Hopf Real Hypersurfaces in the Indefinite Complex Projective Space - PowerPoint PPT Presentation

Symmetry and Shape Santiago de Compostela, October 28–31, 2019 Hopf Real Hypersurfaces in the Indefinite Complex Projective Space Miguel Ortega Partially financed by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund (ERDF), project MTM2016-78807-C2-1-P.

This talk is based on the following joint work with Makoto Kimura (Ibaraki University, Japan) M. Kimura, —, Hopf Real Hypersurfaces in the Indefinite Complex Projective Space , Mediterr. J. Math. (2019) 16: 27. https://doi.org/10.1007/s00009-019-1299-9 https://arxiv.org/abs/1802.05556 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 2 / 30 p

Table Of Contents Introduction 1 Preliminaries 2 Examples 3 A Ridigity Result 4 The shape operator vs the almost contact metric structure 5 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 3 / 30 p

Summary Introduction 1 Preliminaries 2 Examples 3 A Ridigity Result 4 The shape operator vs the almost contact metric structure 5 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 4 / 30 p

Introduction The theory of real hypersurfaces in complex space forms is very well-developed. J. Berdnt , T. Cecil, G. Kaimakamis, M. Kimura, S. Maeda, Y. Maeda, S. Montiel, K. Panagiotidou, Juan de Dios Pérez, P. Ryan, Y. J. Suh, R. Takagi... Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 5 / 30 p

Introduction R. Takagi, On homogeneous real hypersurfaces in a complex projective space . Osaka J. Math. 10 (1973), 495–506 The classification of (extrinsically) homogeneous real hypersurfaces in C P n , n ≥ 2 : Six types of tubes of certain radii over some complex submanifolds [ A 0 , A 1 , B , C , D , E ]. N : a unit normal vector field to M in C P n , J : the complex structure. ξ = − JN ; A : shape operator. All these examples satisfy Aξ = µξ . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 6 / 30 p

Introduction J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space , J. Reine Angew. Math. 395 (1989), 132–141. Theorem A Let M be a real hypersurface in C H n , n ≥ 2 , such that ξ is principal, and M has constant principal curvatures. Then, M is an open subset of one of the following: (A) A tube of radius r > 0 over a totally geodesic C H k , k = 0 , . . . , n − 1 ; (B) a tube of radius r > 0 over a totally geodesic R H n ; (C) a horosphere. Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 7 / 30 p

Introduction Hundreds of works about real hypersurfaces in non-flat complex space forms have appeared, also in the quaternionic space forms, the Grassmanian of 2-complex planes, and the complex quadric. T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces , Springer Monographs in Mathematics, Springer, New York, NY (2015) DOI 10.1007/978-1-4939-3246-7 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 8 / 30 p

Introduction Hundreds of works about real hypersurfaces in non-flat complex space forms have appeared, also in the quaternionic space forms, the Grassmanian of 2-complex planes, and the complex quadric. T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces , Springer Monographs in Mathematics, Springer, New York, NY (2015) DOI 10.1007/978-1-4939-3246-7 Next, we move to real hypersurfaces in the indefinite complex projective space C P n p . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 8 / 30 p

Introduction A. Bejancu, K. L. Duggal, Real hypersurfaces of indefinite Kaehler manifolds , Internat. J. Math. Math. Sci. 16 (1993), no. 3, 545–556. H. Anciaux, K. Panagiotidou, Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms , Diff. Geom. Appl. 42 (2015) 1-14 DOI: 10.1016/j.difgeo.2015.05.004 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 9 / 30 p

Introduction We allow the normal vector to have its own causal character, without changing the metric. We recover the almost contact metric structure ( g, ξ, η, φ ) . Examples: Families of non-degenerate real hypersurfaces whose shape operator is 1 diagonalisable, An example with degenerate metric and non-diagonalisable 2 shape operator . A rigidity result. AX = aX + bη ( X ) ξ , ∀ X ∈ TM . Aφ = φA . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 10 / 30 p

Summary Introduction 1 Preliminaries 2 Examples 3 A Ridigity Result 4 The shape operator vs the almost contact metric structure 5 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 11 / 30 p

