Hemi-slant submersions from almost product Riemannian manifolds

Gulf Journal of Mathematics Vol 4, Issue 3 (2016) 15-27 HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS MEHMET AKIF AKYOL1∗ AND YILMAZ GÜNDÜZALP2 Abstract. In this paper, we introduce hemi-slant submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give an example, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion. We also find necessary and sufficient conditions for a hemi-slant submersion to be totally geodesic. 1. Introduction The theory of smooth maps between Riemannian manifolds has been widely studied in Riemannian geometry. Such maps are useful for comparing geometric structures between two manifolds. In this point of view, the study of Riemannian submersions between Riemannian manifolds was initiated by O’Neill [21] and Gray [12], see also [8] and [34]. Riemannian submersions have several applications in mathematical physics. Indeed, Riemannian submersions have their applications in the Yang-Mills theory ([6], [33]), Kaluza-Klein theory ([7], [16]), supergravity and superstring theories ([17], [20]), etc. Later such submersions were considered between manifolds with differentiable structures, see [8]. Furthermore, we have the following submersions: semi-Riemannian submersion and Lorentzian submersion [8], Riemannian submersion [12], almost Hermitian submersion [32], contact-complex submersion [19], quaternionic submersion [18], etc. Recently, B. Şahin [26] introduced the notion of anti-invariant Riemannian submersions which are Riemannian submersions from almost Hermitian manifolds such that the vertical distributions are anti-invariant under the almost complex structure of the total manifold. Besides there are many other notions related with that of anti-invariant Riemannian submersions. see:([1], [2], [4], [5], [9], [10], [11], [13], [14], [15], [22], [23], [24], [25], [28], [29], [30]). In particular, B. Şahin [27] introduced the notion of semi-invariant Riemannian submersions and slant submersions when the base manifold is an almost Hermitian manifold. He showed that such submersions have rich geometric properties and they are useful for investigating the geometry of the total space. On the other hand, as a generalization of semi-invariant submersions and slant submersions, Taştan, Şahin and Date: Received: Mar 10, 2016; Accepted: May 2, 2016. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 53C15; Secondary 53C40. Key words and phrases. Hemi-slant submersion, Almost product Riemannian manifold, Riemannian submersion. 15 16 MEHMET AKIF AKYOL, YILMAZ GÜNDÜZALP Yanan [31] introduced the notion of hemi-slant submersions. They showed that such submersions have rich geometric properties and they are useful for investigating the geometry of the total space. The present work, we define and study the notion of hemi-slant submersions from almost product Riemannian manifolds. The paper is organized as follows. In section 2, we recall some notions needed for this paper. In section 3, we define hemi-slant submersions from an almost product Riemannian manifold onto a Riemannian manifold. We also investigate the geometry of leaves of the distributions. Finally we give necessary and sufficient conditions for such submersions to be totally geodesic. 