Existence and uniqueness theorem for slant immersions in Sasakian-space-forms

Publ. Math. Debrecen 58 / 4 (2001), 559–574 Existence and uniqueness theorem for slant immersions in Sasakian-space-forms By JOS´ E LUIS CABRERIZO (Sevilla), ALFONSO CARRIAZO (Sevilla), LUIS M. FERN´ ANDEZ (Sevilla) and MANUEL FERN´ ANDEZ (Sevilla) Abstract. In this paper, we present the existence and uniqueness theorems for slant immersions into Sasakian-space-forms. By applying the first result, we prove several existence theorems for slant submanifolds. In particular, we prove the existence theorems for three-dimensional slant submanifolds with prescribed mean curvature or with prescribed scalar curvature. 0. Introduction Slant immersions in complex geometry were defined by B.-Y. Chen as a natural generalization of both holomorphic and totally real immersions [3]. In a recent paper ([7]), A. Lotta has introduced the notion of slant immersion of a Riemannian manifold into an almost contact metric manifold. In [8], he has obtained examples of slant submanifolds in the Sasakian-space-form R2m+1 as the leaves of a harmonic Riemannian threedimensional foliation. On the other hand, in [2], we have also studied and characterized slant submanifolds of K-contact and Sasakian manifolds. In particular, we have paid special attention to three-dimensional slant submanifolds. Mathematics Subject Classification: 53C15, 53C42. Key words and phrases: Sasakian manifold, slant submanifold, prescribed mean curvature, prescribed scalar curvature. The authors are partially supported by the PAI project (Junta de Andaluc´ıa, Spain 1999). The authors wish to be grateful with Bang-Yen Chen for his help and his advice in their research. They also wish to thank Luc Vrancken for his kindness in answering their questions. 560 J. L. Cabrerizo et al. The purpose of the present paper is to establish a general existence and uniqueness theorem for slant immersions in Sasakian-space-forms, which is similar to the result presented by B.-Y. Chen and L. Vrancken for complex-space-forms in [5]. By applying the existence theorem, we prove that there exist infinitely many three-dimensional proper slant submanifolds with prescribed mean curvature (or with prescribed scalar curvature). In [2], we have given examples of slant submanifolds in R2m+1 with its usual Sasakian structure. It is well known that this manifold is a Sasakian-space- form with constant φ-sectional curvature −3. In this paper, we show that there are ample examples of proper slant submanifolds in Sasakian-space- forms with constant φ-sectional curvature c, for any c < −3. In Section 1 we review basic formulas and definitions for almost contact metric manifolds and their submanifolds, which we shall use later. We also review the definition and some properties given in [2], [7]. Moreover, we develop the ground work which will allow us to present the existence and uniqueness theorems in Section 2. In Section 3, we show the applications of the main theorem. 1. Preliminaries Let (f M, g) be an odd-dimensional Riemannian manifold and denote by Tf Mthe Lie algebra of vector fields in f M. Let φbe a (1,1) tensor field, ξa global unit vector field (structure vector field), and ηa 1-form on f M. If we have φ2X=−X+η(X)ξ,g(X, ξ) = η(X) and g(φX, φY ) = g(X, Y )−η(X)η(Y), for any X, Y ∈Tf M, then f Mis said to have an almost contact metric structure (φ, ξ, η, g) and it is called an almost contact metric manifold. Let Φ denote the fundamental 2-form in f M, given by Φ(X, Y ) = g(X, φY ) for all X, Y ∈Tf M. If Φ = dη, then f Mis said to be a contact metric manifold. Moreover, the contact metric structure is called aK-contact structure if (1.1) e ∇Xξ=−φX, for any X∈Tf M, where e ∇denotes the Levi–Civita connection of f M. The structure of f Mis said to be normal if [φ, φ] + 2dη⊗ξ= 0, where [φ, φ] is the Nijenhuis torsion of φ. A Sasakian manifold is a normal contact metric manifold. Every Sasakian manifold is a K-contact manifold. It is Existence and uniqueness theorem for slant immersions . . . 