Conformal anti-invariant Submersions from Sasakian manifolds

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 7 (2017), pp Research India Publications Conformal anti-invariant Submersions from Sasakian manifolds Sushil Kumar Department of Mathematics and Astronomy, University of Lucknow, Lucknow-India. Rajendra Prasad Department of Mathematics and Astronomy, University of Lucknow, Lucknow-India. Abstract In this paper we define conformal anti-invariant submersions from almost contact metric manifolds onto Riemannian manifolds. We obtain some results on conformal anti-invariant submersions from Sasakian manifolds onto Riemannian manifolds. We also give the necessary and sufficient conditions for a conformal anti-invariant submersions to be harmonic and totally geodesic. Moreover, we obtain decomposition theorems by using the existence of conformal anti-invariant submersions from Sasakian manifolds onto Riemannian manifolds. Finally, we give some examples of conformal anti-invariant submersions such that characteristic vector field ξ is vertical. AMS subject classification: 53A30, 53C25, 53C12, 53C22, 53A10. Keywords: Riemannian submersion, Conformal submersion, Anti-invariant submersion, conformal anti-invariant submersion. 1. Introduction The theory of smooth maps between Riemannian manifolds has been extensively studied in Riemannian geometry. Such maps are useful for comparing geometric structures between two manifolds.

2 3578 Sushil Kumar and Rajendra Prasad In 1966, O Neill [19] initiated the study of Riemannian submersion between Riemannian manifolds. It was found beneficial if one should study such submersions between manifolds with differentiable structures. When Watson was studying almost Hermitian submersions between almost Hermitian manifolds he found that the base manifold and each fibre have the same kind of structure as the total space, in most cases [25]. We note that almost Hermitian submersions have been extended to the almost contact metric submersions [7], locally conformal Kahler submersions [16] etc. We have so many submersions. Some of them are: semi-riemannian submersion and Lorentzian submersion [8], semi-invariant submersion [23], slant submersion ([6], [22]), contact-complex submersion [12], anti-invariant Riemannian submersions from Cosymplectic manifold [18] etc. As we know that Riemannian submersions are related with Physics and have their applications in the Yang-Mills theory [24], Kaluza-Klein theory ([5], [13]), supergravity and superstring theories ([14], [17]). In 2010, Sahin defined anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds [21] etc. As a generalization of Riemannian submersions, horizontally conformal submersion are introduced as follows [2]. Let (M, g M ) and (N, g N ) be two Riemannian manifolds of dimension m and n respectively. A smooth map f : (M, g M ) (N, g N ) is called a horizontally conformal submersion, if there is a positive function λ such that λ 2 g M (U, V ) = g N (f U,f V), (1.1) for every U,V (ker f ). It is evident that every Riemannian submersion is a particular horizontally conformal submersion with λ = 1. Let f is a smooth map between given Riemannian manifolds and x M. Then, f is called horizontally weakly conformal map at x if either (i) f x = 0or(ii)f x maps the horizontal space H = (ker f ) conformally onto T f(x) N, i.e., f x is surjective and f satisfies the equation (1.1) for U,V vectors tangent to H x. If a point x is of type (i), then it is called critical point of f.a point x of type (ii) is called regular. The number (x) is called the square dilation which is necessarily non-negative. Its square root λ(x) = (x) is known as the dilation. A horizontally weakly conformal map f to be horizontally homothetic if the gradient of their dilation λ is vertical, i.e., H(gradλ) = 0 at regular points. If a horizontally weakly conformal map f has no critical points, then f is called horizontally conformal submersion [2]. Thus, it follows that a Riemannian submersion is a horizontally conformal submersion with dilation identically one. The horizontal conformal maps were introduced independently by Fuglede in 1978 [9] and by Ishihara 1979 [15]. From the above argument, one can determine that the notion of horizontal conformal maps is a generalization of the concept of Riemannian submersions. Next, we memorize the following description in [10]. Let f : (M, g M ) (N, g N ) be a submersion. A vector field X on M is said to be projectable if there exists a vector field X on N, such that f (X x ) = X f(x) for each x M. In this situation X and X are called f related. We call a horizontal vector field Y on (M, g M ) a basic vector fields if it is projectable. We know that if Ŷ is a vector field on N, then there exists a unique basic vector field Y on M, such that Y and Ŷ are f related. The vector field Y is called the horizontal lift of Ŷ.

3 Conformal anti-invariant Submersions 3579 We denote the kernel space of f by ker f and consider the orthogonal complementary space H = (ker f ) to ker f. Then the tangent bundle of M has the following decomposition TM = (ker f ) (ker f ). (1.2) We also denote the range of f by rangef and consider the orthogonal complementary space (rangef ) to rangef in the tangent bundle TN of N.Thus the tangent bundle TN of N has the following decomposition TN = (rangef ) (rangef ). (1.3) We know that Riemannian submersions are very special maps comparing with conformal submersions. Although conformal maps do not preserve distance between points contrary to isometries, they preserve angles between vector fields. This property enables one to transfer certain properties of a manifold to another manifold by deforming such properties. The concept of Conformal anti-invariant submersions from almost Hermitian manifolds onto Riemannian manifold has been studied in [1]. In this paper, we study conformal anti-invariant submersions from Sasakian manifolds onto Riemannian manifolds. The paper is organized as follows. In section 2, we collect main notions and formulae which we need in this paper. In section 3, we introduce conformal anti-invariant submersions from almost contact metric manifolds onto Riemannian manifolds, investigate the geometry of leaves of the horizontal distribution and the vertical distribution. In section 4, we study the necessary and sufficient conditions for a conformal anti-invariant submersion to be harmonic and totally geodesic. In section 5, we prove that there are certain product structures on the total space of a conformal anti-invariant submersion from Sasakian manifold on Riemannian manifold such that ξ is vertical vector field. Finally in section 6, we give some examples of conformal anti-invariant submersion such that the characteristic vector field ξ is vertical. 2. Preliminaries An n dimensional Riemannian manifold M is said to be an almost contact metric manifold, if there exist on M, a (1, 1) tensor field φ, a vector field ξ, a1 form η and Riemannian metric g such that φ 2 = I + η ξ, φξ = 0, η φ = 0, (2.1) and g(x, ξ) = η(x), (2.2) η(ξ) = 1, (2.3) g(φx, φy) = g(x, Y) η(x)η(y), g(φx, Y) = g(x, φy), (2.4) for any vector fields X, Y on M.

