The contact Whitney sphere

Note di Matematica 20, n. 2, 2000/2001, 125–133. The contact Whitney sphere David E. Blair Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, USA [email protected] Alfonso Carriazo∗ Departamento de Geometr´ıa y Topolog´ıa, Facultad de Matem´aticas, Universidad de Sevilla, Apdo. Correos 1160, 41080–Sevilla, Spain [email protected] Received: 2 March 2001; accepted: 27 March 2001. Abstract. In this paper, we introduce the contact Whitney sphere as an imbedding of the n–dimensional unit sphere as an integral submanifold of the standard contact structure on R2n+1. We obtain a general inequality for integral submanifolds in R2n+1, involving both the scalar curvature and the mean curvature, and we use the equality case in order to characterize the contact Whitney sphere. We also study a similar problem for anti-invariant submanifolds of R2n+1, tangent to the structure vector field. Keywords: Integral submanifolds, Whitney spheres, scalar curvature, mean curvature MSC 2000 classification: 53C15, 53C40, 53D10. Introduction One of the most interesting topics in contact geometry is the study of integral submanifolds, i.e., submanifolds immersed in a contact manifold, such that the contact form restricted to the submanifold vanishes. In particular, 1–dimensional integral submanifolds are called Legendre curves. A well-known property of Legendre curves of the standard contact structure dz−ydxon R3is that the projection ¯γof a closed Legendre curve γin R3 to the xy-plane must have self-intersections and algebraic (signed) area zero. On the other hand, we can think of the pair of γand its projection ¯γin the following terms. Suppose that γitself does not have self-intersections and regard ¯γas a Lagrangian submanifold in C∼ =R2with self-intersections; then think of going from ¯γto γas a way of removing the singularity but preserving the “Lagrangian-Legendre” property. A generalization of this can be done with the Whitney spheres. The Whitney spheres are usually defined as a family of Lagrangian immersions of the unit ∗Partially supported by the PAI project (Junta de Andaluc´ıa, Spain 1999) 126 D. E. Blair, A. Carriazo sphere Sn, centered at the origin of Rn+1,intoCn∼ =R2ngiven by (u0,u 1,...,u n)→r 1+u2 0 (u1,...,u n,u 0u1,...,u 0un)+B, (1) where ris a positive number and Bis a vector of Cn. The number rand the vector Bare called the radius and the center of the Whitney sphere, respectively. From a topological point of view, it is well-known that the sphere can not be imbedded in Cnas a Lagrangian submanifold. The Whitney spheres have the best possible behaviour, because they have only one double point at the poles of Sn. In the contact manifold R2n+1 with its usual contact metric structure, we have the following presentation of the Whitney spheres as a family of imbedded spheres and integral submanifolds of the contact structure (u0,u 1,...,u n)→r 1+u2 0 (u0u1,...,u 0un,u 1,...,u n,ru0 1+u2 0 +C(1 + u2 0)) + B, where ris a positive number, Bis a vector of R2n+1 and Cis a real constant. We refer to these spheres as the contact Whitney spheres. Therefore, with this presentation we have removed the previous singularity at the poles of Sn. On the other hand, the Whitney spheres in Cnhave an interesting geometric property. It was proven by Borrelli, Chen and Morvan [3] and independently by Ros and Urbano [6] that if Mnis a Lagrangian submanifold of Cn,with mean curvature vector Hand scalar curvature τ, then |H|2≥2(n+2) n2(n−1)τ,with equality if and only if Mis either totally geodesic or a (piece of a) Whitney sphere. Moreover, in [4], Castro constructs a one-parameter family of Lagrangian spheres including the Whitney sphere, such that they satisfy a geometric equality of type τ=µ|H|2,withµ>0. In this paper, we establish an analogue of the above result for integral submanifolds in R2n+1 with its standard Sasakian structure, such that the equality case holds for the contact Whitney spheres. We also give a characterization by the second fundamental form, similar to that of Ros and Urbano in [6]. Finally, we study the same problem for anti-invariant submanifolds tangent to the structure vector field on R2n+1 and obtain the corresponding results. 