561 well-known that an almost contact metric manifold is a Sasakian manifold if and only if (e ∇Xφ)Y=g(X, Y )ξ−η(Y)X, for any X, Y ∈Tf M. Given a Sasakian manifold f M, a plane section πin Tpf Mis called a φ-section if it is spanned by Xand φX, where Xis a unit tangent vector field orthogonal to ξ. The sectional curvature K(π) of a φ-section πis called φ-sectional curvature. If a Sasakian manifold f Mhas constant φsectional curvature c,f Mis called a Sasakian-space-form. It can be shown that R2m+1 with its usual Sasakian structure is a Sasakian-space-form with c=−3. Moreover, if we denote the usual contact metric structure on S2m+1 by (φ, ξ, η, g) and we consider the deformed structure given by the D-homothetic deformation (1.2) φ∗=φ, ξ∗=1 aξ, η∗=aη, g∗=ag +a(a−1)η⊗η, where ais a positive constant, then S2m+1 with this structure is a Sasakianspace-form with c= 4/a −3>−3. Given a simply connected bounded domain Bmin Cmand a negative constant k, a different method can be followed to endow Bm×Rwith a Sasakian structure with constant φsectional curvature c=k−3<−3 (see [1, 10]). Actually, it was proved by S. Tanno in [10] that these three types of model spaces are unique up to isomorphisms, where an isomorphism means a diffeomorphism which maps the structure tensors into the corresponding structure tensors, and so, they represent every Sasakian-space-form. We denote by f M2m+1(c) the complete simply-connected Sasakianspace-form with dimension 2m+ 1 and constant φ-sectional curvature c. The curvature tensor e Rof f M2m+1(c) is given by (1.3) e R(X, Y )Z=c+ 3 4(g(Y, Z)X−g(X, Z)Y) + c−1 4(η(X)η(Z)Y −η(Y)η(Z)X+g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ + Φ(Z, Y )φX −Φ(Z, X)φY + 2Φ(X, Y )φZ), for any X, Y, Z ∈Tf M. For more details and background, we refer to the standard reference [1]. Now, let Mbe a submanifold immersed in (f M, φ, ξ, η, g). We also denote by gthe induced metric on M. Let T M be the Lie algebra of vector fields in Mand T⊥Mthe set of all vector fields normal to M. Denote by ∇ 562 J. L. Cabrerizo et al. the Levi–Civita connection of M. Then, the Gauss–Weingarten formulas are given by e ∇XY=∇XY+σ(X, Y ),e ∇XV=−AVX+DXV, for any X, Y ∈TM and any V∈T⊥M, where Dis the connection in the normal bundle, σis the second fundamental form of Mand AVis the Weingarten endomorphism associated with V. Denote by Rthe curvature tensor of Mand by RDthe curvature tensor of the normal connection D. Then the equation of Gauss and the equation of Ricci are given respectively by e R(X, Y ;Z, W) = R(X, Y ;Z, W ) + g(σ(X, Z), σ(Y, W ))(1.4) −g(σ(X, W), σ(Y, Z)), RD(X, Y ;U, V ) = e R(X, Y ;U, V ) + g([AU, AV](X), Y ),(1.5) for any X, Y, Z, W ∈T M and any U, V ∈T⊥M. For the second fundamental form σ, we define the covariant derivative ∇σof σwith respect to the connection on TM ⊕T⊥Mby (1.6) (∇Xσ)(Y, Z) = DX(σ(Y, Z)) −σ(∇XY, Z)−σ(Y, ∇XZ), for any X, Y, Z ∈TM. The equation of Codazzi is given by (1.7) ( e R(X, Y )Z)⊥= (∇Xσ)(Y, Z)−(∇Yσ)(X, Z), for any X, Y, Z ∈TM, where ( e R(X, Y )Z)⊥denotes the normal component of e R(X, Y )Z. For any X∈TM and any V∈T⊥M, we write (1.8) φX =TX +NX, φV =tV +nV, where TX (resp. tV ) is the tangential component of φX (resp. φV ) and NX (resp. nV ) is the normal component of φX (resp. φV ). From now on, we suppose that the structure vector field ξis tangent to M. Hence, if we denote by Dthe orthogonal distribution to ξin TM, we can consider the orthogonal direct decomposition TM =D ⊕ hξi. Existence and uniqueness theorem for slant immersions . . . 