4 3580 Sushil Kumar and Rajendra Prasad An almost contact metric manifold is said to be a contact metric manifold if dη =, where (X, Y) = g(x, φy) is called the fundamental 2 form of M. On the other hand the almost contact metric structure of M is said to be normal if [φ,φ](x, Y ) = 2dη(X, Y )ξ, for any X, Y, where [φ,φ] denotes the Nijenhuis tensor of φ given by [φ,φ] (X, Y ) = φ 2 [X, Y ]+[φx,φy] φ[φx,y] φ[x, φy ]. (2.5) A normal contact metric manifold is called a Sasakian manifold [4]. It can be proved that a Sasakian manifold is K contact, and that an almost contact metric manifold is Sasakian if and only if ( X φ)y = g(x, Y)ξ η(y)x, (2.6) for any vector fields X, Y on M. Moreover, for a Sasakian manifold the following equations satisfy: R(X, Y)ξ = η(y)x η(x)y, (2.7) R(ξ, X)Y = g(x, Y)ξ η(y)x, (2.8) X ξ = φx. (2.9) Definition 2.1. ([2]) Let (M, g M ) and (N, g N ) are two Riemannian manifold with Riemannian metrics g M and g N, respectively. If f : (M, g M ) (N, g N ) be a differentiable map between given Riemannian manifolds, then f is called horizontally weakly conformal or semi-conformal at q if either (i) df q = 0, or (ii) df q maps the horizontal space H q = (ker(df q )) conformally onto T f(q) N i.e., df q is surjective and there exists a number (q) = 0 such that where q M. g N (f U,f V)= (q)g M (U, V ) (U, V H q ), (2.10) Watson introduced the fundamental tensors of a submersion in [19]. It is well known that the fundamental tensor play parallel role to that of the second fundamental form of an immersion. More exactly, O Neill defined tensors A and T for vector fields E and F on M by A E F = V HE HF + H HE VF, (2.11) T E F = H VE VF + V VE HF, (2.12) where V and H are the vertical and horizontal projections [8], and is Riemannian connection on M. On the other hand, from equations (2.11) and (2.12), we have X Y = T X Y + X Y, (2.13)

5 Conformal anti-invariant Submersions 3581 X U = H X U + T X U, (2.14) U X = A U X + V U X, (2.15) U V = H U V + A U V, (2.16) for any X, Y Ɣ(ker f ) and U,V Ɣ(ker f ), where V X Y = X Y. If U is basic, then A X U = H X U. It is Simply seen that for q M,X V q and U H q the linear opretors are skew-symmetric, that is A U, T X : T q M T q M, g M (A U E,F) = g M (E, A U F)and g M (T X E,F) = g M (E, T X F), (2.17) for each E,F T q M.We have also defined the restriction of T to the vertical distribution T V V is precisely the second fundamental form of the fibres of f. Since T V is skewsymmetric we get: f has totally geodesic fibres if and only if T 0. For the special event when f is horizontally conformal we have the following proposition. Proposition 2.2. ([11] (2.1.2)) Let f be horizontal conformal submersion between Riemannian manifolds (M, g M ) and (N, g N ) with dilation λ and U,V be horizontal vector fields, then A U V = 1 ( )} 1λ {V[U,V] λ 2 g M (U, V )grad V 2 2, (2.18) We know that the skew-symmetric part of A H H measures the obstruction integrability of the horizontal distribution H. We also memorize the concept of harmonic maps between Riemannian manifolds. Let f : (M, g M ) (N, g N ) is a smooth map between Riemannian manifolds. Then the differential f of f can be observed a section of the bundle H om(t M, f 1 TN) M, where f 1 TN is the bundle which has fibres (f 1 TN) x = T f(x) N has a connection induced from the Riemannian connection M and the pullback connection. Then the second fundamental form of f is given by ( f )(U, V ) = f U f (V ) f ( M U V), (2.19) for vector fields U,V Ɣ(T M), where f is the pullback connection. We know that the second fundamental form is symmetric. A smooth map f between Riemannian manifolds is said to be harmonic if trace( f ) = 0. On the extra need, the tension field of f is the section τ(f) of Ɣ(f 1 TN)defined by τ(f) = divf = m ( f )(e i,e i ), (2.20) i=1

