Riemannian submersions and slant submanifolds

Publ. Math. Debrecen 61 / 3-4 (2002), 523–532 Riemannian submersions and slant submanifolds By JOS´ E L. CABRERIZO (Sevilla), ALFONSO CARRIAZO (Sevilla), LUIS M. FERN´ ANDEZ (Sevilla) and MANUEL FERN´ ANDEZ (Sevilla) Abstract. We study the relationship between slant submanifolds in both Complex and Contact Geometry through Riemannian submersions. We present some construction procedures to obtain slant submanifolds in the unit sphere and in a Stiefel manifold. We also generalize them by means of the Boothby–Wang fibration. Finally, we show some characterization theorems of three-dimensional slant submanifolds. 0. Introduction The geometry of slant submanifolds has been increasingly studied since B.-Y. Chen defined slant immersions in complex manifolds as a natural generalization of both holomorphic and totally real immersions (see [7]). Later, a similar notion of slant submanifold was introduced in Contact Geometry, which is specially important for submanifolds tangent to the structure vector field of a contact metric manifold. The purpose of the present paper is to study the close relationship between both theories through Riemannian submersions. In particular, we prove that, in some conditions, a submanifold of an almost Hermitian manifold is slant if and only if its lift by a Riemannian submersion is a slant submanifold of an almost contact metric manifold. Mathematics Subject Classification: 53C15, 53C40. Key words and phrases: Riemannian submersion, Kaehlerian manifold, Sasakian manifold, slant submanifold. The authors wish to thank Prof. Manuel Barros for his valuable suggestions and helpful remarks. The authors are partially supported by the PAI project (Junta de Andaluc´ıa, Spain, 2001). 524 Jos´e L. Cabrerizo et al. We use this result as a method to find interesting examples. Examples of proper slant submanifolds of a Sasakian-space-form of constant φ-sectional curvature chave been given in [4], [10] (c=−3) and [6] (c < −3), but, until now, there were no examples in a Sasakian-space-form with c > −3. In fact, in this paper we exhibit a construction procedure to obtain examples of slant submanifolds in the unit sphere with its usual Sasakian structure (c= 1). Afterwards, we extend it in order to get ample examples in Sasakian-space-forms with constant φ-sectional curvature c, for any c > −3. Moreover, we also construct examples of slant immersions into a Stiefel manifold and we generalize both procedures by using the Boothby–Wang fibration. Finally, we present some classifications of three-dimensional slant submanifolds of R5, by attending to their second fundamental form. 1. Preliminaries In this section, we recall some basic formulas and definitions about slant submanifolds in both Complex and Contact Geometry, which we shall use later. For details and background on complex and contact manifolds, we refer to the standard references [1], [13]. A submanifold Nof an almost Hermitian manifold ( e N, g, J) is said to be slant [7] if for each nonzero vector Xtangent to Nat p, the angle θ(X), 0 ≤θ(X)≤π/2, between JX and TpNis a constant, called the slant angle of the submanifold. In particular, holomorphic and totally real submanifolds appear as slant submanifolds with slant angle 0 and π/2, respectively. A slant submanifold is called proper slant if it is neither holomorphic nor totally real. In the case where Nis a Riemann surface and e Nis a Kaehler manifold, S. S. Chern and J. G. Wolfson introduced the notion of Kaehler angle, defined to be the angle between J∂/∂x and ∂/∂y, where z=x+√−1yis a local complex coordinate on N[9]. It is clear that if Nis a surface with constant Kaehler angle α, then it is a slant submanifold with slant angle θsatisfying θ=α(resp. θ=π−α) when α∈[0, π/2] (resp. α∈(π/2, π]). Put JX =PX +FX, for any tangent vector field X, where PX (resp. FX) denotes the tangential (resp. normal) component of JX. Then, θslant submanifolds are characterized by the formula: P2=−cos2θId . Riemannian