On h-Quasi-Hemi-Slant Riemannian Maps

axioms Article On h-Quasi-Hemi-Slant Riemannian Maps Mohd Bilal 1 , Sushil Kumar 2 , Rajendra Prasad 3 , Abdul Haseeb 4, * 1 2 3 4 5 * and Sumeet Kumar 5 Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al Qura University, Makkah 21955, Saudi Arabia Department of Mathematics, Shri Jai Narain Post Graduate College, University of Lucknow (U.P.), Lucknow 226001, India Department of Mathematics and Astronomy, University of Lucknow (U.P.), Lucknow 226007, India Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia Department of Mathematics, Dr. Shree Krishna Sinha Women’s College Motihari, Babasaheb Bhimrao Ambedkar University, Muzaffarpur 845401, India Correspondence: [email protected] or [email protected] 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. Keywords: Riemannian map; hyperkähler manifold; h-quasi-hemi-slant Riemannian map MSC: 53C15; 53C26; 53C43 1. Introduction Citation: Bilal, M.; Kumar, S.; Prasad, R.; Haseeb, A.; Kumar, S. On h-Quasi-Hemi-Slant Riemannian Maps. Axioms 2022, 11, 641. https:// doi.org/10.3390/axioms11110641 Academic Editor: Mica Stankovic Received: 19 October 2022 Accepted: 11 November 2022 Published: 14 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 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 π : ( N1 , g1 ) → ( N2 , g2 ) between Riemannian manifolds ( N1 , g1 ) and ( N2 , g2 ) is called a Riemannian map (0 < rankπ∗ < min{m, n}, where dim N1 = m and dim N2 = n) if it satisfies the following equation: g2 (π∗ W1 , π∗ W2 ) = g1 (W1 , W2 ), for W1 , W2 ∈ Γ(ker π∗ )⊥ , (1) where π∗ is the differentiable map of π. Consequently, isometric immersions and Riemannian submersions are particular cases of Riemannian maps with ker π∗ = 0 and (rangeπ∗ )⊥ = 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]. Axioms 2022, 11, 641. https://doi.org/10.3390/axioms11110641 https://www.mdpi.com/journal/axioms Axioms 2022, 11, 641 2 of 15 We also note that quarternionic manifolds have many applications, including for nonlinear σ 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], hemislant 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 ( N1 , g1 ) and ( N2 , g2 ) be Riemannian manifolds and π : ( N1 , g1 ) → ( N2 , g2 ) be a C ∞ -Riemannian map [1]. We define O’Neill’s tensors T and A [4] by A F1 F2 = H∇H F1 V F2 + V ∇H F1 H F2 , (2) T F1 F2 = H∇V F1 V F2 + V ∇V F1 H F2 , (3) for any vector fields F1 , F2 on N1 , where ∇ is the Levi-Civita connection of g1 . From Equations (2) and (3), we have ∇Y1 Y2 = TY1 Y2 + V ∇Y1 Y2 , (4) ∇Y1 U1 = TY1 U1 + H∇Y1 U1 , (5) ∇U1 Y1 = AU1 Y1 + V ∇U1 Y1 , (6) ∇U1 U2 = H∇U1 U2 + AU1 U2 , (7) )⊥ , for Y1 , Y2 ∈ Γ(ker π∗ ) and U1 , U2 ∈ Γ(ker π∗ where H∇Y1 U1 = AU1 Y1 and U1 is basic. Let π : ( N1 , E, g1 ) → ( N2 , g2 ) be a C ∞ map. The second fundamental form of π is given by N π (∇π∗ )(V1 , V2 ) = ∇V π (V2 ) − π∗ (∇V11 V2 ), (8) 1 ∗ for V1 , V2 ∈ Γ( TN1 ), where ∇π is the pullback connection [27]. The map π is said to be a total geodesic if (∇π∗ )(V1 , V2 ) = 0 for V1 , V2 ∈ Γ( TN1 ). Let ( N1 , E, g1 ) be an almost quaternionic Hermitian manifold, where g1 is a Riemanian metric on the maniifold N1 and E is a rank 3 subbundle of End( TN1 ) such that for any point p ∈ N1 within some neighborhood U, there exists a local basis { J1 , J2 , J3 } of sections of E on U satisfying all α ∈ {1, 2, 3} in which Jα2 = −id, Jα Jα+1 = − Jα+1 Jα = Jα+2 , (9) g1 ( Jα X2 , Jα X1 ) = g1 ( X1 , X2 ), (10) Axioms 2022, 11, 641 3 of 15 for X1 , X2 ∈ Γ( TN1 ), where the indices are taken from {1, 2, 3} modulo 3 and { J1 , J2 , J3 } is called the quaternionic Hermitian basis. The structure ( N1 , E, g1 ) is called a quaternionic kähler manifold if there exist locally defined 1 forms ω1 , ω2 , ω3 such that for α ∈ {1, 2, 3}, we have ∇ X1 Jα = ωα+2 ( X1 ) Jα+1 − ωα+1 ( X1 ) Jα+2 , (11) for X1 ∈ Γ( TN1 ), where the indices are taken from {1, 2, 3} modulo 3. If there exists a global parallel quaternionic Hermitian basis { J1 , J2 , J3 } of sections of E on N1 , then ( N1 , E, g1 ) is called a hyperkähler. The structure { J1 , J2 , J3 , g1 }, where g1 , a hyperkähler metric, is called a hyperkähler structure on N1 . A map π : ( N1 , E1 , g1 ) → ( N2 , E2 , g2 ) is called an ( E1 , E2 )-holomorphic map if for any point p ∈ N1 and J ∈ ( E1 ) p , there exists J ′ ∈ ( E2 )π ( p) such that π∗ ◦ J = J ′ ◦ π∗ . A Riemannian submersion between quaternionic kähler manifolds π : ( N1 , E1 , g1 ) → ( N2 , E2 , g2 ), which is an ( E1 , E2 )-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 ( N1 , E, g1 ) to the Riemannian manifold ( N2 , g2 ) is called an h-semi-slant Riemannian map if, given a point p ∈ N1 with a neighborhood U, there exists a quaternionic Hermitian basis { I, J, K } of sections of E on U such that for any R ∈ { I, J, K }, the following is true: ker π∗ = D1 ⊕ D2 , R( D1 ) = D1 , in which the angle θ R = θ R ( Z1 ) between RZ1 and the space ( D2 )q is constant for a non-zero Z1 ∈ ( D2 )q and q ∈ U, where D2 is an orthogonal complement of D1 in ker π∗ . Furthermore, assume we have θ = θ I = θ J = θK , Then, we call the map π : ( N1 , E, g1 ) → ( N2 , g2 ) a strictly h-semi-slant Riemannian map, the basis { I, J, K } a strictly h-semi-slant basis and the angle θ 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 ( N1 , E, g1 ) to the Riemannian manifold ( N2 , g2 ) is called an h-qhs Riemannian map if, given a point p ∈ N1 with a neighborhood U, there exists a quaternionic Hermitian basis { I, J, K } of sections of E on U such that for any R ∈ { I, J, K }, there is a distribution D ⊂ (ker π∗ ) on U such that ker π∗ = D ⊕ D1 ⊕ D2 , R( D ) = D, R( D2 ) ⊂ (ker π∗ )⊥ , and the angle θ R = θ R ( Z1 ) between RZ1 and the space ( D1 )q is constant for a non-zero Z1 ∈ ( D1 )q and q ∈ U, where ker π∗ admits three orthogonal complementary distributions D, D1 and D2 such that D is invariant, D1 is a slant with an angle θ R and D2 is anti-invariant. We call the basis { I, J, K } an h-qhs basis and the angles {θ I , θ J , θK } h-qhs angles. Furthermore, let us say we have θ = θ I = θ J = θK , Axioms 2022, 11, 641 4 of 15 Then, we call the map π : ( N1 , E, g1 ) → ( N2 , g2 ) a strictly h-qhs Riemannian map, the basis { I, J, K } a strictly quasi-hemi-slant basis and the angle θ a strictly quasi-hemislant angle: Definition 3. A Riemannian map π from the almost quaternionic Hermitian manifold ( N1 , E, g1 ) to the Riemannian manifold ( N2 , g2 ) is called an almost h-qhs Riemannian map if, given a point p ∈ N1 with a neighborhood U, there exists a quaternionic Hermitian basis { I, J, K } of sections of E on U such that for any R ∈ { I, J, K }, there is a distribution D R ⊂ (ker π∗ ) on U such that ker π∗ = D R ⊕ D1R ⊕ D2R , R( D R ) = D R , R( D2R ) ⊂ (ker π∗ )⊥ , and the angle θ R = θ R ( Z1 ) between RZ1 and the space ( D1R )q is constant for a non-zero Z1 ∈ ( D1R )q and q ∈ U, where the vertical distribution ker π∗ admits three orthogonal complementary distributions D R , D1R and D2R such that D R is invariant, D1R is a slant with an angle θ R and D2R is anti-invariant. We call the basis { I, J, K } an almost h-qhs basis and