6 3582 Sushil Kumar and Rajendra Prasad where {e i,...,e m } is orthonormal frame on M. Then it follows that f is harmonic if and only if τ(f) = 0, for facts [2]. Lastly, we recollect the subsequent lemma from [2]. Lemma 2.3. Let (M, g M ) and (N, g N ) are two Riemannian manifolds. If f : (M, g M ) (N, g N ) horizontally conformal submersion between Riemannian manifolds, then for any horizontal vector fields U,V and vertical vector fields X, Y we have (i) df (U, V ) = U(lnλ)df (V ) + V (lnλ)df (U) g M (U, V )df (gradlnλ); (ii) df (X, Y ) = df (A V X Y); (iii) df (U, X) = df ( M U X) = df ((AH ) U X). where (A H ) X is the adjoint of (AH ) characterized by X (A H ) U E,F = E,AH F, ( for E,F Ɣ(T M)). U 3. Conformal anti-invariant submersions admitting vertical structure vector field In this section, we define conformal anti-invariant submersions from an almost contact metric manifold onto Riemannian manifolds. Definition 3.1. Let (M,φ,ξ,η,g M ) be a almost contact metric manifold and (N, g N ) be a Riemannian manifold, where dimm = m and dimn = n. A horizontally conformal submersion f : (M,φ,ξ,η,g M ) (N, g N ) is called a conformal anti-invariant submersion if the distribution ker f is anti-invariant with respect to φ i.e., φ(ker f ) (ker f ). We have φ(ker f ) ker f = {0}. We denote the complementary orthonormal distribution to φ(ker f ) in (ker f ) by µ. Then we have (ker f ) = φ(ker f ) µ. It is clear that µ is an invariant distribution of (ker f ) under the endomorphism φ. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If a map f : (M,φ,ξ,η,g M ) (N, g N ) horizontally conformal submersion admitting vertical structure vector field i.e., (ξ ker f ). Then we have (ker f ) = φ(ker f ) µ. (3.1) It is clear that µ is an invariant distribution of (ker f ), under the endomorphism φ. Thus, for any U Ɣ(ker f ), we have φu = BU + CU, (3.2)

7 Conformal anti-invariant Submersions 3583 where BU Ɣ(ker f ) and CU Ɣ(µ). On the additional point, since f (Ɣ(ker f ) ) = TN and f is a conformal submersion, for every X Ɣ(ker f ) and U (Ɣ(ker f ) ), 1 using equation (3.2) we get λ 2 g N(f φu,f CX) = 0, which denotes that TN = f (φ(ker f )) f (µ). (3.3) For any vector field X Ɣ(ker f ) and V Ɣ(ker f ), using equations (2.1), (2.2), (3.1) and (3.2), we get C 2 V = V φbv,bcv = 0,η(BV)= 0, BφX = X + η(x)ξ, CφX = 0,C 3 V + CV = 0. Lemma 3.2. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then we get T X ξ = φx,a V ξ = CV, and g M (CV, φx) = 0, (3.4) g M ( M U CV,φX) = g M(CV, φa U X) + η(x)g M (U, CV ), (3.5) for X Ɣ(ker f ) and U,V,φX (Ɣ(ker f ) ). Proof. For X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), since BV Ɣ(ker f ) and φx (Ɣ(ker f ) ), using equations (3.2) and (2.4), we have g M (CV, φx) = 0. Now, using equations (2.6), (2.14) and (3.4), we get g M ( U CV,φX) = g M (CV, U φx) = g M (CV, φa U X) + η(x)g M (U, CV ), since φv U X Ɣ(φ(ker f )). Therefore, we obtain the result. Theorem 3.3. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then the followings are equivalent to each other: (i) (ker f ) is integrable, 1 (ii) λ 2 g N( f V f CU f U f CV,f φx) = g M (A U BV A V BU,φX) g M (Hgradlnλ, CV )g M (U, φx) +g M (Hgradlnλ, CU)g M (V, φx) 2g M (Hgradlnλ, φx)g M (CU, V ),

8 3584 Sushil Kumar and Rajendra Prasad for X Ɣ(ker f ) and U,V (Ɣ(ker f ) ). Proof. For X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), since φv (Ɣ ker f µ) and φx (Ɣ(ker f ) ). Using equations (2.1), (2.4), (2.6) and (3.2),weget g M ([U,V ],X)= g M (φ U V,φX)+ η(x)η( U V) g M (φ V U,φX) η(x)η( V U), = g M ( U φv,φx) g M ( V φu,φx) + g M ([U,V], ξ)η(x), = g M ( U BV,φX) + g M ( U CV,φX) g M ( V BU,φX), g M ( V CU,φX) + g M ([U,V], ξ)η(x). Since f is a conformal submersion, using equations (2.14) and (2.15) we get g M ([U,V ],X) = g M (A U BV A V BU,φX) + 1 λ 2 g N(f U CV,f φx) 1 λ 2 g N(f V CU,f φx) + g M ([U,V], ξ)η(x). Using equations (2.23), (3.4) and lemma 1(i), weget g M ([U,V ],X)= g M (A U BV A V BU,φX) g M (Hgradlnλ, CV )g M (U, φx) + g M (Hgradlnλ, CU)g M (V, φx) 2g M (Hgradlnλ, φx)g M (CU, V ) 1 λ 2 g N( f V f CU which implies (i) (ii). f U f CV,f φx) + g M ([U,V], ξ)η(x). Theorem 3.4. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then any two of the following conditions imply the third: (i) (ker f ) is integrable, (ii) f is horizontally homothetic, 1 (iii) λ 2 g N( f V f CU f U f CV,f φx) = g M (A U BV A V BU,φX), for X Ɣ(ker f ) and U,V (Ɣ(ker f ) ). Proof. For X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), using theorem (2), weget g M ([U,V ],X)= g M (A U BV A V BU,φX) g M (U, φx)g M (Hgradlnλ, CV ) + g M (V, φx)g M (Hgradlnλ, CU) 2g M (CU, V )g M (Hgradlnλ, φx) 1 λ 2 g N( f V f CU f U f CV,f φx) + g M ([U,V], ξ)η(x).

