A contact version of B.-Y. Chen's inequality and its applications to slant immersions

KYUNGPOOK Math. J. 39(1999), 465-476 A Contact Version of B.-Y. Chen’s Inequality and Its Applications to Slant Immersions Alfonso Carriazo Departamento de Geometr´ıa y Topolog´ıa, Facultad de Matem´aticas. Universidad de Sevilla, Apartado de Correos 1160, 41080-Sevilla, Spain e-mail: [email protected] (1991 Mathematics Subject Classification : 53C15, 53C40) We establish a version of B.-Y. Chen’s inequality for a submanifold of a Sasakianspace-form, tangent to the structure vector field of the ambient space. We obtain some applications and we study this inequality for slant submanifolds. We also characterize 3–dimensional slant submanifolds satisfying the equality case. 1. Introduction Given a Riemannian manifold M, for each point p∈M, put (inf K)(p) = inf{K(π) : plane sections π⊂TpM}, where K(π) denotes the sectional curvature of Massociated with π. Let δM(p) = τ(p)−inf K(p),(1.1) being τthe scalar curvature of M. Then, δMis a well–defined Riemannian invariant, which was recently introduced by B.-Y. Chen [4, 5]. For submanifolds Min a real-space-form e Rm(c) of constant sectional curvature c, Chen gave the following basic inequality involving the intrinsic invariant δMand the squared mean curvature of the immersion, δM≤n2(n−2) 2(n−1) |H|2+1 2(n+ 1)(n−2)c, (1.2) where ndenotes the dimension of Mand His the mean curvature vector. On the other hand, it was remarked in [8] that the exact proof of (1.2) given in [4] yields the same inequality for totally real submanifolds in a complex-space-form f Mm(4c) with constant holomorphic sectional curvature 4c. Later, Chen generalized the above situation by establishing an inequality for an arbitrary submanifold of dimension greater than 2 in a complex-space-form [6]. By (Received : May 14, 1999) The author is partially supported by the PAI project (Junta de Andaluc´ıa, Spain 1998). 465 466 Alfonso Carriazo applying this inequality, he showed that (1.2) holds for arbitrary submanifolds in the complex hyperbolic space CHm(4c) (c < 0) as well. He also stated a formula for a submanifold in the complex projective space CPm(4c). In contact geometry, Defever, Mihai and Verstraelen obtained an inequality similar to (1.2) for C-totally real submanifolds of a Sasakian-space-form with constant ϕ-sectional curvature c[11]: δM≤n2(n−2) 2(n−1) |H|2+1 2(n+ 1)(n−2)c+ 3 4.(1.3) Several authors have studied the equality cases of the above inequalities (see, for instances, [6, 7, 9, 10, 11, 12, 13]). C-totally real submanifolds have the structure vector field ξof the ambient space as a normal vector field (and so, they are anti-invariant submanifolds if that ambient space is, at least, a contact metric manifold). The purpose of the present paper is to establish a general inequality, similar to that of [6], for submanifolds tangent to the structure vector field of a Sasakianspace-form. We are specially interested in applying the obtained results to slant immersions in contact geometry (see, for references, [2, 3, 15]). Thus, in Section 2, we review basic formulas and definitions for almost contact metric manifolds and their submanifolds, which we shall use later. In Section 3, we establish the mentioned inequality and we adapt our procedures to ξ-tangent situation by introducing a new invariant δD M, closely related to δM. Finally, we show some applications in Sections 4 and 5, by paying a special attention to slant immersions. For example, we characterize 3-dimensional slant submanifolds satisfying our equality case. When this paper was finished, the author learned that in [14], Y.H. Kim and D.