the angles {θ I , θ J , θK } almost h-qhs angles. Let π : ( N1 , E, g1 ) → ( N2 , g2 ) be an almost h-qhs Riemannian map. We can easily observe the following: (a) (b) If dim D R 6= 0, dim D1R 6= 0, 0 < θ R < π2 and dim D2R = 0, then π is an almost proper h-semi-slant Riemannian map with a semi-slant angle θ R ; If dim D R = 0, dim D1R 6= 0, 0 < θ R < π2 and dim D2R 6= 0, then π is an almost h-hemi-slant Riemannian map. We say that the almost h-qhs Riemannian map π : ( N1 , E, g1 ) → ( N2 , g2 ) is proper if D R 6= {0}, D2R 6= {0} and θ R 6= 0, π2 . 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 π∗ )⊥ = ω R ( D1R ) ⊕ R( D2R ) ⊕ µ R . Obviously, µ R is an invariant sub-bundle of (ker π∗ )⊥ with respect to the complex structure R. For V1 ∈ Γ(ker π∗ ), we have V1 = PR V1 + Q R V1 + SR V1 , (12) where PR V1 ∈ Γ( D R ), Q R V1 ∈ Γ( D1R ), SR V1 ∈ Γ( D2R ) and R ∈ { I, J, K }. For Z1 ∈ Γ(ker π∗ ), we obtain RZ1 = φR Z1 + ω R Z1 , (13) where φR Z1 ∈ Γ(ker π∗ ), ω R Z1 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }. For X1 ∈ Γ(ker π∗ )⊥ , we have RX1 = BR X1 + CR X1 , (14) where BR X1 ∈ Γ(ker π∗ ), CR X1 ∈ Γ(µ R ) and R ∈ { I, J, K }. We will denote an almost h-qhs Riemannian map from a hyperkähler manifold ( N1 , I, J, K, g1 ) onto a Riemannian manifold ( N2 , g2 ) such that ( I, J, K ) is an almost h-qhs basis by π. The following lemmas can be easily obtained: Lemma 1. For π : ( N1 , g1 , E1 ) → ( N2 , g2 , E2 ), we get φR D R = D R , ω R D R = 0, φR D2R = 0, ω R D2R ⊂ (ker π∗ )⊥ , Axioms 2022, 11, 641 5 of 15 where R ∈ { I, J, K }. Lemma 2. For π : ( N1 , g1 , E1 ) → ( N2 , g2 , E2 ), we have 2 φR Z1 + BR ω R Z1 φR BR Z2 + BR CR Z2 = − Z1 , ω R φR Z1 + CR ω R Z1 = 0, = 0, ω R BR Z2 + CR2 Z2 = − Z2 , for any Z1 ∈ Γ(ker π∗ ), Z2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }. Proof. Using Equations (9), (13) and (14), we can find all equations of Lemma 2: Lemma 3. With π : ( N1 , I, J, K, g1 ) → ( N2 , g2 ) being an almost h-qhs Riemannian map, we then obtain V ∇ X1 φ R X 2 + T X1 ω R X 2 = B R T X1 X 2 + φ R V ∇ X1 X 2 , (15) T X1 φR X2 + H∇ X1 ω R X2 = CR T X1 X2 + ω R V ∇ X1 X2 , (16) T X1 BR Z1 + H∇ X1 CR Z1 = CR H∇ X1 Z1 + ω R T X1 Z1 , (17) V ∇ X1 BR Z1 + T X1 CR Z1 = BR H∇ X1 Z1 + φT X1 Z1 , (18) V ∇ Z1 φR X1 + A Z1 ω R X1 = BR A Z1 X1 + φR V ∇ Z1 X1 , (19) A Z1 φR X1 + H∇ Z1 ω R X1 = CR A Z1 X1 + ω R V ∇ Z1 X1 , (20) A Z1 BR Z2 + H∇ Z1 CR Z2 = CR H∇ Z1 Z2 + ω R A Z1 Z2 , (21) V ∇ Z1 BR Z2 + A Z1 CR Z2 = BR H∇ Z1 Z2 + φR A Z1 Z2 , (22) for X1 , X2 ∈ Γ(ker π∗ ), Z1 , Z2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }. Proof. Using Equations (4)–(7), (13) and (14), we can easily obtain Equations (15)–(22). Now, we define (∇ X1 φR ) X2 = V ∇ X1 φR X2 − φR V ∇ X1 X2 , (23) (∇ X1 ω R ) X2 = H∇ X1 ω R X2 − ω R V ∇ X1 X2 , (24) (∇ Z1 BR ) Z2 = V ∇ Z1 BR Z2 − BR H∇ Z1 Z2 , (25) (∇ Z1 CR ) Z2 = H∇ Z1 CR Z2 − CR H∇ Z1 Z2 , (26) for X1 , X2 ∈ Γ(ker π∗ ), Z1 , Z2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }. Lemma 4. For π : ( N1 , I, J, K, g1 ) → ( N2 , g2 ), we find (∇ X1 φR ) X2 = BR T X1 X2 − T X1 ω R X2 , (∇ X1 ω R ) X2 = CR T X1 X2 − T X1 φR X2 , (∇ Z1 CR ) Z2 = ω R A Z1 Z2 − A Z1 BR Z2 , (∇ Z1 BR ) Z2 = φR A Z1 Z2 − A Z1 CR Z2 , for all X1 , X2 ∈ Γ(ker π∗ ), Z1 , Z2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }. Proof. Using Equations (15) and (16) as well as Equations (21)–(26), Lemma 4 follows. then If the tensors φR and ω R are parallel with respect to the linear connection ∇ on N1 , B R T X1 X2 = T X1 ω R X2 , C R T X1 X2 = T X1 φ R X2 , for all X1 , X2 ∈ Γ(ker π∗ ) and R ∈ { I, J, K }: Axioms 2022, 11, 641 6 of 15 Lemma 5. Let π : ( N1 , E, g1 ) → ( N2 , g2 ), be an almost h-qhs Riemannian map. Then, we obtain 2 φR V1 = − cos2 θ R V1 , (27) for any non-zero vector field V1 ∈ Γ( D1R ) and R ∈ { I, J, K }, where { I, J, K} is an almost h-qhs basis with the almost h-qhs angles {θ I , θ J , θK }. Proof. For any non-zero vector field V1 ∈ Γ( D1R ) and R ∈ { I, J, K }, we have k φR V1 k , k RV1 k (28) g1 ( RV1 , φR V1 ) , k φR V1 kk RV1 k (29) cos θ R = and cos θ R = where θ R (V1 ) is the h-qhs angle. Using Equations (9) and (13), we obtain cos θ R = − g1 (V1 , φR 2 V1 ) . k φR V1 kk RV1 k (30) From Equations (29) and (30), Equation (27) follows. Theorem 1. Let π be an h-qhs Riemannian map from an almost hyperkahler manifold ( N1 , I, J, K, g1 ) to a Riemannian manifold ( N2 , g2 ). Then, the following cases are equivalent: ( a) D R is integrable; (b) g1 (T Z2 IZ1 − T Z1 IZ2 , ω I Q I U1 + IS I U1 ) = g1 (V ∇ Z1 IZ2 − V ∇ Z2 IZ1 , φ I Q I U1 ) for Z1 , Z2 ∈ Γ( D I ) and U1 ∈ Γ( D1I ⊕ D2I ); (c) g1 (T Z2 JZ1 − T Z1 JZ2 , ω J Q J U1 + JS J U1 ) = g1 (V ∇ Z1 JZ2 − V ∇ Z2 JZ1 , φ J Q J U1 ) J J for Z1 , Z2 ∈ Γ( D J ) and U1 ∈ Γ( D1 ⊕ D2 ); (d) g1 (T Z2 KZ1 − T Z1 KZ2 , ωK QK U1 + KSK U1 ) = g1 (V ∇ Z1 KZ2 − V ∇ Z2 KZ1 , φK QK U1 ) for Z1 , Z2 ∈ Γ( D K ) and U1 ∈ Γ( D1K ⊕ D2K ). Proof. For Z1 , Z2 ∈ Γ( D R ), U1 ∈ Γ( D1R ⊕ D2R ), U2 ∈ (ker π∗ )⊥ and R ∈ { I, J, K }, since [ Z1 , Z2 ] ∈ (ker π∗ ), we have g1 ([ Z1 , Z2 ], U2 ) = 0. Thus, D R is integrable ⇔ g1 ([ Z1 , Z2 ], U1 ) = 0. Now, using Equations (4) and (12)–(14), we have g1 ([ Z1 , Z2 ], U1 ) = = = g1 ( R∇ Z1 Z2 , RU1 ) − g1 ( R∇ Z2 Z1 , RU1 ), g1 (∇ Z1 RZ2 , RU1 ) − g1 (∇ Z2 RZ1 , RU1 ), g1 (T Z1 RZ2 − T Z2 RZ1 , ω R Q R U1 + JRU1 ) − g1 (V ∇ Z1 RZ2 − V ∇ Z2 RZ1 , φR Q R U1 ). Since D R is R-invariant, we have ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Therefore, we obtain the result. Axioms 2022, 11, 641 7 of 15 Theorem 2. The following cases are equivalent for the map π defined in Theorem 1: ( a) D1R is integrable; g1 (TY1 ω I Y2 − TY2 ω I Y1 , φ I PI V1 ) (b) g1 (TY1 ω I φ I Y2 − TY2 ω I φ I Y1 , V1 ) = + g1 (H∇Y1 ω I Y2 − H∇Y2 ω I Y1 , ω I S I V1 ) for all Y1 , Y2 ∈ Γ( D1I ) and V1 ∈ Γ( D I ⊕ D2I ); g1 (TY1 ω J Y2 − TY2 ω J Y1 , φ J PJ V1 ) (c) g1 (TY1 ω J φ J Y2 − TY2 ω J φ J Y1 , V1 ) = J + g1 (H∇Y1 ω J Y2 − H∇Y2 ω J Y1 , ω J S J V1 ) J for all Y1 , Y2 ∈ Γ( D1 ) and V1 ∈ Γ( D J ⊕ D2 ); g1 (TY1 ωK Y2 − TY2 ωK Y1 , φK PK V1 ) (d) g1 (TY1 ωK φK Y2 − TY2 ωK φK Y1 , V1 ) = + g1 (H∇Y1 ωK Y2 − H∇Y2 ωK Y1 , ωK SK V1 ) for all Y1 , Y2 ∈ Γ( D1K ) and V1 ∈ Γ( D K ⊕ D2K ). Proof. For Y1 , Y2 ∈ Γ( D1R ), V1 ∈ Γ( D R ⊕ D2R ), V2 ∈ (ker F∗ )⊥ and R ∈ { I, J, K }, since [Y1 , Y2 ] ∈ (ker π∗ ), we have g1 ([Y1 , Y2 ], V2 ) = 0. Thus, D1R is integrable ⇔ g1 ([Y1 , Y2 ], V1 ) = 0. Using Equations (4), (5), (12) and (13) as well as Lemma 5, we have g1 ([Y1 , Y2 ], V1 ) = = g1 (∇Y1 RY2 , RV1 ) − g1 (∇Y2 RY1 , RV1 ), g1 (∇Y1 φR Y2 , RV1 ) + g1 (∇Y1 ω R Y2 , RV1 ) − g1 (∇Y2 φR Y1 , RV1 ) − g1 (∇Y2 ω R Y1 , RV1 ), = cos2 θ R g1 (∇Y1 Y2 , V1 ) − cos2 θ R g1 (∇Y2 Y1 , V1 ) − g1 (TY1 ω R φR Y2 − TY2 ω R φR Y1 , V1 ) + g1 (H∇Y1 ω R Y2 + TY1 ω R Y2 , RPR V1 + ω R SR V1 ) − g1 (H∇Y2 ω R Y1 + TY2 ω R Y1 , RPR V1 + ω R SR V1 ), which gives sin2 θ1 g1 ([Y1 , Y2 ], V1 ) = g1 (TY1 ω R Y2 − TY2 ω R Y1 , RPR V1 ) + g1 (H∇Y1 ω R Y2 − H∇Y2 ω R Y1 , ω R SR V1 ) − g1 (TY1 ω R φR Y2 − TY2 ω R φR Y1 , V1 ). Since D1R is an R-slant distribution, therefore, we obtain ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Therefore, we find the