2. Preliminaries In this section, we define almost product Riemannian manifolds, recall the notion of Riemannian submersions between Riemannian manifolds and give a brief review of basic facts of Riemannian submersions. Let M be a m-dimensional manifold with a tensor F of a type (1,1) such that F 2 = I, (F 6= I). (2.1) Then, we say that M is an almost product manifold with almost product structure F. We put 1 1 P = (I + F ), Q = (I − F ). 2 2 Then we get P + Q = I, P 2 = P, Q2 = Q, P Q = QP = 0, F = P − Q. Thus P and Q define two complementary distributions P and Q. We easily see that the eigenvalues of F are +1 or −1. If an almost product manifold M admits a Riemannian metric g such that g(F X, F Y ) = g(X, Y ) (2.2) for any vector fields X and Y on M, then M is called an almost product Riemannian manifold, denoted by (M, g, F ). Denote the Levi-Civita connection on M with M respect to g by ∇ . Then, M is called a locally product Riemannian manifold M [34] if F is parallel with respect to ∇ , i.e., M ∇X F = 0, X ∈ Γ(T M ). (2.3) Let (M, g) and (N, g ′ ) be two Riemannian manifolds. A surjective C ∞ −map π : M → N is a C ∞ −submersion if it has maximal rank at any point of M. Putting Vx = kerπ∗x , for any x ∈ M, we obtain an integrable distribution V, which is called vertical distribution and corresponds to the foliation of M determined by the fibres of π. The complementary distribution H of V, determined by the Riemannian metric g, is called horizontal distribution. A C ∞ −submersion π : M → N between two Riemannian manifolds (M, g) and (N, g ′ ) is called a Riemannian submersion if, at each point x of M, π∗x preserves the length of the horizontal vectors. A horizontal vector field X on M is said to be basic if X is π−related to a vector field X ′ on N. It is clear that every vector field X ′ on N HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS 17 has a unique horizontal lift X to M and X is basic. We recall that the sections of V, respectively H, are called the vertical vector fields, respectively horizontal vector fields. A Riemannian submersion π : M → N determines two (1, 2) tensor fields T and A on M, by the formulas: M M T (E, F ) = TE F = H∇VE VF + V∇VE HF (2.4) and M M A(E, F ) = AE F = V∇HE HF + H∇HE HF (2.5) ˆV W; ∇V W = T V W + ∇ (2.6) for any E, F ∈ Γ(T M ), where V and H are the vertical and horizontal projections (see [8]). From (2.4) and (2.5), one can obtain M M M ∇V X = TV X + H(∇V X); M M ∇X V = V(∇X V ) + AX V ; M ⊥ M ∇X Y = AX Y + H(∇X Y ), (2.7) (2.8) (2.9) for any X, Y ∈ Γ((kerπ∗ ) ) and V, W ∈ Γ(kerπ∗ ). Moreover, if X is basic then M M H(∇V X) = V(∇X V ) = AX V. (2.10) We note that for U, V ∈ Γ(kerπ∗ ), TU V coincides with the second fundamental form of the immersion of the fibre submanifolds and for X, Y ∈ Γ((kerπ∗ )⊥ ), AX Y = 12 V[X, Y ] reflecting the complete integrability of the horizontal distribution H. It is known that A is alternating on the horizontal distribution: AX Y = −AY X, for X, Y ∈ Γ((kerπ∗ )⊥ ) and T is symmetric on the vertical distribution: TU V = TV U, for U, V ∈ Γ(kerπ∗ ). We now recall the following result which will be useful for later. Lemma 2.1. (see [8],[21]). If π : M → N is a Riemannian submersion and X, Y basic vector fields on M, π−related to X ′ and Y ′ on N, then we have the following properties (1) H[X, Y ] is a basic vector field and π∗ H[X, Y ] = [X ′ , Y ′ ] ◦ π; M N M N (2) H(∇X Y ) is a basic vector field π−related to (∇X ′ Y ′ ), where ∇ and ∇ are the Levi-Civita connection on M and N ; (3) [E, U ] ∈ Γ(kerπ∗ ), for any U ∈ Γ(kerπ∗ ) and for any basic vector field E. Let (M, gM ) and (N, gN ) be Riemannian manifolds and π : M → N is a smooth map. Then the second fundamental form of π is given by (∇π∗ )(X, Y ) = ∇π∗ X π∗ Y − π∗ (∇X Y ) (2.11) for X, Y ∈ Γ(T M ), where we denote conveniently by ∇ the Levi-Civita connections of the metrics gM and gN . Recall that π is called a totally geodesic map if (∇π∗ )(X, Y ) = 0 for X, Y ∈ Γ(T M ) [3]. It is known that the second fundamental form is symmetric. 