New characterizations of the Whitney spheres and the contact Whitney spheres
aa r X i v : . [ m a t h . DG ] F e b NEW CHARACTERIZATIONS OF THE WHITNEY SPHERESAND THE CONTACT WHITNEY SPHERES
ZEJUN HU AND CHENG XING
Abstract.
In this paper, based on the classical K. Yano’s formula, we firstestablish an optimal integral inequality for compact Lagrangian submanifoldsin the complex space forms, which involves the Ricci curvature in the direction
J ~H and the norm of the covariant differentiation of the second fundamentalform h , where J is the almost complex structure and ~H is the mean curvaturevector field. Second and analogously, for compact Legendrian submanifolds inthe Sasakian space forms with Sasakian structure ( ϕ, ξ, η, g ), we also establishan optimal integral inequality involving the Ricci curvature in the direction ϕ ~H and the norm of the modified covariant differentiation of the second fundamen-tal form. The integral inequality is optimal in the sense that all submanifoldsattaining the equality are completely classified. As direct consequences, weobtain new and global characterizations for the Whitney spheres in complexspace forms as well as the contact Whitney spheres in Sasakian space forms.Finally, we show that, just as the Whitney spheres in complex space forms,the contact Whitney spheres in Sasakian space forms are locally conformallyflat manifolds with sectional curvatures non-constant. Introduction
In this paper, we consider compact Lagrangian submanifolds in the n -dimensionalcomplex space form N n (4 c ) of constant holomorphic sectional curvature 4 c , c ∈{ , , − } ; and analogously, we also consider compact Legendrian submanifolds inthe (2 n + 1)-dimensional Sasakian space form ˜ N n +1 (˜ c ) with constant ϕ -sectionalcurvature ˜ c . As our main achievements, we shall establish an integral inequalityfor either class of such submanifolds. Then, as direct consequences, we can get newand integral characterizations for the Whitney spheres in the complex space formsand also the contact Whitney spheres in the Sasakian space forms.Recall that the complex space form N n (4 c ) with almost complex structure J andRiemannian metric g is the complex Euclidean space C n for c = 0, the complexprojective space C P n (4) for c = 1, and the complex hyperbolic space C H n ( − c = −
1. Let M n ֒ → N n (4 c ) be a Lagrangian immersion of an n -dimensionaldifferentiable manifold M n ( n ≥ J carries each tangent space of M n into itscorresponding normal space. In order to state our first main result, we shall recallthe notion of Whitney spheres in each complex space form.
Example 1.1 : Whitney spheres in C n (cf. [2, 5, 7, 15, 17, 18]). Key words and phrases.
Complex space form, Sasakian space form, Whitney sphere, contactWhitney sphere, integral inequality.2020
Mathematics Subject Classification.
Primary 53C24; Secondary 53C25, 53D12.This project was supported by NSF of China, Grant Number 11771404.
As the most classical notion of
Whitney spheres , these are usually defined as afamily of Lagrangian immersions from the unit sphere S n , centered at the origin O of R n +1 , into the complex Euclidean space C n ∼ = R n , given by Ψ r,B : S n → C n with(1.1) Ψ r,B ( u , . . . , u n +1 ) = r u n +1 ( u , u u n +1 , . . . , u n , u n u n +1 ) + B, where r is a positive number and B is a vector of C n . The number r and thevector B are called the radius and the center of the Whitney spheres, respectively.Up to translation and scaling of C n , all the Whitney spheres are congruent withthe standard one corresponding to r = 1 and B = O . According to Gromov [12],the sphere cannot be embedded into C n as a Lagrangian submanifold. This factimplies that the Whitney spheres in (1.1) have the best possible behavior, becauseit is embedded except at the poles of S n where it has a double points. Indeed, in acertain sense, the Whitney spheres in C n play the role of umbilical hypersurfaces ofthe Euclidean space R n +1 inside the family of Lagrangian submanifolds and havebeen characterized in several ways as done for the Euclidean spheres (cf. [17]). Example 1.2 : Whitney spheres in C P n (4) (cf. [6, 8, 10, 15]).In this case, the Whitney spheres are a one-parameter family of Lagrangiansphere immersions into C P n (4), given by Ψ θ : S n → C P n (4) for θ > θ ( u , . . . , u n +1 ) = Π (cid:16) ( u ,...,u n )cosh θ + i sinh θu n +1 ; sinh θ cosh θ (1+ u n +1 )+ iu n +1 cosh θ +sinh θu n +1 (cid:17) , where Π : S n +1 → C P n (4) is the Hopf projection. We notice that Ψ θ are embed-dings except at the poles of S n where it has a double points, and that Ψ is thetotally geodesic Lagrangian immersion of S n into C P n (4). Example 1.3 : Whitney spheres in C H n ( −