9 Conformal anti-invariant Submersions 3585 Now, using conditions (i) and (ii), we get (iii) 1 λ 2 g N( f V f CU f U f CV,f φx) = g M (A U BV A V BU,φX). Similarly, one can obtain the other assertions. Remark 3.5. Let f be a conformal anti-invariant submersion is conformal Lagrangian submersion, if φ(ker f ) = (ker f ). Then (3.3), we have TN = f (φ(ker f ) ). Corollary 3.6. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then the following assertions are equivalent to each other: (i) (ker f ) is integrable, (ii) A U φv = A V φu, (iii) ( f )(V, φu) = ( f )(U, φv ), for U,V (Ɣ(ker f ) ). Proof. For any X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), from definition (2), φx (Ɣ(ker f ) ) and φv Ɣ(φ(ker f )). From theorem (2), we have g M ([U,V ],X)= g M (A U BV A V BU,φX) g M (Hgradlnλ, CV )g M (U, φx) + g M (Hgradlnλ, CU)g M (V, φx) 2g M (Hgradlnλ, φx)g M (CU, V ) 1 λ 2 g N( f V f CU f U f CV,f φx) + η(x)g M ([U,V],ξ). Since f conformal Lagrangian submersion, we have g M ([U,V ],X)= g M (A U BV A V BU,φX) + η(x)g M ([U,V],ξ), which implies (i) (ii). On the other hand using definition (2) and equation (2.15), we get g M (A U BV A V BU,φX) = g M (A U BV,φX) g M (A V BU,φX), = 1 λ 2 g N(f A U BV,f φx) 1 λ 2 g N(f A V BX,f φx), = 1 λ 2 g N(f ( U BV ), f φx) 1 λ 2 g N(f ( V BU), f φx).

10 3586 Sushil Kumar and Rajendra Prasad Now, using equation (2.23) we have 1 λ 2 g N(f ( U BV ), f φx) 1 λ 2 g N(f ( V BU), f φx) = 1 λ 2 g N( ( f )(U, BV ) + f U f BV,f φx) 1 λ 2 g N( ( f )(V, BU) + f V f BU,f φx), = 1 λ 2 [g N(( f )(V, BU) ( f )(U,BV),f φx)], which proves that (ii) (iii). Theorem 3.7. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then the followings are equivalent to each other: (i) (ker f ) defines a totally geodesic foliation on M, (ii) 1 λ 2 g N( f U f CV,f φx) = g M (A U BV,φX)+g M (U, φx)g M (Hgradlnλ, CV ) g M (U, CV )g M (Hgradlnλ, φx), for any X Ɣ(ker f ) and U,V (Ɣ(ker f ) ). Proof. For any X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), using equations (2.4), (2.6), (2.15), (2.16) and (3.2), we have g M ( U V,X) = g M ( U φv,φx) + η(x)η( U V), = g M (A U BV,φX) + g M (H U CV,φX) + η(x)η( U V). Since f is conformal submersion, using equation (2.23), lemma 1(i), definition (2) and equation (3.4), we get g M ( U V,X) = g M (A U BV,φX) g M (Hgradlnλ, CV )g M (U, φx) + η(x)η( U V)+ g M (Hgradlnλ, φx)g M (U, CV ) + 1 λ 2 g N( f U f CV,f φx), which implies (i) (ii). Theorem 3.8. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then any two of the following conditions imply the third: (i) (ker f ) defines a totally geodesic foliation on M,

11 Conformal anti-invariant Submersions 3587 (ii) f is horizontally homothetic, (iii) g M (A U BV,φX) = 1 λ 2 g N( f U f CV,f φx), U,V (Ɣ(ker f ) ). for any X Ɣ(ker f ) and Proof. For X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), using theorem (4), wehave g M ( U V,X) = g M (A U BV,φX) g M (Hgradlnλ, CV )g M (U, φx) +η(x)η( U V)+ g M (Hgradlnλ, φx)g M (U, CV ) + 1 λ 2 g N( f U f CV,f φx). Using conditions (i) and (ii),weget(iii) g M (A U BV,φX) = 1 λ 2 g N( f U f CV,f φx). Similarly, one can obtain the other assertions. Corollary 3.9. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal Lagrangian submersion, then the followings are equivalent to each other: (i) (ker f ) defines a totally geodesic foliation on M, (ii) A U φv = 0, (iii) ( f )(U, φv ) = 0, for U,V (Ɣ(ker f ) ). Proof. For X Ɣ(ker f ) and U,V (Ɣ(ker f ) ), from definition (2), φv Ɣ(φ(ker f )) and φx Ɣ((ker f ) ). Using theorem (4), wehave g M ( U V,X) = g M (A U BV,φX) g M (Hgradlnλ, CV )g M (U, φx) η(x)η( U V)+ g M (Hgradlnλ, φx)g M (U, CV ) + 1 λ 2 g N( f U f CV,f φx). Since f is conformal Lagrangian submersion, we get g M ( U V,X) = g M (A U BV,φX) + η(x)η( U V), = g M (A U φv,φx) + η(x)η( U V), which implies (i) (ii). Further, using equation (2.15),weget g M (A U BV,φX) = g M ( U BV,φX).

