remark 1. 

Let $\left(M, g, J_0\right)$ be an almost Hermitian manifold, i.e., $\left(M, g\right)$ is a $2n$-dimensional Riemannian manifold admitting an orthogonal almost complex structure $J_0:TM\to TM$. The Riemannian metric defines $♭:TM\to T^{\ast }M$ and $♯:T^{\ast }M⟶TM$ by

$$♭\left(X\right)=i_{X}g, g(♯\left(\eta \right), X)=\eta \left(X\right)$$

for any vector field $X\in \Gamma \left(TM\right)$ and 1-form$\eta \in \Gamma \left(T^{\ast }M\right)$, where $T^{\ast }M$ is the cotangent bundle of M. For any $X\in \Gamma \left(TM\right)$, we denote by $X^{♭}$ the 1-form $g(X, ·)$. It is shown in [13] that the map ${〈·, ·〉}_{J_0}:\Gamma \left(TM\right)\times \Gamma \left(TM\right)\to \Gamma \left(TM\right)$, defined by

$${〈X, Y〉}_{J_0}=-\frac{1}{2}{J}_0\left(\left[X, J_{0}Y\right]+\left[Y, J_{0}X\right]+♯({\mathcal{L}}_X{\left(J_{0}Y\right)}^{♭}+{\mathcal{L}}_Y{\left(J_{0}X\right)}^{♭}+{\mathcal{L}}_{J_{0}X}{Y}^{♭}+{\mathcal{L}}_{J_{0}Y}{X}^{♭})\right)$$

is a symmetric bracket. Here, $\left[·, ·\right]$ is the Lie bracket of vector fields on M and $\mathcal{L}$ is the Lie derivative on M. It is observed in [13] that this symmetric bracket is a totally symmetric part of the connection ${\nabla }^{J_0}$ defined by ${\nabla }_X^{J_0}Y=-J_0{\nabla }_X^g\left(J_{0}Y\right)$, where ${\nabla }^g$ is the Levi–Civita connection with respect to g. Moreover, the affine sum ${\displaystyle \frac{1}{2}}\left({\nabla }^g+{\nabla }^{J_0}\right)$ is Lichnerowicz’s first canonical Hermitian connection [15], which is compatible with both the metric structure and the almost complex structure.

In Section 4.2 of the present paper, we determine the torsion of ∇. If structures I and J are integrable, then $〈·, ·〉$ is a symmetric bracket, i.e., ∇ is torsion-free. Next, we investigate whether the connection is compatible with the given quaternion structures. According to the results, $\nabla J=0$ and $\nabla I$ and $\nabla K$ depend on the Nijenhuis torsion of I. Therefore, we conclude that ∇ is compatible with I, J, and K if I is integrable.

Section 5 shows the uniqueness of a torsion-free connection compatible with the 3-parameter generalized quaternion structure $(I, J, K)$ for non-zero parameters.

The present paper demonstrates that the integrability of K depends on the integrability of I and J. To validate this relation, we utilize the observation that every endomorphism $\phi$ of the tangent bundle $TM$ deforms the Lie bracket $[·, ·]$ of vector fields on M to some bracket ${\left[[·, ·]\right]}^{\phi }$ that, together with $\phi$, determines the structure of the skew–symmetric algebroid $T{M}^{\phi }$ in the tangent bundle with $\phi$ as the anchor. Subsequently, the zero Nijenhuis torsion of the endomorphism $\phi$ implies that $\phi$ is a homomorphism of the skew-symmetric algebroids $T{M}^{\phi }$ and $TM$. Next, we determine how the deformed bracket for $IJ$ depends on the deformed brackets by I and J (Theorem 1). This relation reveals that if I is a homomorphism of algebroids $T{M}^I$ and $TM$ and J is a homomorphism of $T{M}^J$ and $TM$, then $IJ$ is a homomorphism of ${\left(TM\right)}^{IJ}$ and $TM$. Hence, for ${\lambda }_1\ne 0$, the integrability of K follows from the integrability of I and J.

2. Algebroid’s Approach to Nijenhuis Torsion

A skew–symmetric algebroid $\left(A, {\rho }_A, \left[[·, ·]\right]\right)$ over a manifold M is a vector bundle A over M equipped with a homomorphism of vector bundles ${\varrho }_A:A\to TM$ over the identity, which is called an anchor, and an $\mathbb{R}$-bilinear skew–symmetric map $\left[\right[·, ·\left]\right]:\Gamma \left(A\right)\times \Gamma \left(A\right)\to \Gamma \left(A\right)$, which is associated with the anchor through the following derivation law:

$$\left[[X, f·Y]\right]=f·\left[[X, Y]\right]+({\varrho }_A\circ X)\left(f\right)·Y$$

(2)

for $X, Y\in \Gamma \left(A\right)$, $f\in C^{\infty }\left(M\right)$ (cf. [13,16]), where $\Gamma \left(A\right)$ is the module of global sections of the vector bundle A. Kosmann–Schwarzbach and Magri introduced the concept of skew–symmetric algebroids as pre-Lie algebroids in [17] on the level of finitely generated projective modules over commutative and associative algebras with a unit. If the anchor preserves $\left[\right[·, ·\left]\right]$ and the Lie bracket $\left[·, ·\right]$ of vector fields on M, i.e., ${\varrho }_A\circ \left[[X, Y]\right]=\left[{\varrho }_A\circ X, {\varrho }_A\circ Y\right]$ for $X, Y\in \Gamma \left(A\right)$, a skew–symmetric algebroid $\left(A, {\varrho }_A, \left[[·, ·]\right]\right)$ is an almost Lie algebroid (cf. [18,19,20]). According to Pradines [21], any skew–symmetric algebroid in which $\left[\right[·, ·\left]\right]$ satisfies the Jacobi identity is called the Lie algebroid. In particular, skew–symmetric algebroids or Lie algebroids are the generalizations of integrable distributions and Lie algebras.

The Jacobiator ${\mathrm{Jac}}_{\left[\right[·, ·\left]\right]}:\Gamma \left(A\right)\times \Gamma \left(A\right)\times \Gamma \left(A\right)\to \Gamma \left(A\right)$ of the bracket $\left[\right[·, ·\left]\right]$ is given by

$${\mathrm{Jac}}_{\left[\right[·, ·\left]\right]}(X, Y, Z)=\left[[\left[[X, Y]\right], Z]\right]+\left[[\left[[Z, X]\right], Y]\right]+\left[[\left[[Y, Z]\right], X]\right]$$

for $X, Y, Z\in \Gamma \left(A\right)$.