4) (cf. [6, 8, 10, 15]).Let ( · , · ) denote the hermitian form of C n +1 , i.e., ( z, w ) = n P i =1 z i ¯ w i − z n +1 ¯ w n +1 for z, w ∈ C n +1 , and H n +11 ( −
1) = { z ∈ C n +1 : ( z, z ) = − } be the Anti-de Sitterspace of constant sectional curvature −
1. Then, the
Whitney spheres in C H n ( − C H n ( − θ : S n → C H n ( −
4) for θ > θ ( u , . . . , u n +1 ) = Π (cid:16) ( u ,...,u n )sinh θ + i cosh θu n +1 ; sinh θ cosh θ (1+ u n +1 ) − iu n +1 sinh θ +cosh θu n +1 (cid:17) , where Π : H n +11 ( − → C H n ( −
4) is the Hopf projection. We also notice that Φ θ are embeddings except in double points.The remarkable properties of the Whitney spheres are summarized as follows: Theorem 1.1 (cf. [2, 3, 6, 7, 8, 10, 15, 17]) . Let x : M n → N n (4 c ) be an n -dimensional compact Lagrangian submanifold that is neither totally geodesic nor ofparallel mean curvature vector field. Then, x ( M n ) is the Whitney sphere in N n (4 c ) if and only if one of the following pointwise relations holds: (1) The squared mean curvature | ~H | and the scalar curvature R of M n satisfythe relation | ~H | = n +2 n ( n − R − n +2 n c ; (2) The second fundamental form h and the mean curvature vector field ~H of M n satisfy h ( X, Y ) = nn +2 (cid:2) g ( X, Y ) ~H + g ( JX, ~H ) JY + g ( JY, ~H ) JX (cid:3) for X, Y ∈ T M n ; HARACTERIZATIONS OF THE WHITNEY AND CONTACT WHITNEY SPHERES 3 (3)
The second fundamental form h and the mean curvature vector field ~H of M n satisfy k ¯ ∇ h k = n n +2 k∇ ⊥ ~H k . Here, ¯ ∇ h denotes the covariant differentiation of h with respect to the van der Waerden-Bortolotti connection of x : M n → N n (4 c ) . Moreover, Castro-Montealegre-Urbano [6] and Ros-Urbano [17] further provedthat the Whitney spheres in N n (4 c ) can be characterized by some other relationsabout the global geometric and topological invariants.As the first main result of this paper, we have obtained an optimal integralinequality that involves the Ricci curvature Ric ( J ~H, J ~H ) in the direction
J ~H andthe norm of the covariant differentiation ¯ ∇ h of the second fundamental form: Theorem 1.2.
Let x : M n → N n (4 c ) ( n ≥ be an n -dimensional compactLagrangian submanifold. Then, it holds that (1.4) Z M n Ric (
J ~H, J ~H ) dV M n ≤ ( n − n +2)3 n Z M n k ¯ ∇ h k dV M n , where k · k and dV M n denote the tensorial norm and the volume element of M n with respect to the induced metric, respectively.Moreover, the equality in (1.4) holds if and only if either x ( M n ) is of parallelsecond fundamental form, or it is one of the Whitney spheres in N n (4 c ) . Remark 1.1.
The classification of Lagrangian submanifolds with parallel secondfundamental form in N n (4 c ) has been fulfilled for each c , see [11, 14] for details. From Theorem 1.2, we get a new and global geometric characterization of theWhitney spheres in N n (4 c ): Corollary 1.1.
Let x : M n → N n ( c ) ( n ≥ be an n -dimensional compact La-grangian submanifold with non-parallel mean curvature vector field. Then, (1.5) Z M n Ric (
J ~H, J ~H ) dV M n = ( n − n +2)3 n Z M n k ¯ ∇ h k dV M n holds if and only if x ( M n ) is a Whitney sphere in N n ( c ) . Next, before stating our second main result, we shall first review the standardmodels of the Sasakian space form ˜ N n +1 (˜ c ) with Sasakian structure ( ϕ, ξ, η, g )possessing constant ϕ -sectional curvature ˜ c , then for each value ˜ c we introduce thecanonical Legendrian (i.e., the n -dimensional C -totally real , or equivalenly, integral )submanifolds: The contact Whitney spheres in ˜ N n +1 (˜ c ). Example 1.4 : Contact Whitney spheres in ˜ N n +1 ( −
3) = ( R n +1 , ϕ, ξ, η, g ).Here, for the Cartesian coordinates ( x , . . . , x n , y , . . . , y n , z ) of R n +1 , ξ = 2 ∂∂z , η = (cid:16) dz − n X i =1 y i dx i (cid:17) , g = η ⊗ η + n X i =1 ( dx i ⊗ dx i + dy i ⊗ dy i ) ,ϕ (cid:16) n X i =1 (cid:0) X i ∂∂x i + Y i ∂∂y i (cid:1) + Z ∂∂z (cid:17) = n X i =1 (cid:0) Y i ∂∂x i − X i ∂∂y i (cid:1) + n X i =1 Y i y i ∂∂z , define the standard Sasakian structure ( ϕ, ξ, η, g ) on R n +1 .As were introduced by Blair and Carriazo in [1], the contact Whitney spheres in˜ N n +1 ( −