12 3588 Sushil Kumar and Rajendra Prasad Since f is conformal submersion, we get g M (A U BV,φX) = 1 λ 2 g N(f U BV,f φx). Using equation (2.23),weget g M (A U BV,φX) = 1 λ 2 g N(( f )(U,BV),f φx), = 1 λ 2 g N(( f )(U,φV),f φx). which shows (ii) (iii). Theorem Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then the followings are equivalent to each other: (i) (ker f ) defines a totally geodesic foliation on M, (ii) 1 λ 2 g N( f φy f φx,f φcu) = g M (T X φy,bu) + g M (φy, φx)g M (Hgradlnλ, φcu) + η(y)g M (X, φu), for any X, Y Ɣ(ker f ) and U (Ɣ(ker f ) ). Proof. For any X, Y Ɣ(ker f ) and U (Ɣ(ker f ) ), using equations (2.4), (2.6), (2.14) and (3.2),weget g M ( X Y, U) = g M (φ X Y, φu) + η( X Y)η(U) = g M (T X φy,bu) + g M (H X φy,cu) + η(y)g M (X, φu). Since is torsion free and [X, φy ] Ɣ(ker f ), we get g M ( X Y, U) = g M (T X φy,bu) + g M ( φy X, CU) + η(y)g M (X, φu), using equations (2.4) and (2.6),weget g M ( X Y, U) = g M (T X φy,bu) + g M ( φy φx,φcu) + η(y)g M (X, φu), here we have used µ is invariant. Since f is conformal submersion, using equation (2.23) and Lemma 1(i), we get g M ( X Y, U) = g M (T X φy,bu) 1 λ g M(Hgradlnλ, φy )g N (f φx,f φcu) 1 λ g M(Hgradlnλ, φx)g N (f φy,f φcu) + 1 λ g M(φY, φx)g N (f Hgradlnλ, f φcu) + 1 λ 2 g N( f φy f φx,f φcu) + η(y)g M (X, φu).

13 Conformal anti-invariant Submersions 3589 Next, using definition (2) and (3.4), we have g M ( X Y, U) = g M (T X φy,bu) + g M (φy, φx)g M (Hgradlnλ, φcu) + 1 λ 2 g N( f φy f φx,f φcu) + η(y)g M (X, φu), which shows (i) (ii). Theorem Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then any two of the following conditions imply the third: (i) (ker f ) defines a totally geodesic foliation on M, (ii) λ is constant on Ɣ(µ), 1 (iii) λ 2 g N( f φy f φx,f φcu) = g M (T X φy,φu) + η(y)g M (X, φu), for X, Y Ɣ(ker f ) and U (Ɣ(ker f ) ). Proof. For X, Y Ɣ(ker f ) and U (Ɣ(ker f ) ), from theorem (6), wehave g M ( X Y, U) = g M (T X φy,bu) + g M (φy, φx)g M (Hgradlnλ, φcu) Now, using conditions (i) and (iii),wehave + 1 λ 2 g N( f φy f φx,f φcu) + η(y)g M (X, φu). g M (φy, φx)g M (Hgradlnλ, φcu) = 0. From above equation λ is constant on Ɣ(µ). Similarly, one can obtain the other assertions. Corollary Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal Lagrangian submersion, then the following statements are equivalent to each other: (i) (ker f ) defines a totally geodesic foliation on M, (ii) T X φy = η(y)x, or T X φy = η(y)x is parallel to ξ for X, Y Ɣ(ker f ). Proof. For X, Y Ɣ(ker f ) and U Ɣ(ker f ), from theorem (6), wehave g M ( X Y, U) = g M (T X φy,bu) + g M (φy, φx)g M (Hgradlnλ, φcu) + 1 λ 2 g N( f φy f φx,f φcu) + η(y)g M (X, φu). Since f is a conformal Lagrangian submersion, we get which proves (i) (ii). g M ( X Y, U) = g M (T X φy,φu) + η(y)g M (X, φu),

14 3590 Sushil Kumar and Rajendra Prasad 4. Harmonicity of conformal anti-invariant submersion admitting vertical structure vector field In this section, we investigate the necessary and sufficient conditions for a conformal anti-invariant submersions to be harmonic. We also find the necessary and sufficient conditions for such submersions to be totally geodesic. Theorem 4.1. Let (M 2k+2r+1,φ,ξ,η,g M ) be a Sasakian manifold and (N k+2r,g N ) be a Riemannian manifold. If (M 2k+2r+1,φ,ξ,η,g M ) (N k+2r,g N ) be a conformal anti-invariant submersion, then the tension field τ of f is τ(f) = kf (µ ker f ) + (2 k 2r)f (Hgradlnλ), (4.1) where µ ker f is the mean curvature vector field of the distribution of ker f. Proof. Let {e 1,e 2,...,e k,ξ,φe 1,...,φe k,µ 1,...,µ r,φµ 1,...,φµ r } be an orthonormal basis of Ɣ(T M) such that {e 1,e 2,...,e k,ξ} is orthonormal basis of Ɣ(ker f ), {φe 1,...,φe k } is orthonormal basis of Ɣ(φ ker f ) and {µ 1,...,µ r,µ r+1,...,µ 2r } is orthonormal basis of Ɣ(µ). Then the trace of fundamental form (restriction of ker f ker f )isgivenby trace (ker f ) ( f ) = k 2r ( f )(φe i,φe i ) + ( f )(µ j,µ j ). i=1 j=1 Using lemma 1(i), we obtain trace (ker f ) ( f ) = k 2g M (gradlnλ, φe i )f (φe i ) kf (gradlnλ) i=1 2r + 2g M (gradlnλ, µ j )f (µ j ) 2rf (gradlnλ). i=1 Since f is conformal anti-invariant submersion, for x M, and 1 i k, 1 h r 1 { λ(x) f 1 x(φe i ), λ(x) f x(µ h )} is an orthonormal basis of T f(x) N; thus we obtain trace (ker f ) ( f ) = k 2g N (f (gradlnλ), 1 λ f (φe i )) 1 λ f (φe i ) kf (gradlnλ) i=1 2r + 2g N (f (gradlnλ), 1 λ f (µ j )) 1 λ f (µ j ) 2rf (gradlnλ) i=1 trace (ker f ) ( f ) = (2 k 2r)f (Hgradlnλ). (4.2)