Let M be a smooth manifold. For any endomorphism $\phi$ of $TM$, we define its Nijenhuis torsion of $\phi$ as $N_{\phi }:TM\times TM\to TM$ by

$$N_{\phi }(X, Y)=[\phi X, \phi Y]-\phi \left([\phi X, Y]+[X, \phi Y]-\phi [X, Y]\right)$$

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

The bracket ${\left[[·, ·]\right]}^{\phi }$ on $TM$ is defined by

$${\left[[X, Y]\right]}^{\phi }=\left[\phi X, Y\right]+\left[X, \phi Y\right]-\phi \left[X, Y\right].$$

(3)

Kosmann–Schwarzbach and Magri [17] reported this skew–symmetric bracket in the deformation theory of Lie algebras. Thus,

$${\left[[X, f·Y]\right]}^{\phi }=f·{\left[[X, Y]\right]}^{\phi }+\left(\phi X\right)\left(f\right)·Y$$

for $X, Y\in \Gamma \left(TM\right)$ and $f\in C^{\infty }\left(M\right)$. Consequently, the tangent bundle includes a structure of skew–symmetric algebroid, denoted as $T{M}^{\phi }$, with the bracket ${\left[[·, ·]\right]}^{\phi }$ and $\phi$ as an anchor. Note that

$$N_{\phi }(X, Y)=\left[\phi X, \phi Y\right]-\phi {\left[[X, Y]\right]}^{\phi }$$

for $X, Y\in \Gamma \left(TM\right)$. Therefore, the disappearance of the Nijenhuis torsion for a given endomorphism $\phi$ indicates that $\phi$ is a homomorphism of the skew–symmetric algebroids $T{M}^{\phi }$ and $TM$, i.e., $\phi$ preserves anchors ($\phi =\phi \circ {Id}_{TM}$) and brackets ($\phi {\left[[X, Y]\right]}^{\phi }=\left[\phi X, \phi Y\right]$ for $X, Y\in \Gamma \left(TM\right)$). An endomorphism $\phi$ of $TM$ is said to be integrable or the Nijenhuis tensor if $N_{\phi }=0$.

Based on the above-mentioned arguments, if $N_{\phi }=0$, then $\phi {[[{\left[[X, Y]\right]}^{\phi }, Z]]}^{\phi }=\left[\left[\phi X, \phi Y\right], \phi Z\right]$ for $X, Y, Z\in \Gamma \left(TM\right)$; therefore, $\phi \left({\mathrm{Jac}}_{{\left[[·, ·]\right]}^{\phi }}(X, Y, Z)\right)={\mathrm{Jac}}_{[·, ·]}(\phi X, \phi Y, \phi Z)=0$ for $X, Y, Z\in \Gamma \left(TM\right)$. Thus, if $\phi$ is integrable and ${\phi }^2=c{Id}_{TM}$ for some non-zero real number c, then the skew–symmetric algebroid $\left(TM, \phi , {\left[[·, ·]\right]}^{\phi }\right)$ is a Lie algebroid over M.

Let $(I, J, K)$ be the 3-parameter generalized quaternion structure on M satisfying generalized quaternionic conditions (1). First, we show that the integrability of endomorphisms I and J influences the integrability of K if ${\lambda }_1\ne 0$. We demonstrate that the composition of I and J is integrable using the skew–symmetric algebroid approach.

The following theorem could help understand the dependence of the bracket designated by $IJ$ on the brackets of both structures I and J.

Theorem 1. 

Let $I, J$ be endomorphisms of $TM$, such that $JI=-IJ$, $X, Y\in \Gamma \left(TM\right)$. Then

$$\begin{array}{cc}\hfill 2{\left[[X, Y]\right]}^{IJ}& ={\left[[JX, Y]\right]}^I+{\left[[X, JY]\right]}^I-J{\left[[X, Y]\right]}^I\hfill \\ \hfill & -({\left[[IX, Y]\right]}^J+{\left[[X, IY]\right]}^J-I{\left[[X, Y]\right]}^J).\hfill \end{array}$$

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. From

$${\left[[JX, Y]\right]}^I=[IJX, Y]+[JX, IY]-I[JX, Y],$$

$${\left[[X, JY]\right]}^I=[IX, JY]+[X, IJY]-I[X, JY],$$

$${\left[[IX, Y]\right]}^J=[JIX, Y]+[IX, JY]-J[IX, Y]$$

and

$${\left[[X, IY]\right]}^J=[JX, IY]+[X, JIY]-J[X, IY]$$

we obtain the following equalities:

$$[IJX, Y]={\left[[JX, Y]\right]}^I-[JX, IY]+I[JX, Y],$$

(4)

$$[X, IJY]={\left[[X, JY]\right]}^I-[IX, JY]+I[X, JY],$$

(5)

$$[IX, JY]={\left[[IX, Y]\right]}^J-[JIX, Y]+J[IX, Y]$$

(6)

and

$$[JX, IY]={\left[[X, IY]\right]}^J-[X, JIY]+J[X, IY].$$

(7)

By combining (4)–(7) with $IJ=-JI$, we have:

$$\begin{array}{cc}\hfill {\left[[X, Y]\right]}^{IJ}& =[IJX, Y]+[X, IJY]-IJ[X, Y]\hfill \\ \hfill & ={\left[[JX, Y]\right]}^I+{\left[[X, JY]\right]}^I+I{\left[[X, Y]\right]}^J-{\left[[X, IY]\right]}^J-{\left[[IX, Y]\right]}^J\hfill \\ \hfill & -\left[X, IJY\right]-J[X, IY]-[IJX, Y]-J[IX, Y]\hfill \\ \hfill & ={\left[[JX, Y]\right]}^I+{\left[[X, JY]\right]}^I+I{\left[[X, Y]\right]}^J-{\left[[X, IY]\right]}^J-{\left[[IX, Y]\right]}^J\hfill \\ \hfill & -J[X, IY]-J[IX, Y]+JI[X, Y]\hfill \\ \hfill & -\left[X, IJY\right]-[IJX, Y]+IJ[X, Y]\hfill \\ \hfill & ={\left[[JX, Y]\right]}^I+{\left[[X, JY]\right]}^I+I{\left[[X, Y]\right]}^J-{\left[[X, IY]\right]}^J-{\left[[IX, Y]\right]}^J-J{[X, Y]}^I\hfill \\ \hfill & -{\left[[X, Y]\right]}^{IJ}.\hfill \end{array}$$

This implies the claim. □

Theorem 2. 

Let I, J be endomorphisms of $TM$ with zero Nijenhuis torsion, such that $IJ=-JI$. This leads to the following condition:

$$IJ{\left[X, Y\right]}^{IJ}=\left[IJX, IJY\right]$$

for $X, Y\in \Gamma \left(TM\right)$, which implies that $IJ$ is a homomorphism of skew–symmetric algebroids $T{M}^{IJ}$ and $TM$.