-S. Kim obtained a basic inequality for δMfor submanifolds in a Sasakian-space- form. Moreover, they apply it to get a characterization of an odd-dimensional great sphere of an odd-dimensional sphere. On the other hand, they do not study slant immersions and so, they do not modify that inequality in order to consider noninvariant submanifolds satisfying the equality case. Hence, even though δD M≤δM, neither our main pinching result given in Theorem 3.5, nor the applications shown in Section 5, can be obtained from Theorem 3.3 of [14]. 2. 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. A contact version of B.-Y. Chen’s inequality and its applications 467 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, if ξis a Killing vector field with respect to g, the contact metric structure is called a K–contact structure. 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. 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 e K(π) of a ϕ–section πis called ϕ–sectional curvature. If a Sasakian manifold f Mhas constant ϕ–sectional curvature c,f Mis called a Sasakianspace-form and it is denoted by f M(c). 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 TM be the Lie algebra of vector fields in M and T⊥Mthe set of all vector fields normal to M. We denote by σthe second fundamental form of Mand by AVthe Weingarten endomorphism associated with any V∈T⊥M. We put σr ij =g(σ(ei, ej), er), for any ei, ej∈T M and er∈T⊥M. The mean curvature vector His defined by H= (1/dim M) trace σ.Mis said to be minimal if Hvanishes identically. From now on, we denote by n+ 1 (resp. m) the dimension of M(resp. f M). We consider n≥2. We also suppose that the structure vector field ξis tangent to M. Hence, if we denote by Dthe orthogonal distribution to ξin T M, we can consider the orthogonal direct decomposition TM =D ⊕ < ξ >. For any X∈TM, we write ϕX =TX +NX, where T X (resp. NX) is the tangential (resp. normal) component of ϕX. If f Mis a K-contact manifold, it is well-known that σ(X, ξ) = −NX, (2.1) for any X∈TM. Given a local orthonormal frame {e1, . . . , en}of D, we can define the squared norms of Tand Nby |T|2= n ∑ i,j=1 g2(ei, Tej),|N|2= n ∑ i=1 |Nei|2,(2.2) respectively. It is easy to show that both |T|2and |N|2are independent of the choice of the above orthonormal frame. The submanifold Mis said to be invariant if Nis identically zero, that is, ϕX ∈TM, for any X∈TM. On the other hand, Mis said to be an anti–invariant submanifold if Tis identically zero, that is, ϕX ∈T⊥M, for any X∈TM. For each nonzero vector Xtangent to Mat p, such that Xis not proportional to ξp, we denote by θ(X) the angle between ϕX and TpM. Then, Mis said to 468 Alfonso Carriazo be slant [15] if the angle θ(X) is a constant, which is independent of the choice of p∈Mand X∈TpM−< ξp>. The 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 not invariant nor anti–invariant is called a proper slant immersion. In [3] we have proved that a θ-slant submanifold Mof an almost contact metric manifold f Msatisfies g(TX, TY ) = cos2θ(g(X, Y )−η(X)η(Y)),(2.3) g(NX, NY ) = sin2θ(g(X, Y )−η(X)η(Y)).