result. Theorem 3. For the h-qhs Riemannian map π defined in Theorem 1, D2R 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 π∗ )⊥ defines a totally geodesic foliation on N1 ; (b) g1 (A Z1 Z2 , PI W1 + cos2 θ I Q I W1 ) = g1 (H∇ Z1 Z2 , ω I φ I PI W1 + ω I φ I Q I W1 ) − g1 (A Z1 B I Z2 + H∇ Z1 C I Z2 , ω I W1 ) Axioms 2022, 11, 641 8 of 15 for Z1 , Z2 ∈ Γ(ker π∗ )⊥ and W1 ∈ Γ(ker π∗ ); (c) g1 (A Z1 Z2 , PJ W1 + cos2 θ J Q J W1 ) = g1 (H∇ Z1 Z2 , ω J φ J PJ W1 + ω J φ J Q J W1 ) − g1 (A Z1 B J Z2 + H∇ Z1 C J Z2 , ω J W1 ) for Z1 , Z2 ∈ Γ(ker π∗ )⊥ and W1 ∈ Γ(ker π∗ ); (d) g1 (A Z1 Z2 , PK W1 + cos2 θK QK W1 ) = g1 (H∇ Z1 Z2 , ωK φK PK W1 + ωK φK QK W1 ) − g1 (A Z1 BK Z2 + H∇ Z1 CK Z2 , ωK W1 ) for Z1 , Z2 ∈ Γ(ker π∗ )⊥ and W1 ∈ Γ(ker π∗ ). Proof. For Z1 , Z2 ∈ Γ(ker π∗ )⊥ , W1 ∈ Γ(ker π∗ ) and R ∈ { I, J, K }, using Equations (6), (7) and (12)–(14) as well as Lemma 5, we have g1 (∇ Z1 Z2 , W1 ) = = g1 ( R∇ Z1 Z2 , RW1 ), g1 ( R∇ Z1 Z2 , φR PR W1 + φR Q R W1 + ω R Q R W1 + ω R SR W1 ), = − g1 (∇ Z1 Z2 , φR2 PR W1 + ω R φR PR W1 + ω R φR Q R W1 ) + g1 (∇ Z1 BR Z2 , ω R Q R W1 + ω R SR W1 ) + g1 (∇ Z1 CR Z2 , ω R Q R W1 + ω R SR W1 ), = g1 (A Z1 Z2 , PR W1 + cos2 θ R Q R W1 ) − g1 (H∇ Z1 Z2 , ω R φR PR W1 + ω R φR Q R W1 ) + g1 (A Z1 BR Z2 , ω R Q R W1 + ω R SR W1 ) + g1 (H∇ Z1 CR Z2 , ω R Q R W1 + ω R SR W1 ). Thus, we obtain ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Therefore, the result follows. Theorem 5. The following conditions are equivalent for the h-qhs Riemannian map π: ( a) (ker π∗ ) defines a totally geodesic foliation on N1 ; (b) g1 (T X1 PI X2 + cos2 θ I T X1 Q I X2 , Y1 ) = g1 (H∇ X1 ω I φ I PI X2 + H∇ X1 ω I φ I Q I X2 , Y1 ) − g1 (H∇ X1 ω I Q I X2 + H∇ X1 ω I S I X2 , C I Y1 ) − g1 (T X1 ω I Q I X2 + T X1 ω I S I X2 , B I Y1 ) for X1 , X2 ∈ Γ(ker π∗ ) and Y1 ∈ Γ(ker π∗ )⊥ ; (c) g1 (T X1 PJ X2 + cos2 θ J T X1 Q J X2 , Y1 ) = for X1 , X2 ∈ Γ(ker π∗ ) and Y1 ∈ Γ(ker π∗ )⊥ ; (d) g1 (T X1 PK X2 + cos2 θK T X1 QK X2 , Y1 ) = g1 (H∇ X1 ω J φ J PJ X2 + H∇ X1 ω J φ J Q J X2 , Y1 ) − g1 (H∇ X1 ω J Q J X2 + H∇ X1 ω J S J X2 , C J Y1 ) − g1 (T X1 ω J Q J X2 + T X1 ω J S J X2 , B J Y1 ) g1 (H∇ X1 ωK φK PK X2 + H∇ X1 ωK φK QK X2 , Y1 ) − g1 (H∇ X1 ωK QK X2 + H∇ X1 ωK SK X2 , CK Y1 ) − g1 (T X1 ωK QK X2 + T X1 ωK SK X2 , BK Y1 ) for X1 , X2 ∈ Γ(ker π∗ ) and Y1 ∈ Γ(ker π∗ )⊥ . Axioms 2022, 11, 641 9 of 15 Proof. For X1 , X2 ∈ Γ(ker π∗ ), Y1 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }, using Equations (4), (5) and (12)–(14) as well as Lemma 5, we have g1 (∇ X1 X2 , Y1 ) = = g1 ( R∇ X1 X2 , RY1 ), g1 (∇ X1 φR PR X2 , RY1 ) + g1 (∇ X1 φR Q R X2 , RY1 ) + g1 (∇ X1 ω R Q R X2 , RY1 ) + g1 (∇ X1 ω R SR X2 , RY1 ), = g1 (T X1 PR X2 , Y1 ) + cos2 θ R g1 (T X1 Q R X2 , Y1 ) − g1 (H∇ X1 ω R φR PR X2 , Y1 ) − g1 (H∇ X1 ω R φR Q R X2 , Y1 ) + g1 (H∇ X1 ω R Q R X2 + H∇ X1 ω R SR X2 , CR Y1 ) + g1 (T X1 ω R Q R X2 + T X1 ω R SR X2 , BR Y1 ). Thus, we obtain ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Therefore, the result follows. Theorem 6. Let π be an h-qhs Riemannian map from an almost hyperkahler manifold ( N1 , I, J, K, g1 ) to a Riemannian manifold ( N2 , g2 ). Then, any one of the following assertions implies the others: ( a) D R defines a totally geodesic foliation on N1 ; (b) g1 (T Z1 IPI Z2 , ω I Q I Y1 + ω I S I Y1 ) = − g1 (V ∇ Z1 IPI Z2 , φ I Y1 ), g1 (T Z1 IPI Z2 , C I Y2 ) = − g1 (V ∇ Z1 IPI Z2 , B I Y2 ) for Z1 , Z2 ∈ Γ( D I ), Y1 ∈ Γ( D1I ⊕ D2I ) and Y2 ∈ Γ(ker π∗ )⊥ ; (c) g1 (T Z1 JPJ Z2 , ω J Q J Y1 + ω J S J Y1 ) = − g1 (V ∇ Z1 JPJ Z2 , φ J Y1 ), g1 (T Z1 JPJ Z2 , C J Y2 ) = − g1 (V ∇ Z1 JPJ Z2 , B J Y2 ) J J for Z1 , Z2 ∈ Γ( D J ), Y1 ∈ Γ( D1 ⊕ D2 ) and Y2 ∈ Γ(ker π∗ )⊥ ; (d) g1 (T Z1 KPK Z2 , ωK QK Y1 + ωK SK Y1 ) = − g1 (V ∇ Z1 KPK Z2 , φK Y1 ), g1 (T Z1 KPK Z2 , CK Y2 ) = − g1 (V ∇ Z1 KPK Z2 , BK Y2 ) for Z1 , Z2 ∈ Γ( D K ), Y1 ∈ Γ( D1K ⊕ D2K ) and Y2 ∈ Γ(ker π∗ )⊥ . Proof. For Z1 , Z2 ∈ Γ( D R ), Y1 ∈ Γ( D1R ⊕ D2R ), Y2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }, using Equations (4), (12) and (13), we have g1 (∇ Z1 Z2 , Y1 ) = = = g1 (∇ Z1 RZ2 , RY1 ), g1 (∇ Z1 RPR Z2 , RQ R Y1 + RSR Y1 ), g1 (T Z1 φR PR Z2 , ω R Q R Y1 + ω R SR Y1 ) + g1 (V ∇ Z1 φR PR Z2 , φR Q R Y1 ). Moreover, using Equations (4), (12) and (14), we obtain g1 (∇ Z1 Z2 , Y2 ) = = = g1 (∇ Z1 RZ2 , RY2 ), g1 (∇ Z1 RPR Z2 , BR Y2 + CR Y2 ), g1 (V ∇ Z1 RPR Z2 , BR Y2 ) + g1 (T Z1 JPR Z2 , CR Y2 ). Hence, we have ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Axioms 2022, 11, 641 10 of 15 Therefore, the result follows. Theorem 7. With π : ( N1 , I, J, K, g1 ) → ( N2 , g2 ) being an h-qhs Riemannian map, the following conditions are equivalent: ( a) D1R defines a totally geodesic foliation on N1 ; (b) g1 (TY1 ω I φ I Y2 , Z1 ) = g1 (H∇Y1 ω I φ I Y2 , Z2 ) = g1 (TY1 ω I Y2 , φ I PI Z1 ) + g1 (H∇Y1 ω I Y2 , ω I S I Z1 ), g1 (H∇Y1 ω I Y2 , C I Z2 ) + g1 (TY1 ω I Y2 , B I Z2 ) for Y1 , Y2 ∈ Γ( D1I ), Z1 ∈ Γ( D I ⊕ D2I ) and Z2 ∈ Γ(ker π∗ )⊥ ; (c) g1 (TY1 ω J φ J Y2 , Z1 ) = g1 (H∇Y1 ω J φ J Y2 , Z2 ) = g1 (TY1 ω J Y2 , φ J PJ Z1 ) + g1 (H∇Y1 ω J Y2 , ω J S J Z1 ), g1 (H∇Y1 ω J Y2 , C J Z2 ) + g1 (TY1 ω J Y2 , B J Z2 ) J J for Y1 , Y2 ∈ Γ( D1 ), Z1 ∈ Γ( D J ⊕ D2 ) and Z2 ∈ Γ(ker π∗ )⊥ ; (d) g1 (TY1 ωK φK Y2 , Z1 ) = g1 (H∇Y1 ωK φK Y2 , Z2 ) = g1 (TY1 ωK Y2 , φK PK Z1 ) + g1 (H∇Y1 ωK Y2 , ωK SK Z1 ), g1 (H∇Y1 ωK Y2 , CK Z2 ) + g1 (TY1 ωK Y2 , BK Z2 ) for Y1 , Y2 ∈ Γ( D1K ), Z1 ∈ Γ( D K ⊕ D2K ) and Z2 ∈ Γ(ker π∗ )⊥ . Proof. For Y1 , Y2 ∈ Γ( D1R ), Z1 ∈ Γ( D R ⊕ D2R ), Z2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }, using Equations (5), (12) and (13) as well as Lemma 5, we have g1 (∇Y1 Y2 , Z1 ) = = g1 (∇Y1 RY2 , RZ1 ), g1 (∇Y1 φR Y2 , RZ1 ) + g1 (∇Y1 ω R Y2 , RZ1 ), = cos2 θ R g1 (∇Y1 Y2 , Z1 ) − g1 (TY1 ω R φR Y2 , Z1 ) + g1 (TY1 ω R Y2 , φR PR Z1 ) + g1 (H∇Y1 ω R Y2 , ω R SR Z1 ), which gives sin2 θ R g1 (∇Y1 Y2 , Z1 ) = − g1 (TY1 ω R φR Y2 , Z1 ) + g1 (TY1 ω R Y2 , RPR Z1 ) + g1 (H∇Y1 ω R Y2 , ω R SR Z1 ). Moreover, from Equations (5), (13) and (14) as well as Lemma 5, we have g1 (∇Y1 Y2 , Z2 ) = = g1 (∇Y1 RY2 , RZ2 ), g1 (∇Y1 φR Y2 , RZ2 ) + g1 (∇Y1 ω R Y2 , RZ2 ), = cos2 θ R g1 (∇Y1 Y2 , Z2 ) − g1 (H∇Y1 ω R φR Y2 , Z2 ) + g1 (H∇Y1 ω R Y2 , CR Z2 ) + g1 (TY1 ω R Y2 , BR Z2 ). Thus, we find that sin2 θ R g1 (∇Y1 Y2 , Z2 ) = − g1 (H∇Y1 ω R φR Y2 , Z2 ) + g1 (H∇Y1 ω R Y2 , CR Z2 ) + g1 (TY1 ω R Y2 , BR Z2 ). Hence, we have ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Therefore, the result follows. Axioms 2022, 11, 641 11 of 15 Theorem 8. For the h-qhs Riemannian map π defined in Theorem 1, any one of the following assertions implies the others: ( a) D2R defines a totally geodesic foliation on N1 ; (b) g1 (H∇Y1 ω I Y2 , ω I Q I W1 ) = − g1 (TY1 ω I S I Y2 , φ I PI W1 + φ I Q I W1 ), g1 (H∇Y1 ω I S I Y2 , C I W2 ) = − g1 (TY1 ω I S I Y2 , B I W2 ) for Y1 , Y2 ∈ Γ( D2I ), W1 ∈ Γ( D I ⊕ D1I ) and W2 ∈ Γ(ker π∗ )⊥ ; (c) g1 (H∇Y1 ω J Y2 , ω J Q J W1 ) = − g1 (TY1 ω J S J Y2 , φ J PJ W1 + φ J Q J W1 ), g1 (H∇Y1 ω J SY2 , C J W2 ) = − g1 (TY1 ω J SY2 , B J W2 ) J J for Y1 , Y2 ∈ Γ( D2 ), W1 ∈ Γ( D J ⊕ D1 ) and W2 ∈ Γ(ker π∗ )⊥ ; (d) g1 (H∇Y1 ωK Y2 , ωK QK W1 ) = − g1 (TY1 ωK SY2 , φK PK W1 + φK