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. From Theorem 1 and $IJ=-JI$, we have the following equalities:

$$\begin{array}{cc}\hfill 2IJ{\left[X, Y\right]}^{IJ}& =IJ({\left[[JX, Y]\right]}^I+{\left[[X, JY]\right]}^I-J{\left[[X, Y]\right]}^I)\hfill \\ \hfill & +IJ\left(-{\left[[IX, Y]\right]}^J-{\left[[X, IY]\right]}^J+I{\left[[X, Y]\right]}^J\right)\hfill \\ \hfill & =-JI{\left[[JX, Y]\right]}^I-JI{\left[[X, JY]\right]}^I-J^{2}I{\left[[X, Y]\right]}^I\hfill \\ \hfill & -IJ{\left[[IX, Y]\right]}^J-IJ{\left[[X, IY]\right]}^J-I^{2}J{\left[[X, Y]\right]}^J.\hfill \end{array}$$

Because I and J have zero Nijenhuis torsion, we deduce that I is a homomorphism of Lie algebroids $T{M}^I$ and $TM$, while J is a homomorphism of Lie algebroids $T{M}^J$ and $TM$. This can be represented by

$$I{\left[[X, Y]\right]}^I=\left[IX, IY\right]$$

(8)

and

$$J{\left[[X, Y]\right]}^J=\left[JX, JY\right].$$

(9)

Consequently, we arrive at the following relation:

$$\begin{array}{cc}\hfill 2IJ{\left[X, Y\right]}^{IJ}& =-J[IJX, IY]-J[IX, IJY]-J^2[IX, IY]\hfill \\ \hfill & -I[JIX, JY]-I[JX, JIY]-I^2[JX, JY]\hfill \\ \hfill & =J\left(\right[JIX, IY]+[IX, JIY]-J[IX, IY\left]\right)\hfill \\ \hfill & +I\left(\right[IJX, JY]+[JX, IJY]-I[JX, JY\left]\right)\hfill \\ \hfill & =J{[IX, IY]}^J+I{[JX, JY]}^I.\hfill \end{array}$$

By using (8) and (9) again, we obtain

$$2IJ{\left[X, Y\right]}^{IJ}=[JIX, JIY]+[IJX, IJY]=2[IJX, IJY],$$

which validates the claim. □

Corollary 1. 

Let I, J be endomorphisms of $TM$, such that $N_I=N_J=0$, $IJ=-JI$, and $c\in \mathbb{R}$. Then

$$N_{cIJ}=0.$$

Proof. 

Theorem 2 now yields $N_{IJ}=0$. Hence, $N_{cIJ}=c^2·N_{IJ}=0$. □

Corollary 2. 

If $(I, J, K)$ is a 3-parameterized general quaternionic structure for non-zero parameter ${\lambda }_1$, and I and J are integrable, then K is integrable.

The following relation is observed between the Nijenhuis torsion defined by a given endomorphism $\phi$ of $TM$, the torsion of an arbitrary linear connection, and its compatibility with $\phi$.

Lemma 1. 

Let ∇ be any linear connection on M and φ be any endomorphism of $TM$. Then

$$\begin{array}{cc}\hfill N_{\phi }(X, Y)& =(\nabla \phi )(\phi X, Y)-(\nabla \phi )(\phi Y, X)+\phi \left(\right(\nabla \phi \left)\right(Y, X)-(\nabla \phi \left)\right(X, Y\left)\right)\hfill \\ \hfill & -{\phi }^2\left(T^{\nabla }(X, Y)\right)-T^{\nabla }(\phi X, \phi Y)+\phi (T^{\nabla }(X, \phi Y)+T^{\nabla }(\phi X, Y))\hfill \end{array}$$

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

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. By definition, the following conditions are obtained:

$$\begin{array}{cc}\hfill T^{\nabla }(\phi X, \phi Y)& ={\nabla }_{\phi X}\left(\phi Y\right)-{\nabla }_{\phi Y}\left(\phi X\right)-\left[\phi X, \phi Y\right], \hfill \\ \hfill T^{\nabla }(\phi X, Y)& ={\nabla }_{\phi X}Y-{\nabla }_Y\left(\phi X\right)-\left[\phi X, Y\right], \hfill \\ \hfill T^{\nabla }(X, \phi Y)& ={\nabla }_X\left(\phi Y\right)-{\nabla }_{\phi Y}X-\left[X, \phi Y\right].\hfill \end{array}$$

Based on this, we have the following formulas:

$$\left[\phi X, \phi Y\right]=-T^{\nabla }(\phi X, \phi Y)+{\nabla }_{\phi X}\left(\phi Y\right)-{\nabla }_{\phi Y}\left(\phi X\right),$$

(10)

$$-\left[\phi X, Y\right]=T^{\nabla }(\phi X, Y)-{\nabla }_{\phi X}Y+{\nabla }_Y\left(\phi X\right),$$

(11)

$$-\left[X, \phi Y\right]=T^{\nabla }(X, \phi Y)-{\nabla }_X\left(\phi Y\right)+{\nabla }_{\phi Y}X.$$

(12)

Formulas (10)–(12) give us

$$\begin{array}{cc}\hfill & N_{\phi }(X, Y)\hfill \\ \hfill & =[\phi X, \phi Y]-\phi \left([\phi X, Y]+[X, \phi Y]-\phi [X, Y]\right)\hfill \\ \hfill & =-T^{\nabla }(\phi X, \phi Y)+{\nabla }_{\phi X}\left(\phi Y\right)-{\nabla }_{\phi Y}\left(\phi X\right)\hfill \\ \hfill & +\phi \left(T^{\nabla }(\phi X, Y)-{\nabla }_{\phi X}Y+{\nabla }_Y\left(\phi X\right)+T^{\nabla }(X, \phi Y)-{\nabla }_X\left(\phi Y\right)+{\nabla }_{\phi Y}X\right)+{\phi }^2[X, Y]\hfill \\ \hfill & =-T^{\nabla }(\phi X, \phi Y)+\phi \left(T^{\nabla }(\phi X, Y)+T^{\nabla }(X, \phi Y)\right)\hfill \\ \hfill & +{\nabla }_{\phi X}\left(\phi Y\right)-{\nabla }_{\phi Y}\left(\phi X\right)+\phi \left(-{\nabla }_{\phi X}Y+{\nabla }_Y\left(\phi X\right)-{\nabla }_X\left(\phi Y\right)+{\nabla }_{\phi Y}X\right)+{\phi }^2[X, Y].\hfill \end{array}$$

Moreover,

$${\nabla }_X\left(\phi Y\right)=\left(\nabla \phi \right)(X, Y)+\phi \left({\nabla }_{X}Y\right),$$

(13)

$${\nabla }_Y\left(\phi X\right)=\left(\nabla \phi \right)(Y, X)+\phi \left({\nabla }_{Y}X\right),$$

(14)

$${\nabla }_{\phi X}\left(\phi Y\right)=\left(\nabla \phi \right)(\phi X, Y)+\phi \left({\nabla }_{\phi X}Y\right),$$

(15)

$${\nabla }_{\phi Y}\left(\phi X\right)=\left(\nabla \phi \right)(\phi Y, X)+\phi \left({\nabla }_{\phi Y}X\right).$$