563 In particular, from (1.1), (1.4) and (1.8) we obtain ∇Xξ=−TX and σ(X, ξ) = −NX. For each nonzero vector Xtangent to Mat p, such that Xis not proportional to ξp, we denote by θ(X) the Wirtinger angle of X, that is, the angle between φX and TpM. Then, Mis said to be slant ([7]) if the Wirtinger angle θ(X) is a constant, which is independent of the choice of p∈Mand X∈TpM, linearly independent from ξp. The Wirtinger angle θof a slant immersion is called the slant angle of the immersion. Invariant and anti-invariant immersions are slant immersions with slant angle θ= 0 and θ=π/2 respectively. A slant immersion which is neither invariant nor anti-invariant is called a proper slant immersion. Now, suppose Mis θ-slant in f M2m+1(c). Then, for any X, Y ∈TM, we have (cf. [2]): T2X=−cos2θ(X−η(X)ξ),(1.9) g(TX, Y ) + g(X, TY ) = 0,(1.10) (∇XT)Y=tσ(X, Y ) + ANY X+g(X, Y )ξ−η(Y)X,(1.11) DX(NY )−N(∇XY) = nσ(X, Y )−σ(X, TY ).(1.12) If θ6= 0, we will denote, for each X∈TM, (1.13) X∗=1 sin θNX. We define the symmetric bilinear T M-valued form αon Mgiven by (1.14) α(X, Y ) = tσ(X, Y ), for any X, Y ∈TM. In particular, it is easy to prove that, for any X∈ TM, (1.15) α(X, ξ) = sin2θ(X−η(X)ξ). Equations (1.8), (1.13) and (1.14) imply: (1.16) φα(X, Y ) = Tα(X, Y ) + sin θα∗(X, Y ). Moreover, (1.8) and (1.14) imply (1.17) φσ(X, Y ) = α(X, Y ) + β∗(X, Y ), 564 J. L. Cabrerizo et al. where βis a symmetric bilinear D-valued form on M. From (1.16) and (1.17), we have (1.18) −σ(X, Y ) = Tα(X, Y ) + (sin θ)α∗(X, Y ) + φβ∗(X, Y ), since η(σ(X, Y )) = 0. It is easy to see that: (1.19) φβ∗(X, Y ) = −(sin θ)β(X, Y )−(Tβ(X, Y ))∗. Thus, from (1.18) and (1.19) it follows that β(X, Y ) = (csc θ)· Tα(X, Y ) and σ(X, Y ) = −(csc θ)α∗(X, Y ). This second formula is equivalent to: (1.20) σ(X, Y ) = csc2θ(Tα(X, Y )−φα(X, Y )). Given that g(ANY X, Z) = −g(α(X, Z), Y ) for any X, Y, Z ∈TM, we obtain from (1.11) and (1.14): g((∇XT)Y, Z) = g(α(X, Y ), Z)−g(α(X, Z), Y )(1.21) +g(X, Y )η(Z)−g(X, Z)η(Y). For a θ-slant submanifold in f M2m+1(c) with θ6= 0, (1.3), (1.6), (1.8), (1.9)–(1.12), (1.14) and (1.20) imply that the equations of Gauss (1.4) and Codazzi (1.7) of Min f M2m+1(c) are given respectively by R(X, Y ;Z, W) = csc2θ(g(α(X, W ), α(Y, Z)) −g(α(X, Z), α(Y, W))) +c+ 3 4(g(X, W)g(Y, Z)−g(X, Z)g(Y, W )) +c−1 4(η(X)η(Z)g(Y, W)−η(Y)η(Z)g(X, W)(1.22) +η(Y)η(W)g(X, Z)−η(X)η(W)g(Y, Z) +g(TX, W )g(TY, Z)−g(TX, Z)g(TY, W ) + 2g(X, TY )g(TZ, W)), Existence and uniqueness theorem for slant immersions . . . 565 (∇Xα)(Y, Z)−g(α(Y, Z), T X)ξ + csc2θ{Tα(X, α(Y, Z)) + α(X, T α(Y, Z))} + (sin2θ)c−1 4{g(X, TY )(Z−η(Z)ξ) + g(X, TZ)(Y−η(Y)ξ)}(1.23) = (∇Yα)(X, Z)−g(α(X, Z), TY )ξ + csc2θ{Tα(Y, α(X, Z)) + α(Y, Tα(X, Z))} + (sin2θ)c−1 4{g(Y, T X)(Z−η(Z)ξ) + g(Y, TZ)(X−η(X)ξ)}. In the following section we show how equations (1.9), (1.10), (1.15), (1.21), (1.22) and (1.23) allow us to establish the existence theorem for slant immersions into Sasakian-space-forms. We will also need Theorem 1 of [6] (which was previously proved in [11]). We recall its formulation: Theorem 1.1. Let Sbe a manifold with complete connection Dwith parallel torsion and curvature tensors. Let Mbe a simply connected manifold and Ea vector bundle with