3) were the Legendrian imbeddings ˜Ψ
B,a,r : S n → R n +1 defined by(1.6) ˜Ψ B,a,r ( u , u , . . . , u n ) = r u (cid:0) u u , . . . , u u n , u , . . . , ru u + a (1 + u ) (cid:1) + B, ZEJUN HU AND CHENG XING where r is a positive number, a is a real constant and B is a vector of R n +1 . Example 1.5 : Contact Whitney spheres in ˜ N n +1 (˜ c ) = ( S n +1 , ϕ, ξ, η, g ) with˜ c > −
3. Note that the unit sphere S n +1 , as a real hypersurface of the complexEuclidean space C n +1 , has a natural Sasakian structure ( ¯ ϕ, ¯ ξ, ¯ η, ¯ g ): ¯ g is the inducedmetric; ¯ ξ = JN , where J is the natural complex structure of C n +1 and N is theunit normal vector field of the inclusion S n +1 ֒ → C n +1 ; ¯ η ( X ) = ¯ g ( X, ¯ ξ ) and¯ ϕ ( X ) = JX − h JX, N i N for any tangent vector field X of S n +1 , where h· , ·i denotes the standard Hermitian metric on C n +1 . Then, the standard Sasakianstructure ( ϕ, ξ, η, g ) on S n +1 is given by applying a D a -homothetic deformation asfollows: η = a ¯ η, ξ = a ¯ ξ, ϕ = ¯ ϕ, g = a ¯ g + a ( a − η ⊗ ¯ η, where a is a positive real number and ˜ c = a − N n +1 (˜ c ) for˜ c > − θ : S n → S n +1 for θ >
0, that areexplicitly given by(1.7) ˜Ψ θ ( u , u , . . . , u n +1 ) = (cid:16) ( u ,...,u n )cosh θ + i sinh θu n +1 ; sinh θ cosh θ (1+ u n +1 )+ iu n +1 cosh θ +sinh θu n +1 (cid:17) . Example 1.6 : Contact Whitney spheres in ˜ N n +1 (˜ c ) = ( B n × R , ϕ, ξ, η, g )with ˜ c < −
3. Here, B n = { ( z , . . . , z n ) ∈ C n ; k z k = n P i =1 | z i | < } equipped withthe usual complex structure and the canonical Bergman metric˜ g = 4 n −k z k n X i =1 dz i d ¯ z i + −k z k ) n X i,j =1 z i ¯ z j dz j d ¯ z i o is a K¨ahler manifold with constant holomorphic sectional curvature −
1. Let t bethe coordinate of R and ω = √− −k z k n P j =1 (¯ z j dz j − z j d ¯ z j ). Then, B n × R has a Sasakianstructure { ¯ ϕ, ¯ ξ, ¯ η, ¯ g } with constant ¯ ϕ -sectional curvature −
4, defined as follows: ¯ η = ω + dt, ¯ ξ = ∂∂t , ¯ g = ˜ g + ¯ η ⊗ ¯ η, ¯ ϕ (cid:16) n X i =1 ( a i ∂∂z i + b i ∂∂ ¯ z i ) + e ∂∂t (cid:17) = √− n X i =1 ( b i ∂∂z i − a i ∂∂ ¯ z i ) + −k z k n X i =1 ( b i ¯ z i + a i z i ) ∂∂t . Then, ˜ N n +1 (˜ c ) = ( B n × R , ϕ, ξ, η, g ) is given by the D a -homothetic deformation η = a ¯ η, ξ = a ¯ ξ, g = a ¯ g + a ( a − η ⊗ ¯ η, and ˜ c = − a −
3, where a is a positive number.As were introduced in [14], the contact Whitney spheres in ˜ N n +1 (˜ c ) for ˜ c < − θ : S n → B n × R for θ > π ( ˜Φ θ ( u , u , . . . , u n +1 )) = Π (cid:16) ( u ,...,u n )cosh θ + i sinh θu n +1 ; sinh θ cosh θ (1+ u n +1 ) − iu n +1 cosh θ +sinh θu n +1 (cid:17) , HARACTERIZATIONS OF THE WHITNEY AND CONTACT WHITNEY SPHERES 5 where π : ˜ N n +1 (˜ c ) → N n ( c ) with c = ˜ c + 3 is the canonical projection andΠ : H n +11 ( − → C H n ( −
4) is the Hopf projection.According to Proposition 2 of Blair-Carriazo [1] and Theorem 4.2 of Hu-Yin[14], for each contact Whitney sphere M n in any Sasakian space form ˜ N n +1 (˜ c ),the second fundamental form h and the mean curvature vector field ~H satisfy therelation(1.9) h ( X, Y ) = nn +2 (cid:2) g ( X, Y ) ~H + g ( ϕX, ~H ) ϕY + g ( ϕY, ~H ) ϕX (cid:3) for any tangent vector fields X, Y ∈ T M n . Without introducing the notion ofcontact Whitney spheres as canonical examples, Piti¸s [16] proved that results forLagrangian submanifolds of the complex space forms in [6, 17] hold analogouslyfor those Legendrian submanifolds of the Sasakian space forms which satisfy (1.9).Moreover, an analogue of the result for Whitney spheres in complex space formsby Li-Vrancken [15] was established for contact Whitney spheres in Sasakian spaceforms by Hu-Yin [14]. It follows that a corresponding version of Theorem 1.1 forcontact Whitney spheres in Sasakian space forms is already known.Next, along a similar spirit as above, we get the second main result of this paper.Actually, we can show that an optimal integral inequality, as in Theorem 1.2, thatinvolves the Ricci curvature Ric ( ϕ ~H, ϕ ~H ) in the direction ϕ ~H and the norm of themodified covariant differentiation ¯ ∇ ξ h of the second fundamental form holds alsofor compact Legendrian submanifolds in the Sasakian space forms: Theorem 1.3.