1 Preliminaries. Let (R2n+1,φ,ξ,η,g) denote the manifold R2n+1 with its usual Sasakian structure given by η=1 2(dz − n  i=1 yidxi),ξ=2 ∂ ∂z, The contact Whitney sphere 127 g=η⊗η+1 4 n  i=1 (dxi⊗dxi+dyi⊗dyi), φ( n  i=1 (Xi ∂ ∂xi +Yi ∂ ∂yi )+Z∂ ∂z)= n  i=1 (Yi ∂ ∂xi −Xi ∂ ∂yi )+ n  i=1 Yiyi ∂ ∂z, where (xi,y i,z),i=1...n are the cartesian coordinates. It is well-known that (R2n+1,φ,ξ,η,g) is a Sasakian-space-form, with constant φ–sectional curvature equal to −3. Hence, its curvature tensor  Ris given by  R(X,Y )Z=−η(X)η(Z)Y+η(Y)η(Z)X−g(X,Z)η(Y)ξ+ +g(Y,Z)η(X)ξ−g(Z, φY )φX +g(Z, φX)φY −2g(X,φY )φZ, (2) for any vector fields X,Y, Z. For more details and background, we refer to [2]. Let Mbe an n-dimensional Riemannian manifold isometrically immersed in the Sasakian-space-form R2n+1. We also denote by gthe metric on M. Let ∇(resp.  ∇) be the Levi–Civita connection of M(resp. R2n+1). Then, the Gauss–Weingarten formulas are given by  ∇XY=∇XY+σ(X,Y ), ∇XV=−AVX+DXV, for any tangent vector fields Xand Yand any normal vector field V, where Dis the connection in the normal bundle, σis the second fundamental form of Mand Ais the shape operator. The mean curvature vector His defined by H=(1/n) trace σ. A submanifold Mis called an integral submanifold if ηrestricted to M vanishes. It is well-known that the contact subbundle {η=0}admits integral submanifolds up to and including dimension n, but of no higher dimension, see e.g. [2]. A direct consequence of this definition is that φX is a normal vector field, for any tangent vector field X, i.e., Mis an anti-invariant submanifold of R2n+1. Hence, in a neighborhood of every point p∈M,wecanconsideralocal orthonormal frame {e1,...,e n,e 1∗,...,e n∗,ξ}, such that e1,...,e nare tangent to Mand ei∗=φei, for any i=1,...,n. Such a frame is called a Legendre frame. If we put σk ij =g(σ(ei,e j),e k∗), it can be proved that σi jk =σk ji =σj ik,(3) for any i, j, k =1,...,n. On the other hand, we have [2, p. 128] Aξ=0,(4) equivalently σ(X, Y ) is perpendicular to ξ, for any tangent vector fields Xand Y. 128 D. E. Blair, A. Carriazo 2 Characterizing the Contact Whitney Sphere Let Mbe an n–dimensional submanifold of R2n+1. Given a tangent orthonormal frame {e1,...,e n}, the scalar curvature τof Mis defined by τ= i<j K(ei∧ej), where K(ei∧ej) is the sectional curvature of the plane section spanned by ei and ej. By following the same steps as in the proof of Lemma 1 of [3], and by virtue of (2), (3) and (4), we obtain the following inequality for integral submanifolds, involving both curvatures |H|and τ: Proposition 1. Let Mnbe an integral submanifold of R2n+1. Then, the square of its mean curvature |H|2and its scalar curvature τsatisfy at each point the following inequality: |H|2≥2(n+2) n2(n−1)τ. (5) The equality holds if and only if there exists a real function λdefined on M, such that the second fundamental form σof Msatisfies σ(e1,e 1)=3λe1∗,σ(e2,e 2)=... =σ(en,e n)=λe1∗, σ(e1,e j)=λej∗,σ(ej,e k)=0,2≤j=k≤n, (6) where {e1,...,e n,e 1∗,...,e n∗,ξ}is a Legendre frame such that e1∗is parallel to H. On the other hand, we can also characterize the equality case of the above inequality by the behavior of the second fundamental form of the submanifold. We obtain a formula similar to that of [6]: Proposition 2. Let Mnbe an integral submanifold of R2n+1. Then, M satisfies the equality case of (5) at every point, if and only if σ(X,Y )= n n+2{g(X, Y )H+g(φX, H)φY +g(φY, H)φX},(7) for any tangent vector fields Xand Y. Proof. If the equality case of (5) holds for every p∈M, then, in a neighborhood of every point, we can find a Legendre frame with e1∗parallel to Hand such that it satisfies (6), with λ=n n+2|H|. Hence, it follows that (7) holds for any tangent vector fields Xand Y. The converse can be verified directly. QED The contact Whitney sphere 129 We now proceed to show that the only non-trivial example of an integral submanifold satisfying the equality in (5), is the contact Whitney sphere. First, we state the following two lemmas. The first one can be easily proved by straightforward computation. Lemma 1. Let π:R2n+1 →Cnbe the differential map given by: π(x1,...,x n,y 1,...,y n,z)=1 2(y1,...,y n,x 1,...,x n).