submersions and slant submanifolds 525 A special type of proper slant submanifold is that of Kaehlerian slant submanifold, i.e., a proper slant submanifold satisfying ∇0P= 0, where ∇0denotes the Levi–Civita connection on N. It is easy to show that a Kaehlerian slant submanifold is a Kaehlerian manifold with respect to the induced metric and with the almost complex structure given by (sec θ)P. In a similar way, given a submanifold Mtangent to the structure vector field ξof an almost contact metric manifold (f M, φ, ξ, η, G), it is said to be slant [4] if the angle θ(X) between φX and TpMis a constant, which is independent of the choice of p∈Mand X∈Tp(M)\Span(ξp). In particular, for θ= 0 (resp. θ=π/2) we obtain the invariant (resp. anti-invariant) submanifolds. Now, if we denote by TX (resp. NX) the tangential (resp. normal) component of φX, there is an equation which characterizes θ-slant submanifolds: T2=−cos2θ(Id −η⊗ξ). In contact geometry, the similar notion to Kaehlerian slant submanifolds is given by proper θ-slant submanifolds satisfying (∇XT)Y= cos2θ(g(X, Y )ξ−η(Y)X), for any tangent vector fields X, Y , where ∇denotes the Levi–Civita connection on M. This non-trivial fact is shown in [4]. Therefore, by following the complex case notation, we call such a submanifold a Sasakian slant submanifold. On the other hand, the possibility of obtaining an induced contact metric structure on a slant submanifold of a contact metric manifold is studied in [5]. 2. Main results Let f Mbe a (2m+ 1)-dimensional almost contact metric manifold with structure tensors (φ, ξ, η, G) and e Nbe a real 2m-dimensional almost Hermitian manifold with structure (J, g). Let suppose that there exists a Riemannian submersion π:f M→e Nsatisfying the conditions: i) The vertical subspace Vpof the submersion at p∈f Mis equal to the span of ξp, ii) φX∗= (JX)∗, 526 Jos´e L. Cabrerizo et al. for any vector field Xon e N, where ∗denotes the horizontal lift with respect to π. In fact, since πis a Riemannian submersion, we also have iii) G(X∗, Y ∗) = g(X, Y ), for any vector fields X, Y on e N. Now, let Mbe an (n+ 1)-dimensional submanifold tangent to the structure vector field ξof f Mand Nbe an n-dimensional submanifold of e N. Throughout in this section we assume that the following diagram commutes (2.1) M−−−−→ f M   y  yπ N−−−−→ e N where Mis the set of fibres over N. Then, we state the following theorem: Theorem 2.1. In the above conditions, we have: (a) Mis θ-slant in f Mif and only if Nis θ-slant in e N. Moreover, if f Mis a Sasakian manifold, we also have: (b) Mis Sasakian θ-slant in f Mif and only if Nis Kaehlerian θ-slant in e N. Proof. Statement (a) follows directly from i)–iii). In fact, in any almost contact metric manifold, φξ = 0, and then, the condition of M being a slant submanifold is really related to its contact distribution, which is the horizontal subspace of the submersion at any point. Now, suppose that f Mis a Sasakian manifold and denote by ∇(resp. ∇0) the Levi–Civita connection on M(resp. N). It follows from the wellknown O’Neill equations of the submersion that ∇X∗Y∗= (∇0 XY)∗+η(∇X∗Y∗)ξ, η(∇X∗Y∗) = −G(X∗, TY ∗), for any vector fields X, Y on e Ntangent to N. Then, we have (∇X∗T)Y∗= ((∇0 XP)Y)∗−G(X∗, T2Y∗)ξ, (∇ξT)Y∗= 0, which imply (b). ¤ Riemannian submersions and slant submanifolds 527 Notice that, in particular, statement (a) of Theorem 2.1 implies statements (3) and (4) of [13, Proposition 3.2, p. 459]. By using Theorem 2.1, we can show the following construction procedure for giving examples of proper slant submanifolds in the unit sphere. Let π:S2m+1 →CPm(4) be the well-known Hopf fibration, where S2m+1 (resp. CPm(4)) is endowed with its usual Sasakian (resp. Kaehlerian) structure. Given any isometric immersion f:N→CPm(4), then M=π−1(N) is a principal circle bundle over Nwith totally geodesic fibres and the lift ˆ f:M→S2m+1 of fis an isometric immersion such that the following diagram commutes: Mˆ f −−−−→ S2m+1   y  yπ Nf −−−−→ CPm(4). It follows from Theorem 2.1 that, in order to obtain a θ-slant submanifold of S2m+1, it is enough to consider a θ-slant submanifold of CPm(4). For example, we could take the examples given in [11]. The above procedure was first pointed out by B.