connection Dover Mhaving the algebraic structure (R, T)of S. Let F:TM →Ebe a vector bundle homomorphism satisfying equations DVF(W)−DWF(V)−F([V, W ]) = T(F(V), F (W)), DVDWU−DWDVU−D[V,W ]U=R(F(V), F(W))U, for any sections V, W of TM and Uof E. Then there exists a smooth map f:M→Sand a parallel bundle isomorphism Φ : E→f∗TS preserving Tand Rsuch that df = Φ ◦F. If Sis simply connected, then fis unique up to affine diffeomorphisms of S. 2. Existence and uniqueness theorems We have the following existence and uniqueness theorems for slant immersions: Theorem 2.1 (Existence).Let cand θbe two constants with 0< θ ≤ π/2and Ma simply-connected Riemannian manifold with dimension m+1 and metric tensor g. Suppose that there exist a unit global vector field ξ 566 J. L. Cabrerizo et al. on M, an endomorphism Tof the tangent bundle TM and a symmetric bilinear TM-valued form αon Msuch that for X, Y, Z, W ∈TM, we have (2.1) T(ξ) = 0, g(α(X, Y ), ξ) = 0,∇Xξ=−TX, and the equations (1.9), (1.10), (1.15), (1.21), (1.22) and (1.23) are satisfied, where ηdenotes the dual 1-form of ξ. Then, there exists a θ-slant immersion from Minto f M2m+1(c)whose second fundamental form σis given by: (2.2) σ(X, Y ) = csc2θ(Tα(X, Y )−φα(X, Y )). Proof. Let c,θ,M,ξ,Tand αbe in the above conditions. Denote by Dthe orthogonal distribution to ξon Mand consider the Whitney sum TM ⊕ D. For each X∈TM, we identify (X, 0) with X. In particular, we identify b ξ= (ξ, 0) with ξ. Moreover, we denote (0, Z) by Z∗for each Z∈ D. Let bgbe the product metric on TM ⊕D. Hence, if we denote by bηthe dual 1-form of b ξ, then bη(X, Z) = η(X), for any X∈TM and any Z∈ D. Let b φbe the endomorphism on TM ⊕ D defined by (2.3) b φ(X, 0) = (TX, sin θ(X−η(X)ξ)), b φ(0, Z) = (−(sin θ)Z, −TZ), for any X∈TM and Z∈ D. Then, we have b φ2(X, 0) = −(X, 0)+bη(X, 0)b ξ and, similarly, b φ2(0, Z) = −(0, Z). Thus b φ2(X, Z) = −(X, Z) + bη(X, Z)b ξ for any X∈TM and any Z∈ D. By using (1.9), (1.10) and (2.3), it is easy to check that (b φ, bg, b ξ, bη) is an almost contact metric structure on TM ⊕ D. Now we define A,σand Dby AZ∗X= csc θ{(∇XT)Z−α(X, Z)−g(X, Z)ξ},(2.4) σ(X, Y ) = −(csc θ)α∗(X, Y ),(2.5) DXZ∗= (∇XZ−η(∇XZ)ξ)∗ (2.6) + csc2θ{(Tα(X, Z))∗+α∗(X, T Z)}, Existence and uniqueness theorem for slant immersions . . . 567 for any X, Y ∈TM and any Z∈ D. It is easy to verify that each AZ∗is an endomorphism on TM,σis a (D)∗-valued symmetric bilinear form on TM and Dis a metric connection of the vector bundle (D)∗over M. Let b ∇denote the connection on TM ⊕ D induced from equations (2.4)–(2.6). Then, from (1.9), (1.15), (2.1) and (2.3), given X, Y ∈TM and Z∈ D, we have: (b ∇(X,0) b φ)(Y, 0) = bg((X, 0),(Y, 0))b ξ−bη(Y, 0) (X, 0), (b ∇(X,0) b φ)(0, Z) = 0. Let RDdenote the curvature tensor associated with the connection D on (D)∗, i.e. RD(X, Y )Z∗=DXDYZ∗−DYDXZ∗−D[X,Y ]Z∗, for any X, Y ∈TM and any Z∈ D. Then, by virtue of (1.9), (1.10), (1.15), (1.23), (2.1), (2.6) and a simple computation, we may obtain: RD(X, Y )Z∗= (R(X, Y )Z−η(R(X, Y )Z)ξ)∗ +nc−1 4T[g(Y, T Z)X−g(X, TZ)Y−2g(X, TY )Z] +c−1 4[g(Y, T 2Z)(X−η(X)ξ)(2.7) −g(X, T2Z)(Y−η(Y)ξ)−2g(X, T Y )TZ] + csc2θ[(∇XT)α(Y, Z)−(∇YT)α(X, Z)−η(∇XTα(Y, Z))ξ +η(∇YTα(X, Z))ξ−α(X, (∇YT)Z) + α(Y, (∇XT)Z)]o∗ . Also, (1.21), (2.1), (2.4) and (2.5) yield, for any X, Y ∈TM and any Z, W ∈ D: (2.8) sin2θg([AZ∗, AW∗]X, Y ) = g((∇YT)Z, (∇XT)W) −g((∇XT)Z, (∇YT)W) + g((∇XT)Z, α(Y, W )) +g((∇YT)W, α(X, Z)) −g((∇YT)Z, α(X, W)) −g((∇XT)W, α(Y, Z)) +g(α(X, W), α(Y, Z)) −g(α(X, Z), α(Y, W)) + (1 −2 cos2θ)(g(X, W)g(Y, Z)−g(X, Z)g(Y, W)). 574 J. L. 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