15 Conformal anti-invariant Submersions 3591 In a similarly, we get trace (ker f ) ( f ) = k ( f )(e i,e i ) + ( f )(ξ, ξ). i=1 Using equation (2.22) and (2.13), we find trace (ker f ) ( f ) = kf (µ ker f ). (4.3) From equations (4.2) and (4.3), we get τ(f) = kf (µ ker f ) + (2 k 2r)f (Hgradlnλ). Therefore, we obtain the result. Theorem 4.2. Let (M 2k+2r+1,φ,ξ,η,g M ) be a Sasakian manifold and (N k+2r,g N ) be a Riemannian manifold. If (M 2k+2r+1,φ,ξ,η,g M ) (N k+2r,g N ) be a conformal anti-invariant submersion, then any two of the following conditions imply the third: (i) f is harmonic, (ii) The fibres are minimal, (iii) f is a horizontally homothetic map. Proof. Taking equation (4.1),wehave τ(f) = kf (µ ker f ) + (2 k 2r)f (Hgradlnλ). Now, using conditions (i) and (ii), then f is a horizontally homothetic map. Corollary 4.3. Let (M 2k+2r+1,φ,ξ,η,g M ) be a Sasakian manifold and (N k+2r,g N ) be a Riemannian manifold. Let (M 2k+2r+1,φ,ξ,η,g M ) (N k+2r,g N ) be a conformal anti-invariant submersion. If k + 2r = 2, then f is harmonic if and only if the fibres are minimal. Next, we find necessary and sufficient condition for conformal anti-invariant submersion to be totally geodesic. We memorize that a differentiable map f between two Riemannian manifolds is called totally geodesic if ( f )(V, W ) = 0, for all V,W Ɣ(T M). A geometric clarification 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.

16 3592 Sushil Kumar and Rajendra Prasad Theorem 4.4. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then f is a totally geodesic map if and only if f V f W = f (φa V φw 1 + φv V BW 2 + φa V CW 2 + CH V φw 1 +CA V BW 2 + CH V CW 2 + η(w 1 )CV ), for any V,W Ɣ(T M),where W = W 1 +W 2,W 1 Ɣ(ker f ) and W 2 (Ɣ(ker f ) ). Proof. Taking equation (2.22) and using equations (2.1) and (2.6), we get ( f )(V, W ) = f V f W + f (φ V φw + η(w)φv η( V W)ξ), for any V,W Ɣ(T M). Now, using equations (2.15) and (3.2), weget ( f )(V, W ) = f V f W + f (φa V φw 1 + BH V φw 1 + CH V φw 1 +BA V BW 2 + CA V BW 2 + φv V BW 2 + φa V CW 2 +BH V CW 2 + CH V CW 2 + η(w 1 )BV +η(w 1 )CV η( V W)ξ), for W = W 1 + W 2 Ɣ(T M), where W 1 Ɣ(ker f ) and W 2 (Ɣ(ker f ) ). Thus taking into account the vertical terms, we get Thus ( f )(V, W ) = f V f W + f (φ(a V φw 1 + V V BW 2 + A V CW 2 ) +C(H V φw 1 + A V BW 2 + H V CW 2 ) + η(w 1 )CV ). ( f )(V, W ) = 0 f V f W = f (φ(a V φw 1 + V V BW 2 + A V CW 2 ) +C(H V φw 1 + A V BW 2 + H V CW 2 ) + η(w 1 )CV ). Therefore, we obtain the result. Definition 4.5. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then f is called a (φ ker f,µ) totally geodesic map provided ( f )(φx, U) = 0, for X Ɣ(ker f ) and U (Ɣ(ker f ) ). Theorem 4.6. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then f is called a (φ ker f,µ) totally geodesic map if and if f is horizontally homothetic map.

17 Conformal anti-invariant Submersions 3593 Proof. For X Ɣ(ker f ) and U Ɣ(µ), using lemma 1(i), weget ( f )(φx, U) = φx(lnλ)f (U) + U(lnλ)f (φx) g M (φx, U)f (gradlnλ). From above equation, if f is a horizontally homothetic, then ( f )(φx, U) = 0. Conversely, if ( f )(φx, U) = 0,we find φx(lnλ)f (U) + U(lnλ)f (φx) = 0. (4.4) Taking inner product in above equation with f (φx) and since f is conformal submersion, we have g M (Hgradlnλ, φx)g N (f U,f φx) + g M (Hgradlnλ, U)g N (f φx,f φx) = 0. Above equation shows that λ is a constant Ɣ(µ). On the other hand taking inner product in equation (4.4) with f X, we get g M (Hgradlnλ, φx)g N (f U,f φu) + g M (Hgradlnλ, U)g N (f φx,f U) = 0. From above equation shows that λ is a constant on Ɣ(φ(ker f )). Thus λ is a constant on Ɣ((ker f ) ). Theorem 4.7. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. Let f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion. Then f is totally geodesic map if and only if (i) T X φy = η(y)x and H Y φx Ɣ(φ ker f ), (ii) f is horizontally homothetic map, (iii) X BU = T X CU or parallel to ξ and T X BU + H X CU Ɣ(φ ker f ), for all X, Y Ɣ(ker f ) and U,V Ɣ(µ). Proof. For X, Y Ɣ(ker f ) {ξ}, using equations (2.1), (2.6) and (2.22), weget ( f )(X, Y ) = f (φ( X φy) + η(y)φx η( X Y)ξ). Now, using equations (2.14) and (3.2), weget ( f )(X, Y ) = f (φt X φy + CH X φy + η(y)φx). Thus shows that T X φy + η(y)x = 0 and H X φy Ɣ(φ ker f ). On the other hund using lemma 1(i), we get ( f )(U, V ) = U(lnλ)f (V ) + V (lnλ)f (U) g M (U, V )f (gradlnλ), for U,V Ɣ(µ). It is obvious that if f is horizontally homothetic, it follows that ( f )(U, V ) = 0. Conversely, if ( f )(U, V ) = 0, taking V = φu in above equation, we have U(lnλ)f (φu) + φu(lnλ)f (U) = 0. (4.5)