(16)

Hence, we can write the following:

$$\begin{array}{cc}\hfill & N_{\phi }(X, Y)\hfill \\ \hfill & =-T^{\nabla }(\phi X, \phi Y)+\phi \left(T^{\nabla }(\phi X, Y)+T^{\nabla }(X, \phi Y)\right)\hfill \\ \hfill & +\left(\nabla \phi \right)(\phi X, Y)+\phi \left({\nabla }_{\phi X}Y\right)-\left(\nabla \phi \right)(\phi Y, X)-\phi \left({\nabla }_{\phi Y}X\right)\hfill \\ \hfill & -\phi \left({\nabla }_{\phi X}Y\right)+\phi \left(\left(\nabla \phi \right)(Y, X)+\phi \left({\nabla }_{Y}X\right)\right)-\phi \left(\left(\nabla \phi \right)(X, Y)+\phi \left({\nabla }_{X}Y\right)\right)\hfill \\ \hfill & +\phi \left({\nabla }_{\phi Y}X\right)+{\phi }^2[X, Y]\hfill \\ \hfill & =-T^{\nabla }(\phi X, \phi Y)+\phi \left(T^{\nabla }(\phi X, Y)+T^{\nabla }(X, \phi Y)\right)\hfill \\ \hfill & +\left(\nabla \phi \right)(\phi X, Y)-\left(\nabla \phi \right)(\phi Y, X)+\phi \left(\left(\nabla \phi \right)(Y, X)\right)-\phi \left(\left(\nabla \phi \right)(X, Y)\right)\hfill \\ \hfill & +{\phi }^2\left({\nabla }_{Y}X\right)-{\phi }^2\left({\nabla }_{X}Y\right)+{\phi }^2[X, Y]\hfill \\ \hfill & =-T^{\nabla }(\phi X, \phi Y)+\phi \left(T^{\nabla }(\phi X, Y)+T^{\nabla }(X, \phi Y)\right)-{\phi }^2\left(T^{\nabla }(X, Y)\right)\hfill \\ \hfill & +\left(\nabla \phi \right)(\phi X, Y)-\left(\nabla \phi \right)(\phi Y, X)+\phi \left(\left(\nabla \phi \right)(Y, X)\right)-\phi \left(\left(\nabla \phi \right)(X, Y)\right).\hfill \end{array}$$

□

Corollary 3. 

Let φ be an endomorphism of $TM$. If on M there exists a torsion-free connection ∇ satisfying $\nabla \phi =0$, then the Nijenhuis torsion $N_{\phi }$ of φ is equal to zero.

From the last lemma, it follows that the existence of a torsion-free connection on M preserving a given endomorphism $\phi :TM\to TM$ implies the integrability of $\phi$. Here, $\phi$ does not have to be an almost complex structure, nor does it have to satisfy any of the conditions of generalized quaternion relations.

3. Examples of a 3-Parameter Generalized Quaternion Structure

Examples in this section are motivated by the results from [22,23]. In [22], the concept of a hypercomplex structure on a Lie algebra was considered. Given $\mathfrak{g}$, a real Lie algebra, an almost hypercomplex structure on $\mathfrak{g}$ is a family ${\left\{J_{\alpha }\right\}}_{\alpha =1,2}$ of endomorphisms of $\mathfrak{g}$ satisfying the conditions:

$$J_1^2=J_2^2=-Id, J_1{J}_2=-J_2{J}_1.$$

A hypercomplex structure ${\left\{J_{\alpha }\right\}}_{\alpha =1,2}$ on $\mathfrak{g}$ is an almost hypercomplex structure, such that ${\mathcal{N}}_1={\mathcal{N}}_2=0$, where ${\mathcal{N}}_{\alpha }$ is the Nijenhuis tensor corresponding to $J_{\alpha }$, defined by

$${\mathcal{N}}_{\alpha }(x, y)=[J_{\alpha }x, J_{\alpha }y]-J_{\alpha }([x, J_{\alpha }y]+[J_{\alpha }x, y])+J_{\alpha }^2[x, y]$$

for all $x, y\in \mathfrak{g}$. Such a structure is called integrable if ${\mathcal{N}}_1={\mathcal{N}}_2=0$. If G is a Lie group with the Lie algebra $\mathfrak{g}$, a hypercomplex structure on $\mathfrak{g}$ induces an invariant hypercomplex structure on G by left translations. A hypercomplex structure ${\left\{J_i\right\}}_{i=1,2,3}$ on a Lie group G is said to be left invariant if the left multiplication of each element in G is holomorphic with respect to all $J_i$, $i=1, 2, 3$. Since a left-invariant structure on a connected Lie group G is completely determined by its value at the identity element e, the hypercomplex structure on $\mathfrak{g}=T_{e}G$ uniquely determines the hypercomplex structure on G (cf. [22]).

Example 1. 

Let ${\lambda }_1\in \mathbb{R}\setminus \left\{0\right\}$, ${\lambda }_2, {\lambda }_3, a_1, a_2, a_3, a_4\in \mathbb{R}$ and $(a_1, a_2, a_3, a_4)\ne (0, 0, 0, 0)$. Consider, defined in [23], a connected Lie group G with the Lie algebra $\mathfrak{g}$, determined by the following properties for the global basis $\left(X_1, X_2, X_3, X_4\right)$ of left invariant vector fields:

$$\begin{array}{ccc}\left[X_1, X_2\right]=a_2{X}_3-a_1{X}_4, \hfill & & [X_2, X_3]=-a_2{X}_1-a_3{X}_4, \hfill \\ \hfill & & \\ \left[X_1, X_3\right]=a_2{X}_2+a_4{X}_4, \hfill & & [X_2, X_4]=a_1{X}_1+a_3{X}_3, \hfill \\ \hfill & & \\ \hfill [X_1, X_4]=-a_1{X}_2-a_4{X}_3, & & [X_3, X_4]=-a_4{X}_1+a_3{X}_2.\hfill \end{array}$$

Define the endomorphisms $I, J, K$ on the level of left invariant vector fields:

$$\begin{array}{ccccccc}I{X}_1={\lambda }_1{X}_2, \hfill & & I{X}_2=-{\lambda }_2{X}_1, \hfill & & I{X}_3=-{\lambda }_1{X}_4, \hfill & & I{X}_4={\lambda }_2{X}_3, \hfill \\ \hfill & & & & & & \\ J{X}_1={\lambda }_1{X}_3, \hfill & & J{X}_2={\lambda }_1{X}_4, \hfill & & J{X}_3=-{\lambda }_3{X}_1, \hfill & & J{X}_4=-{\lambda }_3{X}_2, \hfill \\ \hfill & & & & & & \\ K{X}_1=-{\lambda }_1{X}_4, \hfill & & K{X}_2={\lambda }_2{X}_3, \hfill & & K{X}_3=-{\lambda }_3{X}_2, \hfill & & K{X}_4={\displaystyle \frac{{\lambda }_2{\lambda }_3}{{\lambda }_1}}{X}_1.\hfill \end{array}$$

The obtained endomorphisms satisfy the following properties: $I^2=-{\lambda }_1{\lambda }_2$, $J^2=-{\lambda }_1{\lambda }_3$, $K^2=-{\lambda }_2{\lambda }_3$, $IJ=-JI={\lambda }_{1}K$, $KI=-IK={\lambda }_{2}J$ and $JK=-KJ={\lambda }_{3}I$. Thus, they determine a 3-parameter generalized quaternion structure on G.