(2.4) for any X, Y ∈TM. On the other hand, Lemma 2.3.8 of [2] implies n ∑ j=1 g2(ei, ϕej) = cos2θ, (2.5) for any i= 1, . . . , n, where {e1, . . . , en, ξ}is a local orthonormal frame of TM. It is well-known that the curvature tensor Rof a submanifold Mof a Sasakianspace-form f M(c) satisfies R(X, Y ;Z, W ) = g(σ(X, W ), σ(Y, Z)) −g(σ(X, Z), σ(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) + η(Y)η(W)g(X, Z)−η(X)η(W)g(Y, Z)+ +g(ϕX, W)g(ϕY, Z)−g(ϕX, Z)g(ϕY, W)+2g(X, ϕY )g(ϕZ, W )),(2.6) for any X, Y, Z, W ∈TM. For an orthonormal basis {e1, . . . , en+1}of the tangent space TpM,p∈M, the scalar curvature τat pis defined by τ=∑ i<j K(ei∧ej),(2.7) where K(ei∧ej) denotes the sectional curvature of Massociated with the plane section spanned by ei, ej. In particular, if we put en+1 =ξp, then (2.7) implies: 2τ= n ∑ i=j K(ei∧ej) + 2 n ∑ i=1 K(ei∧ξ).(2.8) From (2.2), (2.6) and (2.8), we obtain the following relation between the scalar curvature and the mean curvature of M, 2τ= (n+ 1)2|H|2− |σ|2+n(n+ 1)c+ 3 4+ 2n+3(c−1) 4|T|2,(2.9) A contact version of B.-Y. Chen’s inequality and its applications 469 where |σ|denotes the norm of the second fundamental form σ. 3. Chen’s inequality in Sasakian-space-forms Let Mn+1 be a submanifold of f Mm(c), tangent to the structure vector field ξ, and π⊂ Dpa plane section at p∈M, orthogonal to ξp. Then, Φ2(π) = g2(e1, ϕe2) (3.1) is a real number in [0,1] which is independent of the choice of the orthonormal basis {e1, e2}of π. Denote by τand K(π) the scalar curvature of Mand the sectional curvature of Massociated with π, respectively. We first recall an algebraic lemma from [4]: Lemma 3.1.Let a1, . . . ,ak, c be k+ 1 (k≥2) real numbers such that: (k ∑ i=1 ai)2 = (k−1) (k ∑ i=1 a2 i+c). Then 2a1a2≥c, with equality holding if and only if a1+a2=a3=· · · =ak. Now, we can prove the following contact version of Theorem 3 of [6]: Theorem 3.2.Let φ:Mn+1 →f Mm(c)be an isometric immersion from a Riemannian (n+ 1)–manifold into a Sasakian-space-form f Mm(c), such that ξ∈TM. Then, for any point p∈Mand any plane section π⊂ Dp, we have: τ−K(π)≤(n+ 1)2(n−1) 2n|H|2+1 2(n+ 1)(n−2)c+ 3 4+ +n+3 2|T|2c−1 4−3Φ2(π)c−1 4.(3.2) Equality in (3.2) holds at p∈Mif and only if there exist an orthonormal basis {e1, . . . , en+1}of TpMand an orthonormal basis {en+2, . . . , em}of T⊥ pMsuch that (a) en+1 =ξp,(b) πis spanned by e1, e2and (c) the shape operators Ar=Aer, r=n+ 2, . . . , m, take the following forms: An+2 = a0 0 0−a0 0 0 0n−1 ,(3.3) Ar= σr 11 σr 12 0 σr 12 −σr 11 0 0 0 0n−1 , r =n+ 3, . . . , m. (3.4) 470 Alfonso Carriazo Proof. Let Mn+1 be a submanifold of f Mm(c). Put: ε= 2τ−(n+ 1)2(n−1) n|H|2−(n+ 1)(n−2)c+ 3 4−2n−3(c−1) 4|T|2.(3.5) Then, (2.9) and (3.5) yield: (n+ 1)2|H|2=n|σ|2+n(ε−2(c+ 3) 4).