18 3594 Sushil Kumar and Rajendra Prasad Taking inner product in (4.5) with f φu, we get g M (Hgradlnλ, U)g N (f φu,f φu) + g M (Hgradlnλ, φu)g N (f U,f φu) = 0. From above equation λ is constant on Ɣ(µ). On other hand, for X, Y Ɣ(ker f ), from lemma 1(i),weget ( f )(φx, φy ) = φx(lnλ)f (φy ) + φy (lnλ)f (φx) g M (φx, φy )f (gradlnλ), Again if f is horizontally homothetic, then ( f )(φx, φy ) = 0. Conversely, if ( f )(φx, φy ) = 0, putting X = Y in above equation, we get 2φX(lnλ)f (φx) g M (φx, φx)f (gradlnλ) = 0. Taking inner product in above equation with f φx and since f is conformal submersion, we have g M (φx, φx)g M (gradlnλ, φx) = 0. From above equation, λ is constant on Ɣ(φ ker f ).Thus λ is constant on Ɣ((ker f ) ). Now, for X Ɣ(ker f ) and U Ɣ((ker f ) ), using equations (2.1), (2.6) and (2.23) we get ( f )(X, U) = f (φ( X φu) η( X U)ξ). Now, again using (2.14) and (3.2), we get ( f )(X, U) = f (CT X BU + φ X BU + CH X CU + φt X CU). Thus ( f )(X, U) = 0 f (CT X BU + φ X BU + CH X CU + φt X CU) = 0. Therefore, we obtain the result. 5. Decomposition Theorems for a conformal anti-invariant submersion admitting vertical structure vector field In this section, we obtain decomposition theorems by using the existence of conformal anti-invariant submersions. Initial, we memorise the following results from [20]. Let g B be a Riemannian metric tensor on the manifold B = M N and assume that the canonical foliations D M and D N intersect perpendicularly everywhere. Then g B is the metric tensor of (i) a twisted product M F N if and only if D M is totally geodesic foliation and D N is totally umbilical foliation, (ii) a warped product M F N if and only if D M is totally geodesic foliation and D N is a spheric foliation, i.e., it is umbilical and its mean curvature vector field is parallel.

19 Conformal anti-invariant Submersions 3595 We note in this case, from [3] we have X U = X(ln F)U, for X Ɣ(T M) and U Ɣ(T N), where is the Riemannian connection on M N, (iii) a usual product of Riemannian manifolds if and only if D M and D N are totally geodesic foliations. Next, we found a decomposition theorem related to the concept of twisted product manifold. However, we first memorise the adjoint map of a map. Let f : (M, g M ) (N, g N ) be a map between Riemannian manifolds (M, g M ) and (N, g N ). Then the adjoint map f of f is characterized g M (X, f p Y) = g N ( f p X, Y ) by X T p M, Y T f(p) N and p M. Considering f h at each p M as a linear transformation f h p : ((ker f ) (p),g M(p)((ker f ) p ) ) (rangef (q),g N(q)(rangef )(q)), we will denote the adjoint f h (p) by f h (p). Let f h (p) be the adjoint of f h (p) : (T pm,g M(p) ) (T (q) N,g N(q) ). The linear transformation ( f p ) h : (rangef (p) ) (ker f ) (p) defined ( f (p) ) h Y = f h (p) Y, where Y (ranrgef (p)), q = f(p),is an isomorphism and (f h (p) ) 1 = ( f p ) h = f h (p). Our first decomposition theorem for a conformal anti-invariant submersion comes from theorem (4) and theorem (6) in terms of the second fundamental forms of such submersions. Theorem 5.1. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. Let f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion. Then f is a locally product manifold if and only if and 1 λ 2 g N( f U f CV,f φx) (5.1) = g M (A U BV,φX) g M (gradlnλ, CV )g M (U, φx) +g M (gradlnλ, φx)g M (U, CV ), 1 λ 2 g N( f φy f φx,f φcu) (5.2) = g M (T X φy,bu) + g M (φy, φx)g M (Hgradlnλ, φcu) +η(y)g M (X, BU), for X, Y Ɣ(ker f ) and U,V,W (Ɣ(ker f ) ). Again, from Corollary (2) and Corollary (3), we have the following theorem.