18 MEHMET AKIF AKYOL, YILMAZ GÜNDÜZALP 3. Hemi-slant submersions In this section, we define hemi-slant submersions from an almost product Riemannian manifold onto a Riemannian manifold, investigate the integrability of distributions and obtain a necessary and sufficient condition for such submersions to be totally geodesic map. Definition 3.1. Let (M, gM , F ) be an almost product Riemannian manifold and (N, gN ) a Riemannian manifold. A Riemannian submersion π : (M, gM , F ) → (N, gN ) is called a hemi-slant submersion if the vertical distribution kerπ∗ of π admits two orthogonal complementary distributions Dθ and D⊥ such that Dθ is slant and D⊥ is anti-invariant, i.e, we have kerπ∗ = Dθ ⊕ D⊥ . (3.1) In this case, the angle θ is called the hemi-slant angle of the submersion. Suppose the dimension of distribution of D⊥ (resp. Dθ ) is m1 (resp. m2 ). Then we easily see the following particular cases. (a) If m2 = 0, then M is an anti-invariant submersion [10]. (b) If m1 = 0 and θ = 0, then M is an invariant submersion [11]. (c) If m1 = 0 and θ 6= 0, π2 , then M is a proper slant submersion with slant angle θ [11]. (d) If θ = π2 , then M is an anti-invariant submersion. (e) If m1 6= 0 and θ = 0, then M is an semi-invariant submersion. We say that the hemi-slant submersion π : (M, gM , F ) → (N, gN ) is proper if ⊥ D 6= {0} and θ 6= 0, π2 . As we have seen from above argument, anti-invariant submersions, semi-invariant submersions and slant submersions are all examples of hemi-slant submersions. Now, we present an example of proper hemi-slant submersions in locally product Riemannian manifolds and demonstrate that the method presented in this paper is effective. Note that given an Euclidean space R8 with coordinates (x1 , ..., x8 ), we can canonically choose an almost product structure F on R8 as follows: F (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ) = (−a2 , −a1 , a4 , a3 , −a6 , −a5 , a8 , a7 ), where a1 , ..., a8 ∈ R. Example 3.2. Let π be a submersion defined by π: R8 −→ R4 8 7 x6√ , +x ). (x1 , ..., x8 ) (x1 cos α − x4 sin α, x2 sin β − x3 cos β, x5√+x 2 2 Then it follows that kerπ∗ = span{V1 = sin α∂x1 + cos α∂x4 , V2 = cos β∂x2 + sin β∂x3 , and V3 = −∂x5 + ∂x7 , V4 = −∂x6 + ∂x8 } (kerπ∗ )⊥ = span{H1 = cos α∂x1 − sin α∂x4 , H2 = sin β∂x2 − cos β∂x3 , 1 1 H3 = √ (∂x5 + ∂x7 ), H4 = √ (∂x6 + ∂x8 )}. 2 2 HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS 19 Thus it follows that Dθ = span{V1 , V2 } with the slant angle cos θ = sin(β − α) and D⊥ = span{V3 , V4 }. Also by direct computations, we get g(H1 , H1 ) = g ′ (π∗ H1 , π∗ H1 ) and g(H2 , H2 ) = g ′ (π∗ H2 , π∗ H2 ), g(H3 , H3 ) = g ′ (π∗ H3 , π∗ H3 ) and g(H4 , H4 ) = g ′ (π∗ H4 , π∗ H4 ) which show that π is a Riemannian submersion. Thus π is a proper hemi-slant submersion. Let π be a hemi-slant submersion from an almost product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then for V ∈ Γ(ker π∗ ), we put V = PV + QV, (3.2) F V = φV + ωV, (3.3) where PV ∈ Dθ and QV ∈ D⊥ and write where φV ∈ Γ(kerπ∗ ) and ωV ∈ Γ((ker π∗ )⊥ ). Also, for any X ∈ Γ((ker π∗ )⊥ ), we have F X = BX + CX, (3.4) ⊥ where BX ∈ Γ(kerπ∗ ) and CX ∈ Γ((kerπ∗ ) ). We denote the complementary distribution to ωDθ ⊕ F D⊥ in (ker π∗ )⊥ by µ. It is invariant distribution of (kerπ∗ )⊥ with respect to F. Then, the horizontal distribution (kerπ∗ )⊥ is decomposed as (ker π∗ )⊥ = ωDθ ⊕ F D⊥ ⊕ µ. (3.5) From (3.3), (3.4) and (3.5) we have φDθ = Dθ , φD⊥ = {0}, BωDθ = Dθ , BF D⊥ = D⊥ . On the other hand, using (3.3), (3.4) and the fact that F 2 = I, we obtain φ2 + Bω = I, C 2 + ωB = I, ωφ + Cω = 0, BC + φB = 0. Then by using (2.6), (2.7), (3.3) and (3.4) we get (∇U φ)V = BTU V − TU ωV M (3.6) (∇U ω)V = CTU V − TU φV M (3.7) for U, V ∈ Γ(ker π∗ ), where and ˆ U φV − φ∇ ˆ UV (∇U φ)V = ∇ M M ˆ U V. (∇U ω)V = AωV U − ω ∇ The proof of the following theorem is exactly the same as that one for hemi-slant submersions, see Theorem 3.4 of [22]. So, we omit it. Theorem 3.3. Let π be a Riemannian submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then π is a hemi-slant submersion if and only if there exists a constant [0, 1] and a distribution D on kerπ∗ such that (a) D = {V ∈ kerπ∗ | φ2 V = λV }, (b) for any W ∈ ker π∗ orthogonal to D, we have φW = 0. 20 MEHMET AKIF AKYOL, YILMAZ GÜNDÜZALP Moreover, in this case λ = cos2 θ, where θ is the slant angle of π. Thus, from Theorem 3.3, for any Z ∈ Dθ , we conlude that φ2 Z = cosθ Z. (3.8) On the other hand, for any Z, W ∈ Dθ , using (2.1), (3.3) and (3.8), we get gM (φZ, φW ) = cos2 θgM (Z, W ). (3.9) Also, using (2.1), (3.3) and (3.8), we find gM (ωZ, ωW ) = sin2 θgM (Z, W ). (3.10) Next, we easily have the following lemma: Lemma 3.4. Let (M, gM , F ) be a locally product Riemannian manifold and (N, gN ) a Riemannian manifold. Let π : (M, gM , F ) → (N, gN ) be a hemi-slant submersion. Then we have (a) M M AX BY + H∇X CY = CH∇X Y + ωAX Y M M V∇X BY + AX CY = BH∇X Y + φAX Y, (b) ˆ UV TU φV + AωV U = CTU V + ω ∇ ˆ U φV + TU ωV = BTU V + φ∇ ˆ U V, ∇ (c) M M AX φU + H∇U ωV = CAX U + ωV∇X U M M V∇X φU + AX ωU = BAX U + φV∇X U, for X, Y ∈ Γ((ker π∗ )⊥ ) and U, V ∈ Γ(ker π∗ ). We now examine the integrability conditions for the anti-invariant distribution D⊥ and the slant distribution Dθ . Theorem 3.5. Let π be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then the anti-invariant distribution D⊥ is integrable if and only if we have gN ((∇π∗ )(U, F V ) − (∇π∗ )(V, F U ), CZ) = gM (TU F V − TV F U, BZ) for U, V ∈ Γ(D⊥ ) and Z ∈ Γ(Dθ ). Proof. For U, V ∈ Γ(D⊥ ) and Z ∈ Γ(Dθ ), using (2.1), (2.2) and (3.4), we get M M M gM ([U, V ], Z) = gM (∇U F V, φZ) + gM (∇U F V, ωZ) − gM (∇V F U, φZ) M − gM (∇V F U, ωZ). HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS 21 Since π is a hemi-slant submersion, from (2.7) and (2.11), we get M gM ([U, V ], Z) = gM (TU F V, φZ) + gN (π∗ (∇U F V ), π∗ ωZ) − gM (TV F U, φZ) M − gN (π∗ (∇V F U ), ωZ) = gM (TU F V − TV F U, φZ) + gN ((∇π∗ )(V, F U ) − (∇π∗ )(U, F V ), ωZ) which proves assertion.  