Let x : M n → ˜ N n +1 (˜ c ) be an n -dimensional compact Legendriansubmanifold. Then, it holds that (1.10) Z M n Ric ( ϕ ~H, ϕ ~H ) dV M n ≤ ( n − n +2)3 n Z M n k ¯ ∇ ξ h k dV M n , where, ¯ ∇ ξ h denotes the projection of ¯ ∇ h onto T M n ⊕ ϕ ( T M n ) , whereas ¯ ∇ h denotesthe covariant differentiation of h with respect to the van der Waerden-Bortolotticonnection of M n ֒ → ˜ N n +1 (˜ c ) , k · k and dV M n denote the tensorial norm and thevolume element of M n with respect to the induced metric, respectively.Moreover, the equality in (1.10) holds if and only if either ¯ ∇ ξ h = 0 (i.e. x ( M n ) is of C -parallel second fundamental form), or x ( M n ) is one of the contact Whitneyspheres in ˜ N n +1 (˜ c ) . Remark 1.2.
The classification of Legendrian submanifolds with C -parallel secondfundamental form in the Sasakian space forms has been fulfilled. For the details seeTheorem 4.1 in [14] . From Theorem 1.3, we get a new and global geometric characterization of thecontact Whitney spheres in ˜ N n +1 (˜ c ): Corollary 1.2.
Let x : M n → ˜ N n +1 (˜ c ) ( n ≥ be an n -dimensional compactLegendrian submanifold with non- C -parallel mean curvature vector field. Then, (1.11) Z M n Ric ( ϕ ~H, ϕ ~H ) dV M n = ( n − n +2)3 n Z M n k ¯ ∇ ξ h k dV M n holds if and only if x ( M n ) is a contact Whitney sphere in ˜ N n +1 (˜ c ) . ZEJUN HU AND CHENG XING Preliminaries
In this section, we first briefly review some of the basic notions about Lagrangiansubmanifolds in the complex space form N n (4 c ) and Legendrian submanifolds inthe Sasakian space form ˜ N n +1 (˜ c ), respectively. Then, we state a classical formuladue to K. Yano that we need in the proof of our theorems.Let M n ֒ → N n (4 c ) (resp. M n ֒ → ˜ N n +1 (˜ c )) be an isometric immersion froman n -dimensional Riemannian manifold M n into the n -dimensional complex spaceform N n (4 c ) of constant holomorphic sectional curvature 4 c (resp. the (2 n + 1)-dimensional Sasakian space form ˜ N n +1 (˜ c ) of constant ϕ -section curvature ˜ c ). Forsimplicity, we denote by the same notation g the Riemannian metric on M n , N n (4 c )and ˜ N n +1 (˜ c ). Let ∇ (resp. ¯ ∇ ) be the Levi-Civita connection of M n (resp. N n (4 c )and ˜ N n +1 (˜ c )). Then, for both M n ֒ → N n (4 c ) and M n ֒ → ˜ N n +1 (˜ c ), we have theGauss and Weingarten formulas:¯ ∇ X Y = ∇ X Y + h ( X, Y ) , ¯ ∇ X V = − A V X + ∇ ⊥ X V (2.1)for any tangent vector fields X, Y ∈ T M n and normal vector field V ∈ T ⊥ M n .Here, ∇ ⊥ denotes the normal connection in the normal bundle T ⊥ M , h (resp. A V )denotes the second fundamental form (resp. the shape operator with respect to V )of M n ֒ → N n (4 c ) (resp. M n ֒ → ˜ N n +1 (˜ c )). From (2.1), we have the relation(2.2) g ( h ( X, Y ) , V ) = g ( A V X, Y ) . Lagrangian submanifolds of the complex space form N n (4 c ) . The curvature tensor ¯ R ( X, Y ) Z := ¯ ∇ X ¯ ∇ Y Z − ¯ ∇ Y ¯ ∇ X Z − ¯ ∇ [ X,Y ] Z of N n (4 c )has the following expression:(2.3) ¯ R ( X, Y ) Z = c (cid:2) g ( Y, Z ) X − g ( X, Z ) Y + g ( JY, Z ) JX − g ( JX, Z ) JY − g ( JX, Y ) JZ (cid:3) . Let M n ֒ → N n (4 c ) be a Lagrangian immersion. Then, we have (cf. e.g. [15]) ∇ ⊥ X JY = J ∇ ⊥ X Y, A JX Y = − Jh ( X, Y ) = A JY X, (2.4)and thus g ( h ( X, Y ) , JZ ) is totally symmetric in X , Y and Z :(2.5) g ( h ( X, Y ) , JZ ) = g ( h ( Y, Z ) , JX ) = g ( h ( Z, X ) , JY ) . We choose a local adapted Lagrangian frame field { e , ..., e n , e ∗ , . . . , e n ∗ } suchthat e , . . . , e n are orthonormal tangent vector fields, and e ∗ = Je , . . . , e n ∗ = Je n are orthonormal normal vector fields of M n ֒ → N n (4 c ), respectively. In follows weshall make use of the indices convention: i ∗ = n + i, ≤ i, j, k, . . . ≤ n .Denote by { ω , . . . , ω n } the dual frame of { e , . . . , e n } . Let ω ji and ω j ∗ i ∗ denotethe connection 1-forms of T M n and T ⊥ M n , respectively: ∇ e i = n X j =1 ω ji e j , ∇ ⊥ e i ∗ = n X j =1 ω j ∗ i ∗ e j ∗ , ≤ i ≤ n, where ω ji + ω ij = 0 and by (2.4) it holds that ω ji = ω j ∗ i ∗ . Put h k ∗ ij = g ( h ( e i , e j ) , Je k ).From (2.5), we see that(2.6) h k ∗ ij = h j ∗ ik = h i ∗ jk , ≤ i, j, k ≤ n. HARACTERIZATIONS OF THE WHITNEY AND CONTACT WHITNEY SPHERES 7
Let R ijkl := g (cid:0) R ( e i , e j ) e l , e k (cid:1) and R ijk ∗ l ∗ := g (cid:0) R ⊥ ( e i , e j ) e l ∗ , e k ∗ (cid:1) be the com-ponents of the curvature tensors of ∇ and ∇ ⊥ , respectively. Then the equations ofGauss, Ricci and Codazzi of M n ֒ → N n (4 c ) are given by(2.7) R ijkl = c ( δ ik δ jl − δ il δ jk ) + X m ( h m ∗ ik h m ∗ jl − h m ∗ il h m ∗ jk ) , (2.8) R ijk ∗ l ∗ = c ( δ ik δ jl − δ il δ jk ) + n X m =1 ( h m ∗ ik h m ∗ jl − h m ∗ il h m ∗ jk ) = R ijkl ,h l ∗ ij,k = h l ∗ ik,j , (2.9)where h l ∗ ij,k denotes the components of the covariant differentiation of h , namely¯ ∇ h , defined by(2.10) n X l =1 h l ∗ ij,k e l ∗ := ∇ ⊥ e k (cid:0) h ( e i , e j ) (cid:1) − h ( ∇ e k e i , e j ) − h ( e i , ∇ e k e j ) . The mean curvature vector field ~H of M n ֒ → N n (4 c ) is defined by(2.11) ~H := n n X i =1 h ( e i , e i ) =: n X j =1 H j ∗ e j ∗ , H j ∗ = n n X i =1 h j ∗ ii , ≤ j ≤ n. Put ∇ ⊥ e i ~H = n P j =1 H j ∗ ,i e j ∗ , 1 ≤ i ≤ n . From (2.6) and (2.9), we obtain(2.12) H j ∗ ,i = H i ∗ ,j , ≤ i, j ≤ n. Legendrian submanifolds of the Sasakian space form ˜ N n +1 (˜ c ) . The following facts of this subsection are referred to e.g. [14]. The curvaturetensor of the Sasakian space form ˜ N n +1 (˜ c ) is given by¯ R ( X, Y ) Z = ˜ c +34 [ g ( Y, Z ) X − g ( X, Z ) Y ] + ˜ c − (cid:2) η ( X ) η ( Z ) Y − η ( Y ) η ( Z ) X + g ( X, Z ) η ( Y ) ξ − g ( Y, Z ) η ( X ) ξ + g ( ϕY, Z ) ϕX − g ( ϕX, Z ) ϕY + 2 g ( X, ϕY ) ϕZ (cid:3) . (2.13)Moreover, for tangent vector fields X, Y of ˜ N n +1 (˜ c ), the Sasakian structure( ϕ, ξ, η, g ) of ˜ N n +1 (˜ c ) satisfy:(2.14) η ( X ) = g ( X, ξ ) , ϕξ = 0 , η ( ϕX ) = 0 ,ϕ X = − X + η ( X ) ξ, dη ( X, Y ) = g ( X, ϕY ) ,g ( ϕX, ϕY ) = g ( X, Y ) − η ( X ) η ( Y ) , rank ( ϕ ) = 2 n, ¯ ∇ X ξ = − ϕX, ( ¯ ∇ X ϕ ) Y = g ( X, Y ) ξ − η ( Y ) X. Let M n ֒ → ˜ N n +1 (˜ c ) be a Legendrian immersion. Then, we have(2.15) A ϕY X = − ϕh ( X, Y ) , ∇ ⊥ X ϕY = ϕ ∇ X Y + g ( X, Y ) ξ. In follows we shall make the following convention on range of indices: α ∗ = α + n ; 1 ≤ i, j, k, l, m ≤ n ; 1 ≤ α, β ≤ n + 1 . We choose a local