Some of these structures are integrable for certain parameters. The definitions and properties of the bracket and endomorphisms I, J, K show that if ${\lambda }_2={\lambda }_1$ and ${\lambda }_3=-{\lambda }_1$, then $N_I=N_J=0$, and, consequently, I, J, and K are integrable. In this case, we have the following generalized quaternion relations:

$$I^2=-{\lambda }_1^2, J^2=K^2={\lambda }_1^2, IJ={\lambda }_{1}K=-JI, KI={\lambda }_{1}J=-IK, JK=-{\lambda }_{1}I=-KJ.$$

In [23], it was shown that in the classical Hamiltonian case (for ${\lambda }_1={\lambda }_2={\lambda }_3=1$) the pseudo-Hermitian structure on the Lie group G has an additional metric structure. The pseudo-Riemannian metric g on G is defined by

$$g(X_1, X_1)=g(X_2, X_2)=-g(X_3, X_3)=-g(X_4, X_4)=1$$

and $g(X_i, X_j)=0$ if $i\ne j$. Then, $(G, I, J, K)$ is an almost hypercomplex structures, such that

$$g(X, Y)=g(IX, IY)=-g(JX, JY)=-g(KX, KY).$$

Thus, g is Hermitian with respect to I and a Norden with respect to both J and K. Then, $\left(G, I, J, K, g\right)$ is an almost hypercomplex HN-metric structure (cf. [23,24]).

Example 2. 

For the Lie group G from the Example 1, for any real parameters ${\lambda }_1, {\lambda }_2, {\lambda }_3$, we can define a 3-parameter generalized quaternion structure as follows:

$$\begin{array}{ccccccc}I{X}_1=X_2, \hfill & & I{X}_2=-{\lambda }_1{\lambda }_2{X}_1, \hfill & & I{X}_3={\lambda }_1{X}_4, \hfill & & I{X}_4=-{\lambda }_2{X}_3, \hfill \\ \hfill & & & & & & \\ J{X}_1=X_3, \hfill & & J{X}_2=-{\lambda }_1{X}_4, \hfill & & J{X}_3=-{\lambda }_1{\lambda }_3{X}_1, \hfill & & J{X}_4={\lambda }_3{X}_2, \hfill \\ \hfill & & & & & & \\ K{X}_1=X_4, \hfill & & K{X}_2={\lambda }_2{X}_3, \hfill & & K{X}_3=-{\lambda }_3{X}_2, \hfill & & K{X}_4=-{\lambda }_2{\lambda }_3{X}_1.\hfill \end{array}$$

(17)

Consider $a_1\ne 0$ and $a_2=a_3=a_4=0$. Thus, G is a connected Lie group with the Lie algebra $\mathfrak{g}$, defined for the global basis $\left(X_1, X_2, X_3, X_4\right)$ of left invariant vector fields as follows:

$$\begin{array}{ccccc}\left[X_1, X_2\right]=-a_1{X}_4, \hfill & & [X_1, X_4]=-a_1{X}_2, \hfill & & [X_2, X_4]=a_1{X}_1.\end{array}$$

For such a defined Lie group G we obtain integrable endomorphisms $I, J, K$ for the triples $\left({\lambda }_1, {\lambda }_2, {\lambda }_3\right)=\left(1, 1, -1\right)$ or $\left({\lambda }_1, {\lambda }_2, {\lambda }_3\right)=\left(-1, -1, 1\right)$. For the triple $\left({\lambda }_1, {\lambda }_2, {\lambda }_3\right)=\left(1, 1, -1\right)$ we have a 3-parameter generalized quaternions structure satisfying the properties of split quaternions:

$$I^2=-1, J^2=K^2=1, IJ=K=-JI, KI=J=-IK, JK=-I=-KJ,$$

while for $\left({\lambda }_1, {\lambda }_2, {\lambda }_3\right)=\left(-1, -1, 1\right)$ we have

$$I^2=J^2=-1, K^2=1, IJ=-K=-JI, KI=-J=-IK, JK=I=-KJ.$$

Examples of hypercomplex structures (i.e., considered quaternion structures in the classical Hamiltonian case ${\lambda }_1={\lambda }_2={\lambda }_3=1$) can be found in many papers, where examples include various families of manifolds with additional structures. The concept of hypercomplex structures was introduced by Boyer in [1], where he classified compact hypercomplex manifolds of real dimension four. The construction of homogeneous hypercomplex structures on compact Lie groups was provided by Joyce in [12] (we also refer to [10], where this construction is described). Families of hypercomplex or almost hypercomplex structures (including Abelian hypercomplex structures) on Lie groups associated with their Lie algebras are the subject of many papers. Among these works, the following papers by Barberis, Dotti, Fino, and Andrada can be distinguished: [22,25,26,27,28]. For many other interesting examples of hypercomplex manifolds, the reader is reffered to [12,29,30,31].

4. Existence

We further assume that ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are non-zero real numbers. Let M be a manifold equipped with endomorphisms $I, J, K:TM\to TM$ of $TM$ satisfying the properties (1) of 3-parameter generalized quaternions.

4.1. Definition of ∇

First, we define

$$〈·, ·〉:\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(TM\right)$$

(18)

by

$$〈X, Y〉={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, Y]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[X, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[IX, JY].$$

Lemma 2. 

For any $X, Y\in \Gamma \left(TM\right)$ and $f\in C^{\infty }\left(M\right)$, we have the following conditions:

(a) 

$〈X, f·Y〉=f·〈X, Y〉+X\left(f\right)·Y,$

(b) 

$〈f·X, Y〉=f·〈X, Y〉+Y\left(f\right)·X$.