Theorem 3.6. Let π be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then the slant distribution Dθ is integrable if and only if we have gM (TW ωφZ − TZ ωφW , U ) = gN ((∇π∗ )(Z, ωW ) − (∇π∗ )(W, ωZ), F U ) for Z, W ∈ Γ(Dθ ) and U ∈ Γ(D⊥ ). Proof. For Z, W ∈ Γ(Dθ ) and U ∈ Γ(D⊥ ), using (2.1), (2.2) and (3.4), we get M M M gM ([Z, W ], U ) = gM (∇Z ωW, F U ) + gM (∇Z F φW, U ) − gM (∇W ωZ, F Z) M − gM (∇W F φZ, U ). If we take into account that π is a hemi-slant submersion, then from (2.7), (2.11) and (3.9), we get M M M gM ([Z, W ], U ) = gN (π∗ (∇Z ωW ), π∗ F U ) + cos2 θgM (∇Z W, U ) + gM (∇Z ωφW , U ) M M M − gN (π∗ (∇W ωZ), π∗ F U ) − cos2 θgM (∇W Z, U ) − gM (∇W ωφZ, U ) = gN ((∇π∗ )(W, ωZ) − (∇π∗ )(Z, ωW ), F U ) + cos2 θgM ([Z, W ], U ) + gM (TZ ωφW − TW ωφZ, U ) or sin2 θgM ([Z, W ], U ) = gN ((∇π∗ )(W, ωZ) − (∇π∗ )(Z, ωW ), F U ) + gM (TZ ωφW − TW ωφZ, U ) which proves assertion.  Now, we investigate the geometry of the leaves of the distributions D⊥ and Dθ . Theorem 3.7. Let π be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then D⊥ defines a totally geodesic foliation on M if and only if gN ((∇π∗ )(U, F V ), π∗ ωZ) = gM (TU V, ωφZ) and gN ((∇π∗ )(U, F V ), π∗ CX) = gM (TU F V, BZ) for U, V ∈ Γ(D⊥ ), Z ∈ Γ(Dθ ) and X ∈ Γ((kerπ∗ )⊥ ). Proof. From the definition of a hemi-slant submersion, it follows that the antiinvariant distribution D⊥ defines a totally geodesic foliation on M if and only 22 MEHMET AKIF AKYOL, YILMAZ GÜNDÜZALP M M if gM (∇U V, Z) = 0 and gM (∇U V, X) = 0 for U, V ∈ Γ(D⊥ ), Z ∈ Γ(Dθ ) and X ∈ Γ((kerπ∗ )⊥ ), from (2.1) and (2.2), we get M M M gM (∇U V, Z) = gM (∇U V, F φZ) + gM (∇U F V, ωZ). Since π is a hemi-slant submersion, using (2.6), (2.11) and (3.9) we get M M M M gM (∇U V, Z) = gM (∇U V, cos2 θZ) + gM (∇U V, ωφZ) + gN (π∗ (∇U F V ), π∗ ωZ) or M sin2 θgM (∇U V, Z) = gM (TU V, ωφZ) − gN ((∇π∗ )(U, F V ), π∗ ωZ). (3.11) On the other hand, by using (3.4) we have M M M gM (∇U V, X) = gM (∇U F V, BX) + gM (∇U F V, CX). If we take into account that π is a hemi-slant submersion, then by using (2.7) and (2.11) we get M gM (∇U V, X) = gM (TU F V, BZ) − gN ((∇π∗ )(U, F V ), π∗ CX). (3.12) Thus proof follows from (3.11) and (3.12).  For the leaves of Dθ we have the following result. Theorem 3.8. Let π be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then Dθ defines a totally geodesic foliation on M if and only if gN ((∇π∗ )(Z, ωW ), π∗ F U ) = gM (TZ ωφW, U ) and gN ((∇π∗ )(Z, ωφW ), π∗ X) + gN ((∇π∗ )(Z, ωW ), π∗ CX) = gM (TZ BX, ωW ) for Z, W ∈ Γ(Dθ ), U ∈ Γ(D⊥ ) and X ∈ Γ((kerπ∗ )⊥ ). Proof. The slant distribution Dθ defines a totally geodesic foliation on M if and M M only if gM (∇Z W, U ) = 0 and gM (∇Z W, X) = 0 for Z, W ∈ Γ(Dθ ), U ∈ Γ(D⊥ ) and X ∈ Γ((kerπ∗ )⊥ ), from (2.1) and (2.2), we get M M M gM (∇Z W, U ) = gM (∇Z F φW, U ) + gM (∇Z ωW, F U ). Since π is a hemi-slant submersion, using (2.6), (2.11) and (3.9), we obtain M M M M gM (∇Z W, U ) = gM (∇Z cos2 θW, U ) + gM (∇Z ωφW, U ) + gN (π∗ (∇Z ωW ), π∗ F U ) or M sin2 θgM (∇Z W, U ) = gM (TZ ωφW, U ) − gN ((∇π∗ )(Z, ωW ), π∗ F U ). (3.13) On the other hand, by using (3.3) and (3.4) we get M M M M gM (∇Z W, X) = gM (∇Z φW, F X) + gM (∇Z ωW, BX) + gM (H∇Z ωW, CX). HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS 23 Using (2.6), (2.7), (3.3), (3.9) and if we take into account that π is a hemi-slant submersion, we obtain M M M g(∇Z W, X) = cos2 θgM (∇Z W, X) + gN (π∗ (∇Z ωφW ), π∗ X) M + gM (TZ ωW, BX) + gN (π∗ (∇Z ωW ), CX) or M sin2 θg(∇Z W, X) = gM (TZ ωW, BX) + gN ((∇π∗ )(Z, ωφW ), π∗ X) (3.14) + gN ((∇π∗ )(Z, ωW ), π∗ CX). Thus proof follows from (3.13) and (3.14).  