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Lagrangian submanifolds in complex projective space with parallel second fundamental form Lagrangian submanifolds in complex projective space with parallel second fundamental form Xianfeng Wang (Joint work with F. Dillen, H. Li & L. Vrancken) Nankai University, China (Email: wangxianfeng@nankai.edu.cn ) The 10th PRGC 2011 Osaka-Fukuoka, December 1, 2011

Lagrangian submanifolds in complex projective space with parallel second fundamental form Contents Contents Introduction 1 Calabi product Lagrangian immersions in CP n 2 Main result and remarks 3

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction K¨ a hler manifold M n be an K¨ Let ¯ a hler n-manifold, that is, an 2n-dimensional manifold with a M n → T ¯ M n satisfying that almost complex structure J : T ¯ J 2 = − I ,    < Jv , Jw > = < v , w >,  DJ = 0 ,  M n and D is the Levi-Civita connection on ¯ where v , w ∈ T ¯ M n . Complex space forms are the simplest K¨ a hler-Einstein manifold. Let ¯ M n (4 c ) denote an n-dimensional complex space form with constant holomorphic sectional curvature 4 c . When c > 0, ¯ M n (4 c ) = CP n (4 c ) , When c = 0, ¯ M n (4 c ) = C n , When c < 0, ¯ M n (4 c ) = CH n (4 c ) .

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction Lagrangian submanifolds M n be an isometric immersion from an n-dimensional Let φ : M → ¯ a hler n-manifold ¯ M n . Riemannian manifold M into a K¨ Then M is called a Lagrangian submanifold if the almost complex M n carries each tangent space of M into its ¯ structure J of corresponding normal space. Example 1-3: R n → C n , RP n → CP n , RH n → CH n . Example 4: Whitney sphere in C n . It is defined as the Lagrangian immersion of the unit sphere S n , centered at the origin of R n +1 , in C n , given by φ : S n → C n : φ ( x 1 , x 2 , . . . , x n , x n +1 ) = 1 + ix n +1 ( x 1 , . . . , x n ) . (1) 1 + x 2 n +1

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction Example 5: Whitney spheres in CP n . They are a one-parameter family of Lagrangian spheres in CP n , given by φ θ : S n → CP n (4) : ¯ � ( x 1 , . . . , x n ) ; s θ c θ (1 + x 2 n +1 ) + ix n +1 � ¯ φ θ ( x 1 , x 2 , . . . , x n , x n +1 ) = π ◦ , (2) c 2 θ + s 2 θ x 2 c θ + is θ x n +1 n +1 where θ > 0, c θ = cosh θ, s θ = sinh θ , π : S 2 n +1 (1) → CP n (4) is the Hopf fibration. Example 6: Whitney spheres in CH n . They are a one-parameter family of Lagrangian spheres in CH n , given by φ θ : S n → CH n ( − 4) : ¯ � ( x 1 , . . . , x n ) ; s θ c θ (1 + x 2 n +1 ) − ix n +1 � ¯ φ θ ( x 1 , x 2 , . . . , x n , x n +1 ) = π ◦ , (3) s 2 θ + c 2 θ x 2 s θ + ic θ x n +1 n +1 where θ > 0, c θ = cosh θ, s θ = sinh θ , π : H 2 n +1 ( − 1) → CH n (4) is the Hopf 1 fibration.

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction A general method for constructing Lagrangian submanifolds in complex projective space In view of Reckziegel’s results, we have Let φ : M → CP n (4 c ) be a Lagrangian isometric immersion. We consider the Hopf fibration: π : S 2 n +1 ( c ) → CP n (4 c ). Then there exists an isometric covering map τ : ˆ M → M and a C -totally real isometric immersion ˜ φ : ˆ M → S 2 n +1 ( c ) such that φ ◦ τ = π ◦ ˜ φ . Conversely, let ˜ φ : ˆ M → S 2 n +1 ( c ) be a C -totally real isometric immersion. Then φ = π ◦ ˜ φ : M → CP n (4 c ) is an Lagrangian isometric immersoin. Under this correspondence, the second fundamental form h φ and h ˜ φ satisfy h φ = π ∗ h ˜ φ . We shall denote h φ and h ˜ φ simply by h .

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction Parallel submanifolds Let φ : M → ¯ M be an isometric immersion. If at each point p of M , the first derivative of the second fundamental form ∇ h vanishes, i.e., ∇ h ≡ 0, we call M a submanifold with parallel second fundamental form, i.e, a parallel submanifold. Examples: straight lines, circles, planes, round spheres, round cylinders in R 3 ; circles, round spheres, a product of two circles in S 3 , Veronese surface in S 4 . Some examples of Lagrangian parallel submanifolds: R n → C n , RP n → CP n , RH n → CH n . 1 2 k ( k +1) − 1 . SU ( k ) / SO ( k ) ( k ≥ 3) → CP SU ( k ) ( k ≥ 3) → CP k 2 − 1 . SU (2 k ) / Sp ( k ) ( k ≥ 3) → CP 2 k 2 − k − 1 . E 6 / F 4 → CP 26 .