-Y. Chen and Y. Tazawa in [8]. We can also consider the lift of the Veronese sequence in order to get new examples of proper slant immersions into S2m+1. We recall that the Veronese sequence ψ0, . . . , ψmis defined, for any p= 0, . . . , m, by ψp:S2→CPm:ψp[z0, z1] = [gp,0(z0/z1), . . . , gp,m(z0/z1)], where [z0, z1]∈CP1=S2, and gp,j(z) = p! (1 + zz)psµm j¶zj−pX k (−1)kµj p−k¶µm−j k¶(zz)k, for any j= 0, . . . , m. It was shown in [2] that every ψpis a conformal minimal immersion with constant curvature and constant Kaehler angle αpsuch that tan2αp 2=p(m−p+ 1) (p+ 1)(m−p). By combining this procedure and a D-homothetic deformation, we may also obtain the following theorem, similar to [6, Theorem 3.5]: 528 Jos´e L. Cabrerizo et al. Theorem 2.2. Let cbe a constant with c > −3. Then, there exist proper slant submanifolds in a Sasakian-space-form with constant φsectional curvature c. Proof. First, we can choose a proper slant submanifold of S2m+1, given by the above construction procedure. We denote the usual Sasakian structure on S2m+1 by (φ, ξ, η, G). Then, for any c > −3, we consider the constant a= 4/(c+ 3) >0 and the D-homothetic deformation: e φ=φ, e ξ=1 aξ, eη=aη, e G=aG +a(a−1)η⊗η. It was shown in [1] that S2m+1 with this structure is a Sasakian-spaceform with constant φ-sectional curvature (4/a)−3 = c. Finally, it is easy to prove that a D-homothetic deformation maps slant submanifolds into slant submanifolds. ¤ A more elaborate construction procedure for obtaining slant submanifolds in a certain almost contact metric manifold can be shown as follows. Let Hbe the closed connected subgroup in S3×S3given by H={(z, z) : z∈S1}and consider the homogeneous space (S3×S3)/H. Since this is a compact simply connected 5-dimensional spin manifold with H2((S3×S3)/H;Z) = Z, it follows from a classic result of Smale [12] that it is diffeomorphic to S2×S3. On the other hand, if we denote by V(2,4) the Stiefel manifold of orthonormal 2-frames in 4-space, it is known that V(2,4) is diffeomorphic to (S3×S3)/H and then, there is a diffeomorphism f:V(2,4) →S2×S3. Let eπ:S3→S2be the Hopf fibration and put F= (id ×eπ)◦f. Hence, F:V(2,4) →Q2is a submersion, where Q2denotes the complex quadric S2×S2. Now, put S2 ∗=S2\ {(0,0,1)} and let E:S2 ∗→Cbe the corresponding stereographic projection, which preserves the complex structure of C. Then, V(2,4)∗→Q2∗→C2is a submersion, where Q2∗(resp. V(2,4)∗) denotes the manifold S2 ∗×S2 ∗(resp. F−1(Q2∗)). It is clear that, if we consider on C2its usual Kaehler structure, V(2,4)∗can be endowed with a natural almost contact metric structure such that (E, E)◦F|V(2,4)∗ is a Riemannian submersion satisfying the above stated conditions i)–ii). Hence, we obtain ample examples of slant surfaces in V(2,4)∗by considering the lifts of slant surfaces in C2(see, for instance, [7]). Moreover, we can give a generalization of the previous construction procedures. Let f Mbe a (2m+ 1)-dimensional compact regular contact Riemannian submersions and slant submanifolds 529 manifold. According with a classical result of Boothby–Wang [3], one can see f Mas a circle bundle over a 2m-dimensional compact symplectic manifold e N: π:f M−→ e N. Since e Ncarries a global symplectic form Ω, there exist a Riemannian metric gand a tensor field Jof type (1,1) such that (g, J) is an almost Kaehler structure on e Nwith Ω as its fundamental 2-form. Denote by ηthe contact form on f Mwith d η=π∗Ω and ξits characteristic vector field and define a tensor field φand a Riemannian metric Gon f Mby φX = (Jπ∗X)∗ and G=π∗(g)+η⊗η, respectively. Then, it can be proved that (φ, ξ, η, G) is a K-contact structure on f Mand π: (f M, G)→(e N, g) is a Riemannian submersion. Now, we have: Theorem 2.3. In the above conditions, let Mbe a submanifold of f M. Then, Mis a