From Theorem 3.7 and Theorem 3.8, we have the following result. Theorem 3.9. Let π : (M, gM , F ) −→ (N, gN ) be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then the fibers of π are the locally product Riemannian manifold of leaves of D⊥ and Dθ if and only if gN ((∇π∗ )(U, F V ), π∗ ωZ) = gM (TU V, ωφZ) and gN ((∇π∗ )(Z, ωφW ), π∗ X) + gN ((∇π∗ )(Z, ωW ), π∗ CX) = gM (TZ BX, ωW ) for any U, V ∈ Γ(D⊥ ), Z, W ∈ Γ(Dθ ) and X ∈ Γ((kerπ∗ )⊥ ). For the geometry of leaves of the horizontal distribution ((kerπ∗ )⊥ ), we have the following theorem. Theorem 3.10. Let π : (M, gM , F ) −→ (N, gN ) be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then the distribution (kerπ∗ )⊥ defines a totally geodesic foliation on M if and only if M M AX1 BX2 + H∇X1 CX2 ∈ Γ(F D⊥ ⊕ µ), V∇X1 BX2 + AX1 CX2 ∈ Γ(D⊥ ) for any X1 , X2 ∈ Γ((kerπ∗ )⊥ ). Proof. Since M is a locally product Riemannian manifold, from (2.1) and (2.2) M M we have ∇X1 X2 = F ∇X1 F X2 for X1 , X2 ∈ Γ((kerπ∗ )⊥ ). Using (2.8), (2.9) and (3.4) we have M M ∇X1 X2 = F (AX1 BX2 + V∇X1 BX2 ) M + F (H∇X1 CX2 + AX1 CX2 ). Then by using (3.3) and (3.4) we get M M ∇X1 X2 = BAX1 BX2 + CAX1 BX2 + φV∇X1 BX2 M M M + ωV∇X1 BX2 + BH∇X1 CX2 + CH∇X1 CX2 + φAX1 CX2 + ωAX1 CX2 . 24 MEHMET AKIF AKYOL, YILMAZ GÜNDÜZALP M Hence, we have ∇X1 X2 ∈ Γ((kerπ∗ )⊥ ) if and only if M M M B(AX1 BX2 + H∇X1 CX2 ) + φ(V∇X1 BX2 + AX1 CX2 ) = 0. Thus ∇X1 X2 ∈ Γ((kerπ∗ )⊥ ) if and only if M M B(AX1 BX2 + H∇X1 CX2 ) = 0 and φ(V∇X1 BX2 + AX1 CX2 ) = 0, which completes proof.  In the sequel we are going to investigate the geometry of leaves of the vertical distribution kerπ∗ . Theorem 3.11. Let π : (M, gM , F ) −→ (N, gN ) be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then the distribution (kerπ∗ ) defines a totally geodesic foliation on M if and only if ˆ Z1 φZ2 + TZ1 ωZ2 ∈ Γ((kerπ∗ )⊥ ) TZ1 φZ2 + AωZ2 Z1 ∈ Γ(kerπ∗ ), ∇ for any Z1 , Z2 ∈ Γ(kerπ∗ ). Proof. For any Z1 , Z2 ∈ Γ(kerπ∗ ), using (2.2), (2.6), (2.7) and (3.3) we get M M ∇ Z 1 Z2 = F ∇ Z 1 F Z 2 M M = F (∇Z1 φZ2 + ∇Z1 ωZ2 ) ˆ Z1 φZ2 + AωZ2 Z1 + TZ1 ωZ2 ) = F (TZ1 φZ2 + ∇ ˆ Z1 φZ2 + ω ∇ ˆ Z1 φZ2 = BTZ1 φZ2 + CTZ1 φZ2 + φ∇ + BAωZ2 Z1 + CAωZ2 Z1 + φTZ1 ωZ2 + ωTZ1 ωZ2 . From above equation, it follows that (kerπ∗ ) defines a totally geodesic foliation if and only if ˆ Z1 φZ2 + TZ1 ωZ2 ) = 0. C(TZ1 φZ2 + AωZ2 Z1 ) + ω(∇ M Thus ∇Z1 Z2 ∈ Γ(kerπ∗ ) if and only if ˆ Z1 φZ2 + TZ1 ωZ2 ) = 0, C(TZ1 φZ2 + AωZ2 Z1 ) = 0 and ω(∇ which completes proof.  From Theorem 3.10 and Theorem 3.11, we have the following result. Theorem 3.12. Let π : (M, gM , F ) −→ (N, gN ) be a purely hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). Then the total space M is a locally product manifold of the leaves of (kerπ∗ )⊥ and kerπ∗ , i.e., M = M(kerπ∗ )⊥ × Mkerπ∗ , if and only if M and M AX1 BX2 + H∇X1 CX2 ∈ Γ(F D⊥ ⊕ µ), V∇X1 BX2 + AX1 CX2 ∈ Γ(D⊥ ) ˆ Z1 φZ2 + TZ1 ωZ2 ∈ Γ((kerπ∗ )⊥ ) TZ1 φZ2 + AωZ2 Z1 ∈ Γ(kerπ∗ ), ∇ for any X1 , X2 ∈ Γ((kerπ∗ )⊥ ) and Z1 , Z2 ∈ Γ(kerπ∗ ), where M(kerπ∗ )⊥ and Mkerπ∗ are leaves of the distributions (kerπ∗ )⊥ and kerπ∗ , respectively. HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS 25 Now, we give necessary and sufficient conditions for a hemi-slant submersion to be totally geodesic. The Riemannian submersion map π is called totally geodesic map if the map π∗ is parallel with respect to ∇, i.e., ∇π∗ = 0. A geometric interpretation of a totally geodesic map is that it maps every geodesic in the total space into a geodesic in the base space in proportion to