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction Motivation of our research work From the Riemannian geometric point of view, one of the most fundamental problems in the study of Lagrangian submanifolds is: the classification of Lagrangian submanifolds in complex space forms with parallel second fundamental form. In 1980s, H. Naitoh classified the Lagrangian submanifolds with parallel second fundamental form in complex projective space. Prof. Naitoh’s method is based on the theory of Lie groups and symmetric spaces.

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction In the irreducible case, the classification is clear, Naitoh completely classified the Lagrangian submanifolds with parallel second fundamental form and without Euclidean factor in complex projective space. He showed that such a submanifold is always locally symmetric and is one of the symmetric spaces: SO ( k + 1) / SO ( k ) ( k ≥ 2). SU ( k ) / SO ( k ) ( k ≥ 3). SU ( k ) ( k ≥ 3). SU (2 k ) / Sp ( k ) ( k ≥ 3). E 6 / F 4 . However, little information is given on how to construct all reducible examples.

Lagrangian submanifolds in complex projective space with parallel second fundamental form Introduction Question How to determine all reducible parallel Lagrangian submanifolds of complex projective space? Our main result We obtain a complete and explicit classification of all (irreducible and reducible) parallel Lagrangian submanifolds of complex projective space by an elementary geometric method.

Lagrangian submanifolds in complex projective space with parallel second fundamental form Calabi product Lagrangian immersions in CP n Calabi product Lagrangian immersion Definition Let ψ 1 : ( M 1 , g 1 ) → CP n 1 (4) and ψ 2 : ( M 2 , g 2 ) → CP n 2 (4) be two Lagrangian immersions. π : S 2 n +1 ( c ) → CP n (4 c ) is the Hopf fibration. We denote by ˜ ψ i : M i → S 2 n i +1 (1) the horizontal lifts of ψ i , i = 1 , 2, respectively. γ ( t ) = ( r 1 e i ( r 2 r 1 t ) , r 2 e i ( − r 1 r 2 t ) ), be a special Legendre curve, where r 1 and r 2 Let ˜ are positive constants with r 2 1 + r 2 2 = 1, γ 1 ˜ γ 2 ˜ ψ 2 ) : I × M 1 × M 2 → CP n (4) is a Lagrangian Then ψ = π (˜ ψ 1 ; ˜ immersion, where n = n 1 + n 2 + 1. We call ψ a Calabi product Lagrangian immersion of ψ 1 and ψ 2 . When n 1 (or n 2 ) is zero, we call ψ a Calabi product Lagrangian immersion of ψ 2 (or ψ 1 ) and a point.

Lagrangian submanifolds in complex projective space with parallel second fundamental form Calabi product Lagrangian immersions in CP n Characterizations of the Calabi products Theorem (1.6, Li-Wang, Results Math, 2011) Let ψ : M → CP n (4) be a Lagrangian immersion. If M admits two orthogonal distributions T 1 (of dimension 1, spanned by a unit vector E 1 ) and T 2 (of dimension n − 1 , spanned by { E 2 , . . . , E n } ), and there exist local functions λ 1 , λ 2 such that � h ( E 1 , E 1 ) = λ 1 JE 1 , h ( E 1 , E i ) = λ 2 JE i , (4) λ 1 � = 2 λ 2 , i = 2 , . . . , n . If M has parallel second fundamental form, then ψ is locally a Calabi product Lagrangian immersion of a point and an ( n − 1) -dimensional Lagrangian immersion ψ 1 : M 1 → CP n − 1 (4) which has parallel second fundamental form. Conversely, if ψ is locally a Calabi product Lagrangian immersion of a point and an ( n − 1) -dimensional Lagrangian immersion ψ 1 : M 1 → CP n − 1 (4) which has parallel second fundamental form, then M has parallel second fundamental form.

Lagrangian submanifolds in complex projective space with parallel second fundamental form Calabi product Lagrangian immersions in CP n Characterizations of the Calabi products Theorem (4.6, Li-Wang, Results Math, 2011) Let ψ : M → CP n (4) be a Lagrangian immersion. If M admits three mutually orthogonal distributions T 1 (spanned by a unit vector E 1 ), T 2 , and T 3 of dimension 1 , n 1 and n 2 respectively, with 1 + n 1 + n 2 = n, and there three real functions λ 1 , λ 2 and λ 3 ( 2 λ 3 � = λ 1 � = 2 λ 2 � = 2 λ 3 ) such that for all E i ∈ T 2 , E α ∈ T 3 , � h ( E 1 , E 1 ) = λ 1 JE 1 , h ( E 1 , E i ) = λ 2 JE i , (5) h ( E 1 , E α ) = λ 3 JE α . If M has parallel second fundamental form, then ψ is locally a Calabi product Lagrangian immersion of two lower dimensional Lagrangian submanifolds ψ i ( i = 1 , 2) with parallel second fundamental form. Conversely, if ψ is locally a Calabi product Lagrangian immersion of two lower dimensional Lagrangian submanifolds ψ i ( i = 1 , 2) with parallel second fundamental form, then M has parallel second fundamental form.