(8) Then, π:(R2n+1,φ,ξ,η,g)→(Cn,J,G)is a Riemannian submersion, where we denote by (Cn,J,G)the usual Kaehlerian structure on Cn. Moreover, it satisfies the following conditions: i) The vertical subspace Vpof the submersion at p∈R2n+1 is equal to the span of ξp; ii) g=π∗G+η⊗η; iii) φX =(Jπ∗X)∗, for any vector field Xon R2n+1, where ∗denotes the horizontal lift with respect to η. Lemma 2 (Uniqueness of the lift of the Whitney sphere). Let π: R2n+1 →Cnbe the Riemannian submersion given by (8),andw:Sn→Cna Whitney immersion given by (1).Ifψ:Sn→R2n+1 is an integral immersion, such that π◦ψ=w, then ψis the contact Whitney immersion given by ψ(u0,u 1,...,u n)= 2r 1+u2 0 (u0u1,...,u 0un,u 1,...,u n,2ru0 1+u2 0 +C(1+u2 0))+ ˜ B, where Cis a constant and ˜ BisavectorofR2n+1 such that π(˜ B)=B. Proof. Since π◦ψ=wand n i=0 u2 i= 1, we can write ψas ψ(u0,u 1,...,u n)= 2r 1+u2 0 (u0u1,...,u 0un,u 1,...,u n,f(u1,...,u n))+ ˜ B, (9) for any constant vector ˜ Bsuch that π(˜ B)=B. Then, differentiating (9), we obtain, for any i=1,...,n: ∂ ∂ui =2r u2 0(1 + u2 0)−u2 i(1 −u2 0) u0(1 + u2 0)2 ∂ ∂xi − j=i uiuj(1 −u2 0) u0(1 + u2 0)2 ∂ ∂xj + +1+u2 0+2u2 i (1 + u2 0)2 ∂ ∂yi +2 j=i uiuj (1 + u2 0)2 ∂ ∂yj + 2uif+(1+u2 0)∂f ∂ui (1 + u2 0)2 ∂ ∂z    .(10) 130 D. E. Blair, A. Carriazo Since ψis an integral immersion in R2n+1,wehaveη(∂ ∂ui)=0,forany i=1,...,n, where ηis the contact form on R2n+1. Hence, it follows from (10) that the function fmust satisfy the system of partial differential equations ∂ ∂ui (f 1+u2 0 )=2rui(3u2 0−1) u0(1 + u2 0)3,i=1,...,n, which implies that f=2ru0 1+u2 0 +C(1 + u2 0), where Cis a real constant. QED We can now state the main theorem: Theorem 1. Let Mnbe an integral submanifold of the standard contact structure on R2n+1. Then, the equality case of (5) holds at every point p∈M, if and only if either Mis a totally geodesic submanifold or it is a portion of a contact Whitney sphere. Proof. Let ψ:M→R2n+1 be an integral immersion and put  ψ=π◦ψ. Then, it follows from Lemma 1 that  ψ:M→Cnis a Lagrangian immersion, and that the metrics induced on Mby both ψand  ψagree. Denote by σand ∇the second fundamental form of  ψand the Levi–Civita connection of Cn, respectively. Then, it follows from the well-known O’Neill equations [5], that  ∇X∗Y∗=(∇XY)∗+1 2η([X∗,Y∗])ξ for any vector fields X,Y on Cntangent to M.SinceMis normal to ξ,wehave η([X∗,Y∗]) = 0, and therefore σ(X,Y )=(σ(X,Y ))∗and  H∗=H, (11) where  His the mean curvature vector of  ψ. Suppose that the equality case of (5) holds for any p∈M. Then, Proposition 2 implies that σsatisfies (7). Hence, by virtue of (11), we have that σ(X,Y )= n n+2G(X, Y ) H+G(JX,  H)JY +G(JY,  H)JX for any vector fields X, Y on Cntangent to M. Therefore, it follows from [6, Theorem 2], that either  ψis totally geodesic or  ψ(M) is an open portion of a Whitney sphere. If the first case holds, (11) implies that ψis also totally geodesic. On the other hand, if  ψ(M) is a portion of a Whitney sphere, Lemma 2 implies that ψ(M) must be a portion of a corresponding contact Whitney sphere. The converse can be proved by straightforward computation. QED The contact Whitney sphere 131 3 Anti-invariant Submanifolds. We now consider an (n+ 1)–dimensional anti-invariant submanifold Mn+1 of R2n+1; such a submanifold is tangent to the structure vector field ξ. In particular, from the Sasakian condition on R2n+1, ∇Xξ=−φX, and therefore σ(X,ξ)=−φX, (12) for any vector field Xtangent to M. In this case, it is possible to state an inequality similar to (5): Proposition 3. Let Mn+1 be an (n+1)–dimensional