Legendre frame field { e , . . . , e n , e ∗ , . . . , e n ∗ , e n +1 = ξ } along M n ֒ → ˜ N n +1 (˜ c ) such that { e i } ni =1 is an orthonormal frame field of M n , and ZEJUN HU AND CHENG XING { e ∗ = ϕe , . . . , e n ∗ = ϕe n , e n +1 = ξ } are the orthonormal normal vector fieldsof M n ֒ → ˜ N n +1 (˜ c ). Let ω ji and ω β ∗ α ∗ denote the connection 1-forms of T M n and T ⊥ M n , respectively: ∇ e i = n X j =1 ω ji e j , ∇ ⊥ e α ∗ = n +1 X β =1 ω β ∗ α ∗ e β ∗ , ≤ i ≤ n, ≤ α ≤ n + 1 , where ω ji + ω ij = 0 and ω β ∗ α ∗ + ω α ∗ β ∗ = 0. Moreover, by (2.15), we have ω ji = ω j ∗ i ∗ and ω n +1 i ∗ = ω i . Put h k ∗ ij = g ( h ( e i , e j ) , ϕe k ) and h n +1 ij = g ( h ( e i , e j ) , e n +1 ). From(2.14) and (2.15), we have(2.16) h k ∗ ij = h j ∗ ik = h i ∗ jk , h n +1 ij = 0 , ≤ i, j, k ≤ n. Now, with the same notations as in the preceding subsection, the equations ofGauss, Ricci and Codazzi of M n ֒ → ˜ N n +1 (˜ c ) are as follows:(2.17) R ijkl = ˜ c +34 ( δ ik δ jl − δ il δ jk ) + n X m =1 ( h m ∗ ik h m ∗ jl − h m ∗ il h m ∗ jk ) , (2.18) R ijk ∗ l ∗ = ˜ c − ( δ ik δ jl − δ il δ jk ) + n X m =1 ( h m ∗ ik h m ∗ jl − h m ∗ il h m ∗ jk ) , R ijk ∗ (2 n +1) = 0 , (2.19) h α ∗ ij,k = h α ∗ ik,j , where as usual h α ∗ ij,k is defined by(2.20) n +1 X α =1 h α ∗ ij,k e α ∗ := ∇ ⊥ e k (cid:0) h ( e i , e j ) (cid:1) − h ( ∇ e k e i , e j ) − h ( e i , ∇ e k e j ) , ≤ i, j, k ≤ n. Moreover, associated to ∇ , ∇ ⊥ and ¯ ∇ , we can naturally define a modified covariantdifferentiation ¯ ∇ ξ h of the second fundamental form by(2.21) ( ¯ ∇ ξX h )( Y, Z ) := ∇ ⊥ X ( h ( Y, Z )) − h ( ∇ X Y, Z ) − h ( Y, ∇ X Z ) − g ( h ( Y, Z ) , ϕX ) ξ. Recall that the second fundamental form h of M n ֒ → ˜ N n +1 (˜ c ) is said to be C -parallel if and only if ¯ ∇ ξ h = 0 (cf. [14]). Actually, we have g (( ¯ ∇ ξX h )( Y, Z ) , ξ ) = 0for any X, Y, Z ∈ T M n . Thus, we can denote(2.22) ( ¯ ∇ ξe k h )( e i , e j ) := n X l =1 ¯ h l ∗ ij,k e l ∗ , ≤ i, j, k ≤ n. Then, by (2.20), (2.21) and the above discussions, we have(2.23) h ( n +1) ∗ ij,k = h k ∗ ij , h l ∗ ij,k = ¯ h l ∗ ij,k , ∀ i, j, k, l. From (2.16), the mean curvature vector ~H of M n ֒ → ˜ N n +1 (˜ c ) becomes:(2.24) ~H = n n X i =1 h ( e i , e i ) = n X k =1 H k ∗ e k ∗ , H k ∗ := n n X i =1 h k ∗ ii , ≤ k ≤ n. Put ∇ ⊥ e i ~H = n +1 X α =1 H α ∗ ,i e α ∗ , ¯ ∇ ξe i ~H := ∇ ⊥ e i ~H − g ( ~H, e i ∗ ) ξ =: n X k =1 ¯ H k ∗ ,i e k ∗ , ≤ i ≤ n. HARACTERIZATIONS OF THE WHITNEY AND CONTACT WHITNEY SPHERES 9
From (2.16), (2.19) and (2.23), we get(2.25) H j ∗ ,i = H i ∗ ,j , ¯ H j ∗ ,i = H j ∗ ,i , ≤ i, j ≤ n. Yano’s formula.