20 3596 Sushil Kumar and Rajendra Prasad Theorem 5.2. Let (M,φ,ξ,η,g M ) be a Sasakian manifold and (N, g N ) be a Riemannian manifold. If f : (M,φ,ξ,η,g M ) (N, g N ) be a conformal anti-invariant submersion, then f is a locally product manifold if and only if A U φv = 0 and T X φy = η(y)x, for X, Y Ɣ(ker f ) and U,V (Ɣ(ker f ) ). Theorem 5.3. Let f be a conformal anti-invariant submersion from a Sasakian manifolds (M,φ,ξ,η,g M ) to a Riemannian manifold (N, g N ). Then M is a locally twisted product manifold of the form M (ker f ) M (ker f ) if and only if and 1 λ 2 g N( f φy f φx,f φcu) (5.3) = g M (T X φy,bu) + g M (φy, φx)g M (Hgradlnλ, φcu) +η(y)g M (X, φu), g M (U, V )H = BA U BV + CV (lnλ)bu B(Hgradlnλ)g M (U, CV ) (5.4) φ f ( f U f CV), for X, Y Ɣ(ker f ) and U,V (Ɣ(ker f ) ), where M (ker f ) and M (ker f ) are integral manifolds of the distributions (ker f ) and (ker f ) and H is the mean curvature vector field of M (ker f ). Proof. For X, Y Ɣ(ker f ) and U (Ɣ(ker f ) ), using equations (2.4), (2.6), (2.14) and (3.2), we get g M ( X Y, U) = g M (T X φy,bu) + g M (H X φy,cu) + η(y)g M (X, φu). Since is torsion free and [X, φy ] Ɣ(ker f ), we get g M ( X Y, U) = g M (T X φy,bu) + g M (H φy X, CU) + η(y)g M (X, φu). Using equations (2.4), (2.6) and (2.16), wehave g M ( X Y, U) = g M (T X φy,bu) + g M ( φy φx,φcu) + η(y)g M (X, φu). Since f is conformal submersion, using equation (2.23) and lemma 1(i), wefind g M ( X Y, U) = g M (T X φy,bu) 1 λ 2 g M(Hgradlnλ, φy )g N (f φx,f φcu) 1 λ 2 g M(Hgradlnλ, φx)g N (f φy,f φcu) + 1 λ 2 g M(φX, φy )g N (f Hgradlnλ, f φcu) + 1 λ 2 g N( f φy f φx,f φcu) + η(y)g M (U, φx).

21 Conformal anti-invariant Submersions 3597 Next, using definition (2) and equation (3.2), we obtain g M ( V W,X) = g M (T V φw,bx) + g M (φv, φw )g M (Hgradlnλ, φcx) + 1 λ 2 g N( f φw f φv,f φcx) + η(w)g M (X, φv ). Thus shows that M (ker f ) is totally geodesic if and only if 1 λ 2 g N( f φy f φx,f φcu) = g M (T X φy,bu) + g M (φx, φy )g M (Hgradlnλ, φcu) +η(y)g M (U, φx). On the other hand for X, Y Ɣ(ker f ) and U (Ɣ(ker f ) ), using equations (2.4), (2, 4), (2.15), (2.16) and (3.2), we get g M ( U V,X) = g M (A U BV,φX) + g M (H U CV,φX). Since f is conformal submersion, using equation (2.23) and lemma 1(i), we obtain that g M ( U V,X) = g M (T U BV,φX) 1 λ 2 g M(Hgradlnλ, U)g N (f CV,f φx) 1 λ 2 g M(Hgradlnλ, CV )g N (f U,f φx) + 1 λ 2 g M(U, CV )g N (f Hgradlnλ, f φx) + 1 λ 2 g N( f φu f CV,f φx) + η(x)η( U V). Moreover, using definition (2) and equation (3.4), weget g M ( U V,X) = g M (T U BV,φX) g M (Hgradlnλ, CV )g M (U, φx) + η(x)η( U V) +g M (U, CV )g N (Hgradlnλ, φx) + 1 λ 2 g N( f φu f CV,f φx). Then, we have g M (U, V )H = BA U BV + CV (lnλ)bu B(Hgradlnλ)g M (U, CV ) φf ( f U f CV) + η(a U V)ξ, which proves.

22 3598 Sushil Kumar and Rajendra Prasad 6. Example Note that given an Euclidean space (x 1,...,x 2m,x 2m+1 ) with coordinates we can canonically choose an almost contact structure φ on R 2m+1 as follows: φ(a 1 + a 2 + +a 2m 1 + a 2m + a 2m+1 ) x 1 x 2 x 2m 1 x 2m x 2m+1 = ( a 2 + a 1 + a 2m + a 2m 1 ) x 1 x 2 x 2m 1 x 2m where ξ = and a 1,a 2,...,a 2m,a 2m+1 are C real valued functions in R. Let x 2m+1 η = dx 2m+1 and (,,...,, ) is orthogonal basis of vector fields on x 1 x 2 x 2m x 2m+1 R 2m+1. Example 6.1. Define a map f : R 5 R 2 by Then we have ker f =< f(x 1,...,x 5 ) = (e x 1 sin x 2,e x 1 cos x 2 ) x 3, x 4, x 5 > and (ker f ) =< Thus, f is a conformal anti-invariant submersion with λ = e x 1. Example 6.2. Define a map f : R 5 R 2 by Then we have ker f =< f(x 1,...,x 5 ) = (e x 3 cos x 4,e x 3 sin x 4 ) x 1, x 2, x 5 > and (ker f ) =< Thus, f is a conformal anti-invariant submersion with λ = e x 3. x 1, x 3, x 2 > x 4 > References [1] M. A. Akyol and B. Sahin, Conformal anti-invariant submersions from almost Hermitian manifolds. Turk J. Math., 40 (2016), [2] P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds. London Mathematical Society Monographs, 29, Oxford University Press, The Clarendon Press. Oxford, (2003).

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