anti-invariant submanifold of R2n+1. Then, the square of its mean curvature |H|2and its scalar curvature τsatisfy at each point the following inequality: |H|2≥2(n+2) (n+1) 2(n−1)τ. (13) The equality holds if and only if there exists a real function λdefined on M, such that the second fundamental form σof Msatisfies σ(e1,e 1)=3λe1∗,σ(e2,e 2)=... =σ(en,e n)=λe1∗, σ(e1,e j)=λej∗,σ(ej,e k)=0,2≤j=k≤n, where {e1,...,e n,ξ,e 1∗,...,e n∗}is a local orthonormal frame such that e1,..., en,ξare tangent to M,ei∗=φei, for any i=1,...,n,ande1∗is parallel to H. We can also describe the second fundamental form of the submanifolds satisfying the equality case of (13) at every point: Proposition 4. Let Mn+1 be an (n+1)–dimensional anti-invariant submanifold of R2n+1. Then, Msatisfies the equality case of (13) at every point, ifandonlyif σ(X,Y )=n+1 n+2{(g(X, Y )−η(X)η(Y))H+ +(g(φX, H)−n+2 n+1η(X))φY +(g(φY, H)−n+2 n+1η(Y))φX},(14) for any tangent vector fields Xand Y. A submanifold Mof R2n+1, tangent to ξ,issaidtobetotally contact geodesic [1, p.110] if σ(X,Y )=η(X)σ(Y,ξ)+η(Y)σ(X,ξ),(15) for any tangent vector fields Xand Y. It is clear that every totally contact geodesic submanifold is minimal. Then, it follows from (12), (14) and (15) that 132 D. E. Blair, A. Carriazo every anti-invariant totally contact geodesic submanifold satisfies the equality case of (13). With an additional condition, we can characterize the non-trivial anti-invariant submanifolds satisfying that equality. Let π:R2n+1 →Cnbe the Riemannian submersion given by (8), and suppose that there exists a Lagrangian submanifold Nof Cn, such that the following diagram commutes M−→ R2n+1 ↓↓π N−→ Cn, where Mis the set of fibres over N. Now, we state the following theorem: Theorem 2. Under the above conditions, the equality case of (13) holds for any p∈M, if and only if either Mis a totally contact geodesic submanifold or is locally isometric to the Riemannian product of a portion of a Whitney sphere and R. Proof. As in the proof of Theorem 1, it follows from the O’Neill equations of the Riemannian submersion πthat σ(X∗,Y∗)=(σ(X, Y ))∗and  H∗=n+1 nH, (16) where σand  Hare the second fundamental form and the mean curvature vector of N, respectively. On the other hand, we also know that η(∇X∗Y∗)=−g(∇X∗ξ,Y ∗)=g(φX∗,Y∗)=0, since Mis anti-invariant. This implies that Mis locally isometric to the Riemannian product of Nand R. Suppose now that the equality case of (13) holds for any p∈M. Then, Proposition 4 implies that σsatisfies (14). Hence, it follows from (11) that σ satisfies the equation of [6, Theorem 2], and so, either Nis totally geodesic or is a portion of a Whitney sphere. Finally, it only remains to remark that, if Nis totally geodesic, then, we obtain from (16) that σ(X,Y ) = 0, for any tangent vector fields Xand Y, perpendicular to ξ, and therefore, Mis totally contact geodesic. The converse can be proved by straightforward computation. QED Acknowledgements. This article was written while the second author was visiting Michigan State University in 1999. He would like to express his deep appreciation to professors D.E. Blair and B.-Y. Chen for the warm hospitality he received during his visit. The contact Whitney sphere 133 References [1] A. Bejancu: Geometry of CR-Submanifolds, Mathematics and Its Applications, D. Reidel Publishing Company, Dordrecht, 1986. [2] D. E. Blair: Riemannian Geometry of Contact and Symplectic Manifolds, Birkh¨auser Boston, 2002. [3] V.Borrelli,B.Y.Chen,J.M.MorvanUne caract´erisation g´eom´etrique de la sph`ere de Whitney,C.R.Acad.Sci.Paris,S´erie I 321 (1995), 1485–1490. [4] I. Castro:Lagrangian spheres in the complex Euclidean space satisfying a geometric equality, Geom. Dedicata, 70 n. 2 (1998), 197–208. [5] B. O’Neill:The fundamental equations of a submersion, Michigan Math. J., 13 (1966), 459–469. [6] A. Ros, F. Urbano:Lagrangian submanifolds of Cnwith conformal Maslov form and the Whitney sphere, J. Math. Soc. Japan, 50 n. 1 (1998), 203–226.