arc lengths. Theorem 3.13. Let π : (M, gM , F ) −→ (N, gN ) be a hemi-slant submersion from a locally product Riemannian manifold (M, gM , F ) onto a Riemannian manifold (N, gN ). π is a totally geodesic map if and only if M ωTU1 ωV1 + CH∇U1 ωV1 = 0, ˆ V φW + TV ωW ) = 0, C(TU φW + AωW V ) + ω(∇ ˆ V BX + TV CX) = 0, C(TV BX + ACX V ) + ω(∇ for any U1 , V1 ∈ Γ(D1 ), W ∈ Γ(D2 ), U ∈ Γ(kerπ∗ ) and X ∈ Γ((kerπ∗ )⊥ ). Proof. For X1 , X2 ∈ Γ((kerπ∗ )⊥ ), since π is a Riemannian submersion, from (2.11) we obtain (∇π∗ )(X1 , X2 ) = 0. ⊥ For U1 , V1 ∈ Γ(D ), using (2.3) and (2.11) we have M (∇π∗ )(U1 , V1 ) = −π∗ (F ∇U1 ωV1 ). Then from (2.6) we arrive at M (∇π∗ )(U1 , V1 ) = −π∗ (F (TU1 ωV1 + H∇U1 ωV1 )). Using (3.3) and (3.4) in above equation we obtain M M (∇π∗ )(U1 , V1 ) = −π∗ (φTU1 ωV1 + ωTU1 ωV1 + BH∇U1 ωV1 + CH∇U1 ωV1 ). M Since φTU1 ωV1 + BH∇U1 ωV1 ∈ Γ(kerπ∗ ), we derive M (∇π∗ )(U1 , V1 ) = −π∗ (ωTU1 ωV1 + CH∇U1 ωV1 ). Then, since π is a linear isomorphism between (kerπ∗ )⊥ and T M , (∇π∗ )(U1 , V1 ) = 0 if and only if M (3.15) ωTU1 ωV1 + CH∇U1 ωV1 = 0. For U ∈ Γ(kerπ∗ ) and W ∈ Γ(Dθ ), using (2.3), (2.11) and (3.3), we have M (∇π∗ )(U, W ) = ∇πU π∗ W − π∗ (∇U W ) M = −π∗ (F ∇U F W ) M Then from (2.7) we arrive at = −π∗ (F ∇U (φW + ωW )). ˆ V φW ) + F (AωW V + TV ωW )). (∇π∗ )(U, W ) = −π∗ (F (TU φW + ∇ Using (3.3) and (3.4) in above equation we obtain ˆ V φW + ω ∇ ˆ V φW ) (∇π∗ )(U, W ) = −π∗ ((BTU φW + CTU φW ) + (φ∇ + (BAωW V + CAωW V ) + (φTV ωW + ωTV ωW )). 26 MEHMET AKIF AKYOL, YILMAZ GÜNDÜZALP Thus (∇π∗ )(V, W ) = 0 if and only if ˆ V φW + TV ωW ) = 0. C(TU φW + AωW V ) + ω(∇ (3.16) On the other hand, using (2.3), (2.6), (2.7) and (3.4) for any V ∈ Γ(kerπ∗ ) and X ∈ Γ((kerπ∗ )⊥ ), we get M (∇π∗ )(V, X) = ∇πV π∗ X − π∗ (∇V X) M = −π∗ (F ∇V F X) M = −π∗ (F ∇V (BX + CX)) ˆ V BX + ω ∇ ˆ V BX = −π∗ (BTV BX + CTV BX + φ∇ + BACX V + CACX V + φTV CX + ωTV CX). Thus (∇π∗ )(V, X) = 0 if and only if ˆ V BX + TV CX) = 0. C(TV BX + ACX V ) + ω(∇ The result then follows from (3.15), (3.16) and (3.17). (3.17)  Acknowledgement. The authors would like to express their warm thanks to the referee for his/her valuable comments and suggestions. References 1. S. Ali and T. Fatima, Generic Riemannian submersions, Tamkang Journal of Mathematics, 44 (2013), 395-409. 2. S. Ali and T. Fatima, Anti-invariant Riemannian submersions from nearly Kählerian manifolds, Filomat 27 (2013), 1219-1235. 3. P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press. Oxford, 2003. 4. A. Beri, İ. K. Erken and C. Murathan, Anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds, Turkish Journal of Mathematics, 40 (2016), 540-552. 5. M. Cengizhan and İ. K. Erken, Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions, Filomat. 29 (2015), 1429-1444. 6. J. P. Bourguignon and H. B. Lawson, Stability and isolation phenomena for Yang-mills elds, Commun. Math. Phys. 79 (1981), 189-230. 7. J. P. Bourguingnon and H. B. Lawson, A mathematician’s visit to Kaluza-Klein theory, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue (1989), 143-163. 8. M. Falcitelli, S. Ianus and A. M. Pastore, Riemannian submersions and Related Topics. World Scientific, River Edge, NJ, 2004. 9. Y. Gündüzalp, Anti-invariant semi-Riemannian submersions from almost para-Hermitian manifolds, Journal of Function Spaces and Applications, Volume 2013, Article ID 720623, 7 pages. 