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$, $f\in C^{\infty }\left(M\right)$. By using the Leibniz rules of the Lie bracket of vector fields on M and the properties $J^2=-{\lambda }_1{\lambda }_3$ and $KJ=-{\lambda }_{3}I$, we can write

$$\begin{array}{cc}\hfill & 〈X, f·Y〉\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, f·Y]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[X, f·JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[IX, f·JY]\hfill \\ \hfill & =f·〈X, Y〉+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left(IX\right)\left(f\right)·\left(IY\right)-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}X\left(f\right)·\left(J^{2}Y\right)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}\left(IX\right)\left(f\right)·\left(KJY\right)\hfill \\ \hfill & =f·〈X, Y〉+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left(IX\right)\left(f\right)·\left(IY\right)+X\left(f\right)·Y-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left(IX\right)\left(f\right)·\left(IY\right)\hfill \\ \hfill & =f·〈X, Y〉+X\left(f\right)·Y.\hfill \end{array}$$

Next, by using $I^2=-{\lambda }_1{\lambda }_2$ and $KI={\lambda }_{2}J$, we obtain

$$\begin{array}{cc}\hfill & 〈f·X, Y〉\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[f·IX, Y]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[f·X, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[f·IX, JY]\hfill \\ \hfill & =f·〈X, Y〉-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}Y\left(f\right)·\left(I^{2}X\right)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}\left(JY\right)\left(f\right)·\left(JX\right)-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}\left(JY\right)\left(f\right)·\left(KIX\right)\hfill \\ \hfill & =f·〈X, Y〉+Y\left(f\right)·X+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}\left(JY\right)\left(f\right)·\left(JX\right)-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}\left(JY\right)\left(f\right)·\left(JX\right)\hfill \\ \hfill & =f·〈X, Y〉+Y\left(f\right)·X.\hfill \end{array}$$

□

Based on the properties of the map $〈·, ·〉$ written in Lemma 2, we have

$$\nabla :\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(TM\right)$$

defined by

$${\nabla }_{X}Y={\displaystyle \frac{1}{2}}\left(\left[X, Y\right]+〈X, Y〉\right),$$

which is a linear connection on M with torsion $T^{\nabla }$ given by

$$T^{\nabla }(X, Y)={\displaystyle \frac{1}{2}}(〈X, Y〉-〈Y, X〉)$$

(19)

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

4.2. Torsion of ∇

Next, we determine the torsion of ∇.

Theorem 3. 

For any $X, Y\in \Gamma \left(TM\right)$, we have the following condition:

$$〈X, Y〉=〈Y, X〉-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}{N}_I(X, Y)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}{N}_J(X, Y)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3^2}}{N}_K(JX, JY).$$

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. First, based on the definition of the Nijenhuis torsion for I and J, and $I^2=-{\lambda }_1{\lambda }_2$, we obtain the following formulas:

$$I[IX, Y]=\left[IX, IY\right]-I[X, IY]-{\lambda }_1{\lambda }_2[X, Y]-N_I(X, Y)$$

(20)

and

$$-J[X, JY]=-\left[JX, JY\right]+J[JX, Y]+{\lambda }_1{\lambda }_3[X, Y]+N_J(X, Y).$$

(21)

Moreover, according to the definition of the Nijenhuis torsion and $KJ=-{\lambda }_{3}I$ and $K^2=-{\lambda }_2{\lambda }_3$,

$$\begin{array}{cc}\hfill N_K(JX, JY)& =\left[KJX, KJY\right]-K[KJX, JY]-K[JX, KJY]+K^2[JX, JY]\hfill \\ \hfill & =\left[-{\lambda }_{3}IX, -{\lambda }_{3}IY\right]-K[-{\lambda }_{3}IX, JY]-K[JX, -{\lambda }_{3}IY]-{\lambda }_2{\lambda }_3[JX, JY]\hfill \\ \hfill & ={\lambda }_3^2\left[IX, IY\right]+{\lambda }_{3}K[IX, JY]+{\lambda }_{3}K[JX, IY]-{\lambda }_2{\lambda }_3[JX, JY].\hfill \end{array}$$

Thus, we obtain the following formula:

$$K[IX, JY]=-K[JX, IY]-{\lambda }_3\left[IX, IY\right]+{\lambda }_2[JX, JY]+{\displaystyle \frac{1}{{\lambda }_3}}{N}_K(JX, JY).$$

(22)

From (20)–(22), we conclude that

$$\begin{array}{cc}\hfill & 〈X, Y〉\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, Y]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[X, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[IX, JY]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left(\left[IX, IY\right]-I[X, IY]-{\lambda }_1{\lambda }_2[X, Y]-N_I(X, Y)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}\left(-\left[JX, JY\right]+J[JX, Y]+{\lambda }_1{\lambda }_3[X, Y]+N_J(X, Y)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}\left(-K[JX, IY]-{\lambda }_3\left[IX, IY\right]+{\lambda }_2[JX, JY]+{\displaystyle \frac{1}{{\lambda }_3}}{N}_K(JX, JY)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left[IX, IY\right]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[X, IY]-[X, Y]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}{N}_I(X, Y)\hfill \\ \hfill & -{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}\left[JX, JY\right]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[JX, Y]+[X, Y]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}{N}_J(X, Y)\hfill \\ \hfill & -{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[JX, IY]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left[IX, IY\right]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}[JX, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3^2}}{N}_K(JX, JY).\hfill \end{array}$$

Subsequently, based on the reduction of opposite terms, we obtain

$$\begin{array}{cc}\hfill 〈X, Y〉& ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IY, X]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[Y, JX]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[, IY, JX]\hfill \\ \hfill & -{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}{N}_I(X, Y)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}{N}_J(X, Y)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3^2}}{N}_K(JX, JY)\hfill \\ \hfill & =〈Y, X〉\hfill \\ \hfill & -{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}{N}_I(X, Y)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}{N}_J(X, Y)+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3^2}}{N}_K(JX, JY), \hfill \end{array}$$

which validates the claim. □

Taken together with (19), we can deduce the following:

Corollary 4. 

For $X, Y\in \Gamma \left(TM\right)$, we have

$$T^{\nabla }(X, Y)=-{\displaystyle \frac{1}{2{\lambda }_1{\lambda }_2}}{N}_I(X, Y)+{\displaystyle \frac{1}{2{\lambda }_1{\lambda }_3}}{N}_J(X, Y)+{\displaystyle \frac{1}{2{\lambda }_1{\lambda }_2{\lambda }_3^2}}{N}_K(JX, JY).$$

Corollary 5. 

If structures I and J are integrable, i.e., $N_I=N_J=0$, then ∇ is a torsion-free connection.

Proof. 

Lemma 1 shows that the integrability of I and J implies the integrability of K. Corollary 4 now yields $T^{\nabla }=0$. □

4.3. Compatibility of ∇ with the Generalized Quaternionic Structure

According to

$$\left({\nabla }_{X}IJ\right)\left(Y\right)=\left({\nabla }_{X}I\right)\left(JY\right)+I\left(\left({\nabla }_{X}J\right)\left(Y\right)\right)$$

(23)

for $X, Y\in \Gamma \left(TM\right)$, if ∇ is compatible with I and J, then ∇ is also compatible with $K={\displaystyle \frac{1}{{\lambda }_1}}IJ$.

Theorem 4. 

Let $X, Y\in \Gamma \left(TM\right)$. Then, we can write the following accordingly

(a) 

$2\left({\nabla }_{X}I\right)\left(Y\right)={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I\left(N_I(X, Y)\right)-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}J\left(N_I(X, KY)\right),$

(b) 

$\nabla J=0,$

(c) 

$2\left({\nabla }_{X}K\right)\left(Y\right)={\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2}}I\left(N_I(X, JY)\right)+{\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2{\lambda }_3^2}}J\left(N_I(X, IY)\right).$

Proof. 