Preliminaries See [2] (Barros-Romero) for more details. C n +1 the Euclidean complex space endowed with the following p pseudo-Riemannian metric of index 2 p : z = ( z 1 , . . . , z n +1 ) , w = ( w 1 , . . . , w n +1 ) ∈ C n +1 ,   p n +1 � �  , g ( z, w ) = Re  − z j ¯ w j + z j ¯ w j j =1 j = p +1 where ¯ w is the complex conjugate of w ∈ C . S 1 = { a ∈ C : a ¯ a = 1 } = { e iθ : θ ∈ R } . S 2 n +1 = { z ∈ C n +1 : g ( z, z ) = 1 } . 2 p p Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 12 / 30 p

x ∼ y ⇔ ∃ a ∈ S 1 : x = a y. x, y ∈ S 2 n +1 , 2 p π : S 2 n +1 → S 2 n +1 / ∼ =: C P n p . 2 p 2 p The manifold C P n p is called the Indefinite Complex Projective Space . Let g be the metric on C P n p such that π becomes a semi-Riemannian submersion. Let ¯ ∇ be its Levi-Civita connection. C P n p admits a complex structure J induced by π . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 13 / 30 p

M : a connected, orientable, immersed real hypersurface in C P n p . N : a unit normal vector field such that ε = g ( N, N ) = ± 1 . ξ = − JN : The structure vector field on M . Clearly, g ( ξ, ξ ) = ε . Given X ∈ TM , we decompose JX in its tangent and normal parts, namely JX = φX + ε η ( X ) N, where φX is the tangential part, and η is the 1-form on M . Given X, Y ∈ TM , η ( X ) = g ( X, ξ ) , φξ = 0 , η ( ξ ) = ε, φ 2 X = − X + εη ( X ) ξ, η ( φX ) = 0 , g ( φX, φY ) = g ( X, Y ) − εη ( X ) η ( Y ) , g ( φX, Y ) + g ( X, φY ) = 0 . ( g, φ, η, ξ ) is called an almost contact metric structure on M . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 14 / 30 p

Next, if ∇ is the Levi-Civita connection of M , we have the Gauss and Weingarten formulae: ¯ ¯ ∇ X Y = ∇ X Y + εg ( AX, Y ) N, ∇ X N = − AX, for any X, Y ∈ TM , where A is the shape operator associated with Definition 1 Let M be a real hypersurface in C P n p . We will say that M is Hopf when its structure vector field ξ is everywhere principal, i. e., it is an eigenvector of A . Its associated principal curvature µ = εg ( Aξ, ξ ) will be called the Hopf curvature : Aξ = µξ . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 15 / 30 p

Summary Introduction 1 Preliminaries 2 Examples 3 A Ridigity Result 4 The shape operator vs the almost contact metric structure 5 Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 16 / 30 p

Recall the projection π : S 2 n +1 → C P n p . 2 p ˜ M 2 n → S 2 n +1 − − − − 2 p     � � M 2 n − 1 − → C P n − − − p Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 17 / 30 p

Given 0 ≤ q ≤ p ≤ m ≤ n + 2 , m > q + 1 , the case q = 0 and m = n + 2 is not considered. We define the following map pr : C n +1 → C n +1 : p p if 1 ≤ q and m ≤ n + 1 , pr ( z ) = ( z 1 , . . . , z q , 0 , . . . , 0 , z m , . . . , z n +1 ) , if q = 0 and m ≤ n + 1 , pr ( z ) = (0 , . . . , 0 , z m , . . . , z n +1 ) , if 1 ≤ q and m = n + 2 , pr ( z ) = ( z 1 , . . . , z q , 0 , . . . , 0) . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 18 / 30 p

Examples Type A Consider t ∈ R , t � = 0 , 1 , and 0 ≤ q ≤ p ≤ m ≤ n + 2 , m > q + 1 . With this notation, we define � � M m ˜ z = ( z 1 , . . . , z n ) ∈ S 2 n +1 q ( t ) = : g ( pr ( z ) , pr ( z )) = t 2 p M m q ( t ) = π ( ˜ M m q ( t )) ⊂ C P n p Aξ = µξ For a suitable r > 0 , ( A + ) ε = +1 , 0 < t = cos 2 ( r ) < 1 , µ = 2 cot(2 r ) , λ 1 = − tan( r ) , λ 2 = cot( r ) . ( A − ) ε = − 1 , 1 < t = cosh 2 ( r ) , µ = 2 coth(2 r ) , λ 1 = − tanh( r ) , λ 2 = coth( r ) . dim V λ 1 = 2( m − q − 2) , dim V λ 2 = 2( n + q − m + 1) . Miguel Ortega (Univ. Granada) Hopf Real Hypersurfaces in C P n Santiago de Compostela 19 / 30 p