S1-invariant θ-slant submanifold if and only if M=π−1(N), where Nis a θ-slant submanifold of e N. Proof. Let Nbe a submanifold of e Nand denote by M=π−1(N). Then, Mis a submanifold of f Mand the characteristic vector field ξis tangent to M, in particular, Mis S1-invariant. The converse of the above stated fact also holds, that is, if Mis a S1-invariant submanifold of f M, then ξis tangent to Mand there exists a submanifold Nin e Nwith M=π−1(N). Hence, the proof concludes by applying Theorem 2.1. ¤ 3. Some applications We now proceed to show some applications of the above stated relationship between slant submanifolds and Riemannian submersions, by considering the differential map given by π:R5−→ C2; (x1, x2, y1, y2, z)7−→ 1 2(y1, y2, x1, x2). It is easy to see that, if we have on R5(resp. C2) its usual Sasakian (resp. Kaehlerian) structure, then πis a Riemannian submersion satisfying conditions i)–ii). Therefore, by using this submersion, we can obtain examples of slant submanifolds in R5by taking the lifts of Examples 2.1, 2.3, 2.4 530 Jos´e L. Cabrerizo et al. and 2.5 of [7]. Notice that those examples will be similar to Examples 3.7– 3.10 of [4]. Now, suppose that we have a 3-dimensional submanifold Mtangent to the structure vector field on R5and a surface Nin C2satisfying diagram (2.1). Then, we have the following classification theorem: Theorem 3.1. In the above conditions, Mis a 3-dimensional slant submanifold of R5with parallel mean curvature vector if and only if Mis one of the following submanifolds: (a) a submanifold locally isometric to an open portion of the product of a plane circle and a circular cylinder. (b) a submanifold locally isometric to an open portion of the product of a circular cylinder and R. (c) a minimal slant submanifold in R5. Moreover, if either case (a) or case (b) occurs, then Mis an anti-invariant submanifold. Proof. First, it is known that if the mean curvature vector of Mis parallel then the mean curvature vector of Nis also parallel, and that M is minimal if and only if Nis minimal (see, for instance, [13, p. 462–463]). Hence, the proof of this theorem follows from Theorem 1.1 of [7, p. 50] and by taking into account that, if Mis an anti-invariant submanifold, then η(∇X∗Y∗) = 0, for any X,Ytangent to N, which means that, in this case, Mis locally isometric to the Riemannian product of Nand R.¤ In the same conditions, we can also classify the submanifold Mattending to a particular behaviour of its second fundamental form σ: Theorem 3.2. Mis a 3-dimensional slant submanifold of R5satisfying (3.1) (∇Xσ)(Y, Z) = G(Y, TX)NZ +G(Z, TX)NY for any tangent vector fields X, Y, Z orthogonal to ξ, if and only if Mis one of the following submanifolds: (a) a submanifold locally isometric to an open portion of the product of a plane circle and a circular cylinder. (b) a submanifold locally isometric to an open portion of the product of a circular cylinder and R. Riemannian submersions and slant submanifolds 531 (c) a lift by πof an open portion of a plane in C2. Moreover, if either case (a)or case (b)occurs, then Mis an anti-invariant submanifold. Proof. It follows from the O’Neill equations that (∇X∗σ)(Y∗, Z∗) = ((∇Xσ0)(Y, Z))∗+G(Y∗, TX∗)NZ∗+G(Z∗, TX∗)NY ∗, for any X, Y, Z tangent to N, where σ0denotes the second fundamental form of N, and so, Msatisfies (3.1) if and only if σ0is parallel. Therefore, this proof works as that of Theorem 3.1, by applying now Theorem 1.2 of [7, p. 51]. ¤ A sufficient condition for a submanifold M, in the above conditions, to satisfy equation (3.1) is to be totally contact geodesic, i.e., such that σ(X, Y ) = η(X)σ(Y, ξ) + η(Y)σ(X, ξ), for any tangent vector fields Xand Y. In fact there are examples of totally contact geodesic slant submanifolds in R5(see, for instance, Example 3.7 of [4]). References [1] D. E. Blair , Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, 509, Springer-Verlag,New York, 1976. [2] J. Bolton, G. R. Jensen, M. Rigoli and L. M. 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