10. Y. Gündüzalp, Anti-invariant Riemannian submersions from almost product Riemannian manifolds, Mathematical Science and Applications E-notes. 1 (2013), 58-66. 11. Y. Gündüzalp, Slant submersions from almost product Riemannian manifolds, Turkish J. Math. 37 (2013) 863-873. 12. A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715-737. 13. J. W. Lee, Anti-invariant ξ ⊥ Riemannian submersions from almost contact manifolds, Hacettepe Journal of Mathematics and Statistic. 42(3), (2013), 231-241. HEMI-SLANT SUBMERSIONS FROM ALMOST PRODUCT RIEMANNIAN MANIFOLDS 27 14. J. W. Lee and B. Şahin, Pointwise slant submersions, Bull. Korean Math. Soc. 51(4), (2014), 1115-1126. 15. J. C. Lee, J. H. Park, B. Şahin and D. Y. Song, Einstein conditions for the base space of anti-invariant Riemannian submersions and Clairaut submersions, Taiwanese J. Math. 19 (2015), 1145-1160. 16. S. Ianus and M. Visinescu, Kaluza-Klein theory with scalar elds and generalized Hopf manifolds, Class. Quantum Gravity 4 (1987), 1317-1325. 17. S. Ianus and M. Visinescu, Space-time compactication and Riemannian submersions, In: Rassias, G.(ed.) The Mathematical Heritage of C. F. Gauss, (1991), 358-371, World Scientic, River Edge. 18. S. Ianus, R. Mazzocco and G. E. Vilcu, Riemannian submersions from quaternionic manifolds, Acta Appl. Math. 104 (2008), 83-89. 19. S. Ianus, M. Ionescu, R. Mazzocco and G. E. Vilcu, Riemannian submersions from almost contact metric manifolds, Abh. Math. Semin. Univ. Hamb. 81 (2011), 101-114. 20. M. T. Mustafa, Applications of harmonic morphisms to gravity, J. Math. Phys., 41 (2000), 6918-6929. 21. B. O’Neill, The fundamental equations of a submersion, Mich. Math. J., 13 (1966), 458-469. 22. K. S. Park and R. Prasad, Semi-slant submersions, Bull. Korean Math. Soc. 50 (2013), 951-962. 23. K. S. Park, h-semi-invariant submersions, Taiwanese J. Math. 16 (2012), 1865-1878. 24. K. S. Park, h-slant submersions, Bull. Korean Math. Soc. 49 (2012), 329-338. 25. S. A. Sepet and M. Ergut, Pointwise slant submersions from cosymplectic manifolds, Turkish Journal of Mat. 40 (2016), 582-593. 26. B. Şahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds, 8 central European J.Math, 3 (2010), 437-447. 27. B. Şahin, Semi-invariant Riemannian submersions from almost Hermitian manifolds, Canad. Math. Bull, 56 (2011), 173-183. 28. B. Şahin, Slant submersions from almost Hermitian manifolds, Bull. Math. Soc. Sci. Math. Roumanie, 1 (2011), 93-105. 29. B. Şahin, Riemannian submersions from almost Hermitian manifolds, Taiwanese J. Math., 17 (2012), 629-659. 30. H. M. Taştan, On Lagrangian submersions, Hacettepe J. Math. Stat. 43 (2014), 993-1000. 31. H. M. Tastan, B. Şahin and Ş. Yanan, Hemi-slant submersions, Mediterr. J. Math. DOI 10.1007/s00009-015-0602-7. 32. B. Watson, Almost Hermitian submersions, J. Differential Geometry, 11 (1976), 147-165. 33. B. Watson, G, G’-Riemannian submersions and nonlinear gauge eld equations of general relativity, In: Rassias, T. (ed.) Global Analysis - Analysis on manifolds, dedicated M. Morse. Teubner-Texte Math., 57 (1983), 324-349, Teubner, Leipzig. 34. K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore, 1984. 1 Faculty of Science and Arts, Department of Mathematics, University of Bingöl, 12000, Bingöl, TURKEY E-mail address: [email protected] 2 Faculty of Science and Arts, Department of Mathematics, University of Dicle, 21280, Diyarbakır, TURKEY E-mail address: [email protected]