(a) We observe that

$$〈X, IY〉={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, IY]+{\displaystyle \frac{1}{{\lambda }_3}}J[X, KY]-{\displaystyle \frac{1}{{\lambda }_2{\lambda }_3}}K[IX, KY],$$

(24)

$$I〈X, Y〉=-[IX, Y]-{\displaystyle \frac{1}{{\lambda }_3}}K[X, JY]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[IX, JY],$$

(25)

$$I\left(N_I(X, Y)\right)=I[IX, IY]+{\lambda }_1{\lambda }_2\left[X, IY\right]+{\lambda }_1{\lambda }_2[IX, Y]-{\lambda }_1{\lambda }_{2}I\left[X, Y\right]$$

(26)

and

$$J\left(N_I(X, KY)\right)={\lambda }_{1}K[IX, KY]-{\lambda }_{2}J[IX, JY]-{\lambda }_1{\lambda }_{2}K[X, JY]-{\lambda }_1{\lambda }_{2}J[X, KY].$$

(27)

From (24) and (25), we deduce:

$$\begin{array}{cc}\hfill 2\left({\nabla }_{X}I\right)\left(Y\right)& =2{\nabla }_X\left(IY\right)-2I\left({\nabla }_{X}Y\right)\hfill \\ \hfill & =\left[X, IY\right]-I\left[X, Y\right]+〈X, IY〉-I〈X, Y〉\hfill \\ \hfill & =\left[X, IY\right]-I\left[X, Y\right]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, IY]+{\displaystyle \frac{1}{{\lambda }_3}}J[X, KY]-{\displaystyle \frac{1}{{\lambda }_2{\lambda }_3}}K[IX, KY]\hfill \\ \hfill & +[IX, Y]+{\displaystyle \frac{1}{{\lambda }_3}}K[X, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[IX, JY]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, IY]+\left[X, IY\right]+[IX, Y]-I\left[X, Y\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[IX, JY]-{\displaystyle \frac{1}{{\lambda }_2{\lambda }_3}}K[IX, KY]+{\displaystyle \frac{1}{{\lambda }_3}}K[X, JY]+{\displaystyle \frac{1}{{\lambda }_3}}J[X, KY]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}\left(I[IX, IY]+{\lambda }_1{\lambda }_2\left[X, IY\right]+{\lambda }_1{\lambda }_2[IX, Y]-{\lambda }_1{\lambda }_{2}I\left[X, Y\right]\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}\left({\lambda }_{1}K[IX, KY]-{\lambda }_{2}J[IX, JY]-{\lambda }_1{\lambda }_{2}K[X, JY]-{\lambda }_1{\lambda }_{2}J[X, KY]\right).\hfill \end{array}$$

Subsequently, by using (26) and (27), we obtain the equality (a).

(b) Additionally, because $J^2=-{\lambda }_1{\lambda }_3$, we have

$$\begin{array}{cc}\hfill 〈X, JY〉& ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, JY]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}J[X, J^{2}Y]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}K[IX, J^{2}Y]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, JY]+J[X, Y]-{\displaystyle \frac{1}{{\lambda }_2}}K[IX, Y].\hfill \end{array}$$

Moreover, based on the properties (1) of a generalized quaternionic structure,

$$\begin{array}{cc}\hfill J〈X, Y〉& ={\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}JI[IX, Y]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_3}}{J}^2[X, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2{\lambda }_3}}JK[IX, JY]\hfill \\ \hfill & =-{\displaystyle \frac{1}{{\lambda }_2}}K[IX, Y]+[X, JY]+{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, JY].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill 2\left({\nabla }_{X}J\right)\left(Y\right)& =\left[X, JY\right]-J\left[X, Y\right]+〈X, JY〉-J〈X, Y〉\hfill \\ \hfill & =\left[X, JY\right]-J\left[X, Y\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, JY]+J[X, Y]-{\displaystyle \frac{1}{{\lambda }_2}}K[IX, Y]\hfill \\ \hfill & +{\displaystyle \frac{1}{{\lambda }_2}}K[IX, Y]-[X, JY]-{\displaystyle \frac{1}{{\lambda }_1{\lambda }_2}}I[IX, JY]\hfill \\ \hfill & =0.\hfill \end{array}$$

(c) From (23), (a) and (b), we obtain

$$\begin{array}{cc}\hfill 2\left({\nabla }_{X}K\right)\left(Y\right)& ={\displaystyle \frac{2}{{\lambda }_1}}\left({\nabla }_{X}IJ\right)\left(Y\right)={\displaystyle \frac{2}{{\lambda }_1}}\left({\nabla }_{X}I\right)\left(JY\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2}}I\left(N_I(X, JY)\right)-{\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2{\lambda }_3}}J\left(N_I(X, KJY)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2}}I\left(N_I(X, JY)\right)-{\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2{\lambda }_3}}J\left(N_I(X, -{\lambda }_{3}IY)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2}}I\left(N_I(X, JY)\right)+{\displaystyle \frac{1}{{\lambda }_1^2{\lambda }_2{\lambda }_3^2}}J\left(N_I(X, IY)\right).\hfill \end{array}$$

□

Corollary 6. 

If $N_I=0$, then $\nabla I=\nabla J=\nabla K=0$.

Taken together with Corollary 5, we can deduce the following:

Corollary 7. 

If $N_I=N_J=0$, then ∇ is a torsion-free connection on M compatible with I, J and K.

5. Uniqueness

Here, we demonstrate the uniqueness of the torsion-free connection compatible with the given generalized hypercomplex structure $(I, J, K)$, i.e., $\nabla I=\nabla J=\nabla K=0$.

We take two torsion-free connections ∇ and ${\nabla }^{\prime }$ on M that are compatible with I, J, and K. We define $\omega$ as follows:

$$\omega :\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(TM\right),$$

$$\omega (X, Y)={\nabla }_{X}Y-{\nabla }_X^{\prime }Y.$$

Because ∇ and ${\nabla }^{\prime }$ are torsion-free connections, the following symmetric products are deduced:

$$〈·, ·〉:\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(TM\right)$$

and

$${〈·, ·〉}^{\prime }:\Gamma \left(TM\right)\times \Gamma \left(TM\right)⟶\Gamma \left(TM\right)$$

such that

$${\nabla }_{X}Y={\displaystyle \frac{1}{2}}\left(\left[X, Y\right]+〈X, Y〉\right)$$

and

$${\nabla }_X^{\prime }Y={\displaystyle \frac{1}{2}}\left(\left[X, Y\right]+{〈X, Y〉}^{\prime }\right)$$

for $X, Y\in \Gamma \left(TM\right)$. Therefore,

$$\omega (X, Y)=〈X, Y〉-{〈X, Y〉}^{\prime }, X, Y\in \Gamma \left(TM\right),$$

and $\omega$ is a symmetric form with values in the tangent bundle $TM$ as a difference of two symmetric products, i.e., $\omega \in \Gamma ({⨀}^2{T}^{\ast }M\otimes TM)$.