(3.6) Let π⊂ Dpbe a plane section. We choose an orthonormal frame {e1, . . . , en+1} of TpMand an orthonormal basis {en+2, . . . , em}of T⊥ pMsuch that en+1 =ξp,π is spanned by e1, e2and en+2 is in the direction of the mean curvature vector H. Hence, (3.6) gives (n+1 ∑ i=1 σn+2 ii )2 =n   n+1 ∑ i=1 (σn+2 ii )2+∑ i=j (σn+2 ij )2+ m ∑ r=n+3 ∑ i,j (σr ij)2+ε−2(c+ 3) 4  , and so, by applying Lemma 3.1, we obtain: 2σn+2 11 σn+2 22 ≥∑ i=j (σn+2 ij )2+ m ∑ r=n+3 ∑ i,j (σr ij)2+ε−2(c+ 3) 4.(3.7) On the other hand, from (2.6) we find: K(π) = σn+2 11 σn+2 22 −(σn+2 12 )2+ m ∑ r=n+3 (σr 11σr 22 −(σr 12)2)+ +c+ 3 4+3(c−1) 4g2(e1, ϕe2).(3.8) Then, from (3.7) and (3.8) we get: K(π)≥ m ∑ r=n+2 ∑ j>2 {(σr 1j)2+ (σr 2j)2}+1 2∑ i=j>2 (σn+2 ij )2+1 2 m ∑ r=n+3 ∑ i,j>2 (σr ij)2+ +1 2 m ∑ r=n+3 (σr 11 +σr 22)2+ε 2+3(c−1) 4g2(e1, ϕe2)≥ε 2+3(c−1) 4g2(e1, ϕe2).(3.9) Finally, combining (3.1), (3.5) and (3.9), we obtain (3.2). If the equality in (3.2) holds, then the inequalities in (3.7) and (3.9) become equalities. Thus, we have: σn+2 1j=σn+2 2j=σn+2 ij = 0, i =j > 2; σr 1j=σr 2j=σr ij = 0, r =n+ 3, . . . , m;i, j = 3, . . . , n + 1; σn+3 11 +σn+3 22 =· · · =σm 11 +σm 22 = 0. A contact version of B.-Y. Chen’s inequality and its applications 471 Furthermore, we may choose e1, e2such that σn+2 12 = 0. Moreover, by applying Lemma 3.1 and (2.1), we also have: σn+2 11 +σn+2 22 =σn+2 33 =···=σn+2 n+1 n+1 = 0. Therefore, with respect to the chosen orthonormal basis {e1, . . . , em}, the shape operators of Mtake the forms (3.3) and (3.4). The converse follows from a direct calculation. Now, for each point p∈M, we define: (infDK)(p) = inf{K(π) : plane sections π⊂ Dp}. Then, infDKis a well–defined function on M. Let δD Mdenote the difference between the scalar curvature and infDK, i.e.: δD M(p) = τ(p)−infDK(p).(3.10) From (1.1) and (3.10), it is clear that: δD M≤δM.(3.11) If c= 1, then we obtain directly from (3.2) and (3.10) the following result: Corollary 3.3.Let φ:Mn+1 →f Mm(1) be an isometric immersion from a Riemannian (n+1)-manifold into a Sasakian-space-form with constant ϕ-sectional curvature 1, such that ξ∈T M. Then, we have: δD M≤(n+ 1)2(n−1) 2n|H|2+1 2(n+ 2)(n−1).(3.12) Note that, in fact, (3.12) also follows from (3.11) and (1.2) with c= 1, since a Sasakian-space-form with constant ϕ-sectional curvature 1 is a real-space-form of constant sectional curvature 1. This seems to point out that (3.2) may be a natural contact version of (1.2). Nevertheless, by using (2.1), (3.3) and (3.4), we can state the following result: Corollary 3.4.If equality in (3.2) holds at any p∈M, then φis an invariant immersion. Now, we are going to modify (3.2) in order to consider non-invariant submanifolds (for example, proper slant submanifolds) satisfying a similar equality. We can prove the following theorem: Theorem 3.5.Let φ:Mn+1 →f Mm(c)be an isometric immersion from a Riemannian (n+ 1)–manifold into a Sasakian-space-form f Mm(c), such that ξ∈TM. Then, for any point p∈Mand any plane section π⊂ Dp, we have: τ−K(π)≤(n+ 1)2(n−1) 2n|H|2+1 2(n+ 1)(n−2)c+ 3 4+ 472 Alfonso Carriazo +n+3 2|T|2c−1 4−3Φ2(π)c−1 4− |N|2.(3.13) Equality in (3.13) holds at p∈Mif and only if there exist an orthonormal basis {e1, . . . , en+1}of TpMand an orthonormal basis {en+2, . . . , em}of T⊥ pMsuch that (a) en+1 =ξp,(b) πis spanned by e1, e2and (c) the shape operators Ar=Aer, r=n+ 2, . . . , m, take the following forms: An+2 =     a0 0 µn+2 1 0−a0. . . 