In order to prove Theorem 1.2 and Theorem 1.3, we still need the following usefulformula due to K. Yano [19]. A simply proof is referred also to [13].
Lemma 2.1 (cf. Lemma 5.1 of [13]) . Let ( M, g ) be a Riemannian manifold withLevi-Civita connection ∇ . Then, for any tangent vector field X on M , it holds that (2.26) div( ∇ X X − (div X ) X ) = Ric ( X, X ) + kL X g k − k∇ X k − (div X ) , where L X g is the Lie derivative of g with respect to X and k · k denotes the lengthwith respect to g . Proof of Theorem 1.2
First of all, we state the following simple fact without proof.
Lemma 3.1.
Let x : M n → N n (4 c ) be an n -dimensional Lagrangian submanifoldwith mean curvature vector field ~H . Then, it holds that (3.1) k∇ J ~H k ≥ n (div J ~H ) . Moreover, the equality in (3.1) holds if and only if ∇ J ~H = n (div J ~H ) id , i.e.,
J ~H is a conformal vector field on M n , or equivalently, M n is a Lagrangian submanifoldwith conformal Maslov form. We also need the following result due to Li-Vrancken [15]:
Lemma 3.2 (cf. Lemma 3.2 in [15]) . Let x : M n → N n (4 c ) be an n -dimensionalLagrangian submanifold with mean curvature tensor ~H . Then, it holds that (3.2) k ¯ ∇ h k ≥ n n +2 k∇ ⊥ ~H k . Moreover, the equality in (3.2) holds if and only if (3.3) ( ¯ ∇ Z h )( X, Y ) = nn +2 (cid:2) g ( Y, Z ) ∇ ⊥ X ~H + g ( X, Z ) ∇ ⊥ Y ~H + g ( X, Y ) ∇ ⊥ Z ~H (cid:3) . Now, we are ready to complete the proof of Theorem 1.2.
Proof of Theorem 1.2.
Let M n ֒ → N n (4 c ) be a compact Lagrangian submanifoldand { e , ..., e n , e ∗ , . . . , e n ∗ } be a local adapted Lagrangian frame field along M n .From (2.4) and that ∇ ⊥ e i ~H = n P j =1 H j ∗ ,i e j ∗ , we have(3.4) k∇ J ~H k = k∇ ⊥ ~H k = n X i,j =1 ( H j ∗ ,i ) . Then, by (2.12), calculating the squared length of the Lie derivative L J ~H g of g with respect to J ~H , we obtain(3.5) kL J ~H g k = n X i,j =1 (cid:2) ( L J ~H g )( e i , e j ) (cid:3) = n X i,j =1 (cid:0) H j ∗ ,i + H i ∗ ,j (cid:1) = 4 k∇ ⊥ ~H k . Thus, we can apply Lemma 2.1 and (3.1) to obtain that(3.6) div( ∇ J ~H
J ~H − (div J ~H ) J ~H ) = Ric (
J ~H, J ~H ) + k∇ J ~H k − (div J ~H ) ≥ Ric (
J ~H, J ~H ) − ( n − k∇ ⊥ ~H k , where the equality in (3.6) holds if and only if ∇ J ~H = n (div J ~H ) id, or equivalently, M n is a Lagrangian submanifold with conformal Maslov form.From (3.6), by further applying Lemma 3.2, we get(3.7) div( ∇ J ~H
J ~H − (div J ~H ) J ~H ) ≥ Ric (
J ~H, J ~H ) − ( n − n +2)3 n k ¯ ∇ h k , where the equality holds if and only if both ∇ J ~H = n (div J ~H ) id and (3.3) hold.By the compactness of M n , we can integrate the inequality (3.7) over M n . Then,applying for the divergence theorem, we obtain the integral inequality (1.4).It is easily seen that the equality holds in (1.4) if and only if the equality in (3.2)holds identically. Thus, according to Main Theorem in [15], equality in (1.4) holdsif and only if either x ( M n ) is of parallel second fundamental form, or x ( M n ) is oneof the Whitney spheres in N n (4 c ).This completes the proof of Theorem 1.2. (cid:3) Proof of Theorem 1.3
Let x : M n → ˜ N n +1 (˜ c ) be an n -dimensional Legendrian submanifold in theSasakian space form ˜ N n +1 (˜ c ) with Sasakian structure ( ϕ, ξ, η, g ). First of all,similar to Lemma 3.1, we have the following simple result. Lemma 4.1.