QK W1 ), g1 (H∇Y1 ωK SY2 , CK W2 ) = − g1 (TY1 ωK SK Y2 , BK W2 ) for Y1 , Y2 ∈ Γ( D2K ), W1 ∈ Γ( D K ⊕ D1K ) and W2 ∈ Γ(ker π∗ )⊥ . Proof. For Y1 , Y2 ∈ Γ( D2R ), W1 ∈ Γ( D R ⊕ D1R ), W2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }, using Equations (5), (12) and (13), we have g1 (∇Y1 Y2 , W1 ) = = = g1 (∇Y1 RY2 , RW1 ) g1 (∇Y1 ω R SR Y2 , φR PR W1 + φR Q R W1 + ω R Q R W1 ), g1 (TY1 ω R SR Y2 , φR PR W1 + φR Q R W1 ) + g1 (H∇Y1 ω R SR Y2 , ω R Q R W1 ). Again, using Equations (5), (13) and (14), we have g1 (∇Y1 Y2 , W2 ) = = = g1 (∇Y1 RY2 , RW2 ) g1 (∇Y1 ω R SR Y2 , BR W2 + CR W2 ), g1 (TY1 ω R SR Y2 , BR W2 ) + g1 ( H∇Y1 ω R RY2 , CR W2 ). Hence, we have ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Therefore, the result follows. Theorem 9. Let π be an h-qhs Riemannian map from an almost hyperkahler manifold ( N1 , I, J, K, g1 ) to a Riemannian manifold ( N2 , g2 ). Then, the following conditions are equivalent: ( a) π is a totally geodesic map; (b) g1 (TY1 PI Y2 + cos2 θ I TY1 Q I Y2 − H∇Y1 ω I φ I PI Y2 − H∇Y1 ω I φ I Q I Y2 , W1 ) = g1 (TY1 ω I Q I Y2 + TY1 ω I S I Y2 , B I W1 ) + g1 (H∇Y1 ω I φ I Q I Y2 + H∇Y1 ω I φ I S I Y2 , W1 ), g1 (AW1 PI Y1 + cos2 θ I AW1 Q I Y1 − H∇W1 ω I φ I PI Y1 − H∇W1 ω I φ I Q I Y1 , W2 ) = g1 (AW1 ω I Q I Y1 + AW1 ω I S I Y1 , B I W2 ) + g1 (H∇W1 ω I Q I Y1 + H∇W1 ω I S I Y1 , C I W2 ) for Y1 , Y2 ∈ Γ(ker π∗ ) and W1 , W2 ∈ Γ(ker π∗ )⊥ ; (c) g1 (TY1 PJ Y2 + cos2 θ J TY1 Q J Y2 − H∇Y1 ω J φ J PJ Y2 − H∇Y1 ω J φ J Q J Y2 , W1 ) = g1 (TY1 ω J Q J Y2 + TY1 ω J S J Y2 , B J W1 ) + g1 (H∇Y1 ω J φ J Q J JY2 + H∇Y1 ω J φ J S J Y2 , W1 ), Axioms 2022, 11, 641 12 of 15 g1 (AW1 PJ Y1 + cos2 θ J AW1 Q J Y1 − H∇W1 ω J φ J PJ Y1 − H∇W1 ω J φ J Q J Y1 , W2 ) = g1 (AW1 ω J Q J Y1 + AW1 ω J S J Y1 , B J W2 ) + g1 (H∇W1 ω J Q J Y1 + H∇W1 ω J S J Y1 , C J W2 ) for Y1 , Y2 ∈ Γ(ker π∗ ) and W1 , W2 ∈ Γ(ker π∗ )⊥ ; (d) g1 (TY1 PK Y2 + cos2 θK TY1 QK Y2 − H∇Y1 ωK φK PK Y2 − H∇Y1 ωK φK QK Y2 , W1 ) = g1 (TY1 ωK QK Y2 + TY1 ωK SK Y2 , BK W1 ) + g1 (H∇Y1 ωK φK QK Y2 + H∇Y1 ωK φK SK Y2 , W1 ), g1 (AW1 PK Y1 + cos2 θK AW1 QK Y1 − H∇W1 ωK φK PK Y1 − H∇W1 ωK φK QK Y1 , W2 ) = g1 (AW1 ωK QK Y1 + AW1 ωK SK Y1 , BK W2 ) + g1 (H∇W1 ωK QK Y1 + H∇W1 ωK SK Y1 , CK W2 ) for Y1 , Y2 ∈ Γ(ker π∗ ) and W1 , W2 ∈ Γ(ker π∗ )⊥ . Proof. Since π is a Riemannian map, therefore, we have (∇π∗ )(W1 , W2 ) = 0, for W1 , W2 ∈ Γ(ker π∗ )⊥ . For Y1 , Y2 ∈ Γ(ker π∗ ), W1 , W2 ∈ Γ(ker π∗ )⊥ and R ∈ { I, J, K }, using Equations (4), (5) and (12)–(14) as well as Lemma 5, we have = = = = g2 ((∇π∗ )(Y1 , Y2 ), π∗ (W1 )) − g1 (∇Y1 Y2 , W1 ) − g1 (∇Y1 RY2 , RW1 ) − g1 (∇Y1 RPR Y2 , RW1 ) − g1 (∇Y1 RQ R Y2 , RW1 ) − g1 (∇Y1 RSR Y2 , RW1 ), − g1 (∇Y1 φR PR Y2 , RW1 ) − g1 (∇Y1 φR Q R Y2 , RW1 ) − g1 (∇Y1 ω R Q R Y2 , RW1 ) − g1 (∇Y1 ω R SR Y2 , RW1 ), = − g1 (TY1 PR Y2 + cos2 θ R TY1 Q R Y2 − H∇Y1 ω R φR PR Y2 − H∇Y1 ω R φR Q R Y2 , W1 ) − g1 (TY1 ω R Q R Y2 + TY1 ω R SR Y2 , BR W1 ) − g1 (H∇Y1 ω R φR Q R Y2 + H∇Y1 ω R φR SR Y2 , W1 ). Moreover, using Equations (4), (5) and (12)–(14) as well as Lemma 5, we have = = = = g2 ((∇π∗ )(W1 , Y1 ), π∗ (W2 )) − g1 (∇W1 Y1 , W2 ), − g1 (∇W1 RY1 , RW2 ), − g1 (∇W1 RPR Y1 , RW2 ) − g1 (∇W1 RQ R Y1 , RW2 ) − g1 (∇W1 RSR Y1 , RW2 ), − g1 (∇W1 φR PR Y1 , RW2 ) − g1 (∇W1 φR Q R Y1 , RW2 ) − g1 (∇W1 ω R Q R Y1 , RW2 ) − g1 (∇W1 ω R SR Y1 , RW2 ), = − g1 (AW1 PR Y1 + cos2 θ R AW1 Q R Y1 − H∇W1 ω R φR PR Y1 − H∇W1 ω R φR Q R Y1 , W2 ) − g1 (AW1 ω R Q R Y1 + AW1 ω R SR Y1 , BR W2 ) − g1 (H∇W1 ω R Q R Y1 + H∇W1 ω R SR Y1 , CR W2 ). Hence, we obtain ( a ) ⇔ ( b ), ( a ) ⇔ ( c ), ( a ) ⇔ ( d ). Thus, the theorem is proven. Axioms 2022, 11, 641 13 of 15 4. Example Note that given a Euclidean space R4n with coordinates ( x1 , x2 , ....., x4n ), we can canonically choose complex structures I, J and K on R4n as follows: I( ∂ ) = ∂x4s+1 ∂ I( ) ∂x4s+4 ∂ ) J( ∂x4s+3 ∂ ) K( ∂x4s+2 = = = ∂ , I( ∂ )=− ∂ , I( ∂ )= ∂ , ∂x4s+2 ∂x4s+2 ∂x4s+1 ∂x4s+3 ∂x4s+4 ∂ ∂ ∂ ∂ ∂ − , J( )= , J( )=− , ∂x4s+3 ∂x4s+1 ∂x4s+3 ∂x4s+2 ∂x4s+4 ∂ ∂ ∂ ∂ ∂ − , J( )= , K( )= , ∂x4s+1 ∂x4s+4 ∂x4s+2 ∂x4s+1 ∂x4s+4 ∂ ∂ ∂ ∂ ∂ , K( )=− , K( )=− , ∂x4s+3 ∂x4s+3 ∂x4s+2 ∂x4s+4 ∂x4s+1 for s ∈ {0, 1, 2, ...., ..., n − 1}. Then, we can easily check that ( I, J, K, h, i) is a hyperkähler structure on R4n , where h, i denotes the Euclidean metric on R4n . Throughout this section, we will use these notations. Example 1. Define a map π : R12 → R6 by π ( x1 , x2 , ........., x12 ) = (2020, x2 , x6 , x8 − x9 √ , x10 , 2022). 2 Then, the map π is an almost h-qhs Riemannian map such that   ∂ ∂ ∂ ∂ ∂ ∂ 1 ∂ ∂ ∂ ker π∗ = , , , , ,√ ( + ), , , ∂x1 ∂x3 ∂x4 ∂x5 ∂x7 ∂x9 ∂x11 ∂x12 2 ∂x8   ∂ ∂ ∂ 1 ∂ ∂ (ker π∗ )⊥ = , ,√ ( − ), , ∂x2 ∂x6 ∂x9 ∂x10 2 ∂x8       ∂ ∂ ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ I I I , , , ,√ ( + ) , D2 = , , , D1 = D = ∂x3 ∂x4 ∂x11 ∂x12 ∂x7 ∂x9 ∂x1 ∂x5 2 ∂x8       1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ J J J , , , + ), , , D1 = √ ( , D2 = , D = ∂x1 ∂x3 ∂x5 ∂x7 ∂x9 ∂x11 ∂x4 ∂x12 2 ∂x8       ∂ ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ ∂ K K K D = , , D1 = ,√ ( + ), , D2 = , , , ∂x1 ∂x4 ∂x5 ∂x9 ∂x12 ∂x3 ∂x7 ∂x11 2 ∂x8 with the almost h-qhs angles {θ I = θ J = θK = π 4 }. Example 2. Define a map π : R16 → R8 by π ( x1 , x2 , ........., x16 ) = (101, √ 3x5 − x9 , x6 , x8 , x11 , x14 , 202, x15 ). 2 Then, the map π is an almost h-qhs Riemannian map such that   √ ∂ ∂ ∂ ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ ∂ ker π∗ = , , , , ( + 3 ), , , , , , ∂x1 ∂x2 ∂x3 ∂x4 2 ∂x5 ∂x9 ∂x7 ∂x10 ∂x12 ∂x13 ∂x16   1 √ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (ker π∗ )⊥ = − ), , , , , , , ( 3 2 ∂x5 ∂x9 ∂x6 ∂x8 ∂x11 ∂x14 ∂x15      √ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 1 ∂ ∂ , , , , , ( + 3 ), , , D1I = , D2I = , ∂x1 ∂x2 ∂x3 ∂x4 2 ∂x5 ∂x9 ∂x10 ∂x7 ∂x12 ∂x13 ∂x16       √ ∂ ∂ ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ J J , D1 = , , , + 3 ), , , , ( , D2 = , DJ = ∂x1 ∂x2 ∂x3 ∂x4 ∂x10 ∂x12 2 ∂x5 ∂x9 ∂x7 ∂x13 ∂x16 DI =  Axioms 2022, 11, 641 14 of 15 DK =       √ ∂ ∂ 1 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , D1K = , D2K = , , , , , , ( + 3 ), , ∂x1 ∂x2 ∂x3 ∂x4 ∂x13 ∂x16 2 ∂x5 ∂x9 ∂x12 ∂x7 ∂x10 with the almost h-qhs angles {θ I = π 6, θJ = π 3, θK = π 6 }. Author Contributions: Conceptualization, M.B., R.P., A.H. and S.K. (Sushil Kumar); methodology, M.B., R.P., A.H. and S.K. (Sumeet Kumar); investigation, R.P., A.H., S.K. (Sushil Kumar) and S.K. (Sumeet Kumar); writing original draft preparation, M.B., A.H., S.K. (Sushil Kumar) and S.K. (Sumeet Kumar); writing—review and editing, M.B., R.P., S.K. (Sushil Kumar) and S.K. (Sumeet Kumar). All authors have read and agreed to the published version of the manuscript. Funding: The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work under Grant Code 22UQU4330007DSR04. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are thankful to the editor and anonymous referees for the constructive comments to improve the quality of this paper. Conflicts of Interest: The authors declare no conflict of interest. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. Fischer, A.E. 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