0 0 0n−2µn+2 n µn+2 1· · · µn+2 n0     ,(3.14) Ar=     σr 11 σr 12 0µr 1 σr 12 −σr 11 0. . . 0 0 0n−2µr n µr 1· · · µr n0     , r =n+ 3, . . . , m, (3.15) where µr i=g(ϕei, er), for any i= 1, . . . , n;r=n+ 2, . . . , m. Proof. We follow the first steps of the proof of Theorem 3.2 and we state equations (3.5)-(3.9). Then, inequality (3.9) can now be written as: K(π)≥ m ∑ r=n+2 n ∑ j=3 {(σr 1j)2+ (σr 2j)2}+1 2 n ∑ i=j>2 (σn+2 ij )2+1 2 m ∑ r=n+3 n ∑ i,j=3 (σr ij)2+ +1 2 m ∑ r=n+3 (σr 11 +σr 22)2+ε 2+3(c−1) 4g2(e1, ϕe2) + m ∑ r=n+2 n ∑ i=1 (σr i n+1)2≥ ≥ε 2+3(c−1) 4g2(e1, ϕe2) + m ∑ r=n+2 n ∑ i=1 (σr i n+1)2.(3.16) But, from (2.1) and (2.2) we find: m ∑ r=n+2 n ∑ i=1 (σr i n+1)2=|N|2.(3.17) Hence, combining (3.1), (3.5), (3.16) and (3.17), we obtain (3.13). If the equality in (3.13) holds, then the inequalities in (3.7) and (3.16) become equalities. By using this fact, (2.1) and Lemma 3.1, we have: σm+2 1j=σn+2 2j=σn+2 ij = 0,2< i =j < n; σr 1j=σr 2j=σr ij = 0, r =n+ 3, . . . , m;i, j = 3, . . . , n; σn+3 11 +σn+3 22 =···=σm 11 +σm 22 = 0; σn+2 11 +σn+2 22 =σn+2 33 =· · · =σn+2 n+1 n+1 = 0. A contact version of B.-Y. Chen’s inequality and its applications 473 Hence, if we also choose e1, e2such that σn+2 12 = 0, then we obtain (3.14) and (3.15). As in the proof of Theorem 3.2, the converse can be verified by straight-forward computation. Moreover, it is obvious that (3.2) follows from (3.13), since |N|2≥0. On the other hand, it is also clear that, if φis an anti-invariant immersion, then |T|2= 0, |N|2=nand Φ2(π) = 0, for any plane section πorthogonal to ξ. Hence, from (3.13) we obtain: Corollary 3.6.Let Mn+1 be an anti-invariant submanifold of a Sasakian-space- form f Mm(c), such that ξ∈TM. Then, we have: δD M≤(n+ 1)2(n−1) 2n|H|2+1 2(n+ 1)(n−2)c+ 3 4.(3.18) Note that inequality (3.18) is the ξ-tangent version of (1.3), with the logical differences about the dimensions. 4. Some applications By using Theorem 3.5, we can find some general pinching results for δD Mif either c > 1 or c < 1. Theorem 4.1.Let φ:Mn+1 →f Mm(c)be an isometric immersion from a Riemannian (n+ 1)–manifold (n > 2) into a Sasakian-space-form f Mm(c), with c > 1, such that ξ∈TM. Then: δD M≤(n+ 1)2(n−1) 2n|H|2+1 2(n2+ 2n−2)c+ 3 4−n 2.(4.1) Equality in (4.1) holds identically if and only if nis even and Mn+1 is immersed as an invariant, totally geodesic submanifold of f Mm(c). Theorem 4.2.Let φ:Mn+1 →f Mm(c)be an isometric immersion from a Riemannian (n+ 1)–manifold (n > 2) into a Sasakian-space-form f Mm(c), with c < 1, such that ξ∈TM. Then: δD M≤(n+ 1)2(n−1) 2n|H|2+1 2(n+ 1)(n−2)c+ 3 4+n− |N|2.(4.2) Equality in (4.2) holds at a point pof Mif and only if there exist an orthonormal basis {e1, . . . , en+1}of TpMand an orthonormal basis {en+2, . . . , em}of T⊥ pMsuch that (a) en+1 =ξp,(b) the subspace spanned by e3, . . . , en+1 is anti-invariant, (c) K(e1∧e2) = infDKat p, and (d) the shape operators Ar=Aer,r=n+ 2, . . . , m, take the forms (3.14) and (3.15). Theorems 4.1 and 4.2 can be proved by following the same steps as in the proofs of Theorems 3 and 4 of [6], respectively.