Lemma 3. 

For any $X, Y\in \Gamma \left(TM\right)$, we have

(a) 

$\omega (X, IY)=I\left(\omega (X, Y)\right)+2({\nabla }_{X}I-{\nabla }_X^{\prime }I)\left(Y\right),$

(b) 

$\omega (X, JY)=J\left(\omega (X, Y)\right)+2({\nabla }_{X}J-{\nabla }_X^{\prime }J)\left(Y\right),$

(c) 

$\omega (X, KY)=K\left(\omega (X, Y)\right)+2({\nabla }_{X}K-{\nabla }_X^{\prime }K)\left(Y\right)$.

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. Subsequently, we obtain

$$\begin{array}{cc}\hfill 2\left({\nabla }_{X}I\right)\left(Y\right)& =2\left({\nabla }_{X}IY\right)-2I\left({\nabla }_{X}Y\right)\hfill \\ \hfill & =\left[X, IY\right]-〈X, IY〉.\hfill \end{array}$$

(28)

Likewise,

$$2\left({\nabla }_{X}I\right)\left(Y\right)=\left[X, IY\right]-{〈X, IY〉}^{\prime }.$$

(29)

From (28) and (29), we immediately conclude (a). Based on a similar argument, we have (b) and (c). □

Corollary 8. 

Let ∇ be compatible with structures $I, J$ and K. Then, for all $X, Y\in \Gamma \left(TM\right)$, we obtain

(a) 

$\omega (X, IY)=\omega (IX, Y),$

(b) 

$\omega (X, JY)=\omega (JX, Y),$

(c) 

$\omega (X, KY)=\omega (KX, Y)$.

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. Based on (a) in Lemma 3, the compatibility of ∇ with I, and the symmetry of $\omega$, we obtain

$$\omega (X, IY)=I\left(\omega \right(X, Y\left)\right)=I\left(\omega \right(Y, X\left)\right)=\omega (IX, Y).$$

We also prove (b) and (c) analogously. □

Theorem 5. 

Let ∇ be compatible with the structures $I, J$ and K. Then, for any $X, Y\in \Gamma \left(TM\right)$, we can write

(a) 

$\omega (IX, IY)=-{\lambda }_1{\lambda }_2\omega (X, Y),$

(b) 

$\omega (JX, JY)=-{\lambda }_1{\lambda }_3\omega (X, Y),$

(c) 

$\omega (KX, KY)=-{\lambda }_2{\lambda }_3\omega (X, Y).$

Proof. 

Let $X, Y\in \Gamma \left(TM\right)$. Based on Corollary 8 and the properties of I, J, and K given in (1), we can now write

$$\begin{array}{cc}\hfill \omega (IX, IY)& =\omega (I^{2}X, Y)=-{\lambda }_1{\lambda }_2\omega (X, Y), \hfill \\ \hfill \omega (JX, JY)& =\omega (J^{2}X, Y)=-{\lambda }_1{\lambda }_3\omega (X, Y), \hfill \\ \hfill \omega (KX, KY)& =\omega (K^{2}X, Y)=-{\lambda }_2{\lambda }_3\omega (X, Y).\hfill \end{array}$$

□

Let $X, Y\in \Gamma \left(TM\right)$. By using the equality $IJ={\lambda }_{1}K$ and Theorem 5, we have

$${\lambda }_1^2·\omega (KX, KY)=\omega (IJX, IJY)=-{\lambda }_1{\lambda }_2\omega (JX, JY)={\lambda }_1^2{\lambda }_2{\lambda }_3\omega (X, Y).$$

Moreover, according to (c) in Theorem 5 and

$${\lambda }_1^2·\omega (KX, KY)={\lambda }_1^2{\lambda }_2{\lambda }_3\omega (X, Y),$$

it follows that

$$-{\lambda }_1^2{\lambda }_2{\lambda }_3\omega (X, Y)={\lambda }_1^2{\lambda }_2{\lambda }_3\omega (X, Y).$$

Because ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$ are non-zero numbers, we have

$$-\omega (X, Y)=\omega (X, Y).$$

This yields

$$\omega (X, Y)=0$$

which implies that $〈X, Y〉={〈X, Y〉}^{\prime }$ for $X, Y\in \Gamma \left(TM\right)$. In equivalent terms, $\nabla ={\nabla }^{\prime }$. Thus, we validate the uniqueness of the torsion-free linear connection compatible with the given structures I, J, K.

6. Conclusions

In this paper, we show that a manifold endowed with a 3-parameter generalized quaternion structure has a unique torsion-free connection compatible with these structures if they are integrable (with zero Nijenhuis torsions) and have nonzero parameters. This generalizes a classical result obtained by Obata [8]. Some algebraic properties of quaternionic structures were also obtained. We claim that the proposed generalization can contribute to the study of new structures on manifolds and their algebroid generalizations. This is evidenced by the fact that the literature on hypercomplex structures considers manifolds on the one hand, and Lie algebras on the other. The main goal of considering hypercomplex structures on Lie algebras was originally to find a corresponding structure on Lie groups. Lie algebras, on the other hand, are certain Lie algebroids, similar to the tangent bundle to a manifold. The series of papers deals not only with hypercomplex structures on Lie algebras, but also with structures equipped with torsion. In particular, Refs. [32,33] are devoted to hyperkähler structures on Lie algebras with torsion 3-form (HKT). Lie algebras equipped with HKT structures induce left-invariant HKT structures on the corresponding simply connected Lie groups. If we equip a Lie algebroid with a metric, we can adopt some of the concepts and methods known for smooth manifolds and those for Lie algebras to any Lie algebroid. In both cases, for manifolds and Lie algebras with an HKT structure, we can define the Lee form and further study the holonomy group of the Obata connections using important ideas from the theory of connections with skew–symmetric torsion contained in [34,35] or study the properties of algebroids using ideas from Hodge theory [11,36]. Verbitsky proved in [11] that a compact HKT manifold has a holomorphically trivial canonical bundle exactly when the holonomy of the Obata connection is a subgroup of $SL(n, \mathbb{H})$. It is shown there that a compact hypercomplex manifold with a holomorphically trivial canonical bundle has the holonomy group of the Obata connection inside $SL(n, \mathbb{H})$ if it admits an HKT structure. Furthermore, Ivanov and Petkov studied in [35] the existence of an HKT structure depending on the cohomology properties of the Lee form, which under certain conditions (cf. [35]) test whether the holonomy group of the Obata connection is contained in a special quaternionic linear group. From these considerations, it follows that it is important to investigate the Obata connection for various geometric structures.