Let x : M n → ˜ N n +1 (˜ c ) be an n -dimensional Legendian submanifoldwith mean curvature vector field ~H . Then, it holds that (4.1) k∇ ( ϕ ~H ) k ≥ n (div ϕ ~H ) . Moreover, the equality in (4.1) holds if and only if ∇ ( ϕ ~H ) = n (div ϕ ~H ) id , i.e., ϕ ~H is a conformal vector field on M n . We also need the following result:
Lemma 4.2 (cf. Lemma 3.3 in [14]) . Let x : M n → ˜ N n +1 (˜ c ) be an n -dimensionalLegendrian submanifold with second fundamental form h and mean curvature vectorfield ~H . Then, it holds that (4.2) k ¯ ∇ ξ h k ≥ n n +2 k ¯ ∇ ξ ~H k , where, with respect to a local Legendre frame field { e A } n +1 A =1 , k ¯ ∇ ξ h k = n X i,j,k,l =1 ( h l ∗ ij,k ) , k ¯ ∇ ξ ~H k = n X i,j =1 ( H j ∗ ,i ) . Moreover, the equality in (4.2) holds if and only if (4.3) h l ∗ ij,k = nn +2 (cid:0) H l ∗ ,i δ jk + H l ∗ ,j δ ik + H l ∗ ,k δ ij (cid:1) , ≤ i, j, k, l ≤ n. Now, we are ready to complete the proof of Theorem 1.3.
HARACTERIZATIONS OF THE WHITNEY AND CONTACT WHITNEY SPHERES 11
Proof of Theorem 1.3.
Let x : M n → ˜ N n +1 (˜ c ) be a compact n -dimensional Leg-endrian submanifold and { e , . . . , e n , e ∗ , . . . , e n ∗ , e n +1 = ξ } be a local adaptedLegendre frame field along M n . By definition, we have(4.4) k∇ ( ϕ ~H ) k = n X i,j =1 ( g ( ∇ e i ( ϕ ~H ) , e j )) = n X i,j =1 ( ¯ H j ∗ ,i ) = k ¯ ∇ ξ ~H k . Then, by (2.25), calculating the squared length of the Lie derivative L ϕ ~H g of g with respect to ϕ ~H , we obtain(4.5) kL ϕ ~H g k = n X i,j =1 (cid:2) ( L ϕ ~H g )( e i , e j ) (cid:3) = n X i,j =1 (cid:0) H j ∗ ,i + H i ∗ ,j (cid:1) = 4 k∇ ( ϕ ~H ) k . Thus, we can apply Lemma 2.1 and (4.1) to obtain that(4.6) div( ∇ ϕ ~H ( ϕ ~H ) − (div ϕ ~H ) ϕ ~H ) = Ric ( ϕ ~H, ϕ ~H ) + k∇ ( ϕ ~H ) k − (div ϕ ~H ) ≥ Ric ( ϕ ~H, ϕ ~H ) − ( n − k∇ ( ϕ ~H ) k , where the equality in (4.6) holds if and only if ∇ ( ϕ ~H ) = n (div ϕ ~H ) id.From (4.6) and that k∇ ( ϕ ~H ) k = k ¯ ∇ ξ ~H k , by further applying Lemma 4.2, weget(4.7) div( ∇ ϕ ~H ( ϕ ~H ) − (div ϕ ~H ) ϕ ~H ) ≥ Ric ( ϕ ~H, ϕ ~H ) − ( n − n +2)3 n k ¯ ∇ ξ h k , where the equality holds if and only if both ∇ ( ϕ ~H ) = n (div ϕ ~H ) id and (4.3) hold.By the compactness of M n , we can integrate the inequality (4.7) over M n . Then,applying for the divergence theorem, we obtain the integral inequality (1.10).It is easily seen from the above arguments that the equality in (1.10) holds ifand only if the equality in (4.2) holds identically. Thus, according to Theorem 1.1in [14], equality in (1.10) holds if and only if either x ( M n ) is of C -parallel secondfundamental form, or x ( M n ) is one of the contact Whitney spheres in ˜ N n +1 (˜ c ).This completes the proof of Theorem 1.3. (cid:3) As final remarks, we would mention that all the Whitney spheres in the complexspace forms are conformally equivalent to the round sphere (cf. [17] and [6]).Now, for any one of the contact Whitney spheres, x : S n → ˜ N n +1 (˜ c ), its secondfundamental form h has the expression (1.9). Thus, by using the Gauss equationand direct calculations, we can immediately obtain the following Theorem 4.1.
The sectional curvatures of the contact Whitney spheres are notconstant. Nevertheless, all the contact Whitney spheres in each of the Sasakianspace forms are conformally equivalent to the round sphere.
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