Geometry of CR submanifolds MIRJANA DJORI C, University of - PowerPoint PPT Presentation

Geometry of CR submanifolds MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Symmetry and shape Celebrating the 60th birthday of Prof. J. Berndt Santiago de Compostella October 30, 2019. MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

One of the aims of submanifold geometry is to understand geometric invariants of submanifolds and to classify submanifolds according to given geometric data. In Riemannian geometry, the structure of a submanifold is encoded in the second fundamental form. We are interested in certain submanifolds, called contact CR -submanifolds, of S 7 (1), which are (nearly) totally geodesic. We study certain conditions on the structure F and on h of CR submanifolds of maximal CR dimension in complex space forms and we characterize several important classes of submanifolds in complex space forms. We also show some results on CR submanifolds of the nearly K¨ ahler six sphere. MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Let ( M , g ) be an ( n + p )–dimensional Riemannian manifold with Levi Civita connection ∇ and let M be an n –dimensional submanifold of M with the immersion ı of M into M , whose metric g is induced from ¯ g in such a way that g ( X , Y ) = g ( ı X , ı Y ) , X , Y ∈ T ( M ) . We denote by T ( M ) and T ⊥ ( M ) the tangent bundle of M and the normal bundle of M , respectively. MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Then, for all X , Y ∈ T ( M ), we have ∇ ı X ı Y = ı ∇ X Y + h ( X , Y ) , The tangent part defines the the Levi-Civita connection ∇ with respect to the induced Riemannian metric g , The normal part h defines the second fundamental form , symmetric covariant tensor field of degree two with values in T ⊥ ( M ). MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

We have further, for all ξ ∈ T ⊥ ( M ) ∇ ı X ξ = − ı A ξ X + D X ξ , It is a easy to check that A ξ (the shape operator with respect to the normal ξ ) is a linear mapping from the tangent bundle T ( M ) into itself and that D defines a linear connection on the normal bundle T ⊥ ( M ). We call D the normal connection of M in M . h and A ξ are related by g ( h ( X , Y ) , ξ ) = g ( A ξ X , Y ) . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

M. Djori´ c, M. Okumura, Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space, Ann. Glob. Anal. Geom. , 39, (2011), 1-12. MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

J. Berndt, ¨ Uber untermannifaltigkeiten von komplexen Raumformen , Dissertation, Universit¨ at zu K¨ oln, 1989. J. Berndt, J. C. Diaz-Ramos, Real hypersurfaces with constant principal curvatures in complex hyperbolic space , J. London Math. Soc. , (2) 74 , 778–798, (2006). J. Berndt, J. C. Diaz-Ramos, Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane , Proc. Amer. Math. Soc. , (10) 135 , 3349–3357, (2007). MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Main Theorem Let M be a complete n–dimensional CR submanifold of maximal CR dimension of a complex hyperbolic n + p 2 . If the condition space CH h ( FX , Y ) − h ( X , FY ) = g ( FX , Y ) η, η ∈ T ⊥ ( M ) is satisfied, where F is the induced almost contact structure and h is the second fundamental form of M, respectively, then F is a contact structure and M is an invariant submanifold of ˜ M by the almost contact structure ˜ F of ˜ M, where ˜ M is a geodesic hypersphere or a horosphere, or M is congruent to one of the following: (i) a tube of radius r > 0 around a totally geodesic, totally real n +1 2 ( − 1) ; hyperbolic space form H (ii) a tube of radius r > 0 around a totally geodesic complex n − 1 2 ( − 4) ; hyperbolic space form CH (iii) a geodesic hypersphere of radius r > 0 ; (iv) a horosphere; n +1 2 . (v) a tube over a complex submanifold of CH MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Let M be an almost Hermitian manifold with the structure ( J , ¯ g ). J is the endomorphism of the tangent bundle T ( M ) satisfying J 2 = − I g is the Riemannian metric of M satisfying the Hermitian condition ¯ g ( J ¯ X , J ¯ g ( ¯ X , ¯ X , ¯ ¯ ¯ Y ) = ¯ Y ) , Y ∈ T ( M ) . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

The fundamental 2-form, (K¨ ahler form) Ω of M is defined by Ω( X , Y ) = g ( JX , Y ) for all vector fields X and Y on M . If a complex manifold ( M , J ) with Hermitian metric g satisfies d Ω = 0, then ( M , J ) is called a K¨ ahler manifold . A necessary and sufficient condition that a complex manifold ( M , J ) with Hermitian metric is a K¨ ahler manifold is ∇ X J = 0 for any X ∈ T ( M ). Here ∇ is the Levi-Civita connection with respect to the Hermitian metric g . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Let M ′ be a real hypersurface of M and let ξ be the unit normal local field to M ′ . Then ı 1 F ′ X ′ + u ′ ( X ′ ) ξ, J ı 1 X ′ = − ı 1 U ′ , J ξ = where F ′ is a skew symmetric endomorphism acting on T ( M ′ ), U ′ ∈ T ( M ′ ), u ′ is a one form on M ′ . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Y.Tashiro, On contact structure of hypersurfaces in complex manifold I, Tˆ ohoku Math. J. , 15 , 62–78, (1963). By iterating J on i 1 X ′ and on ξ , we easily see F ′ 2 X ′ = − X ′ + u ′ ( X ′ ) U ′ , g ′ ( U ′ , X ′ ) = u ′ ( X ′ ) , u ′ ( U ′ ) = 1 , F ′ U ′ = 0 . u ′ ( F ′ X ′ ) = 0 , Thus the real hypersurface M ′ is equipped with an almost contact structure ( F ′ , u ′ , U ′ ), naturally induced by the almost Hermitian structure on M . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

CR submanifolds of maximal CR dimension H x ( M ) = T x ( M ) ∩ JT x ( M ) is called the holomorphic tangent space of M . H x ( M ) is the maximal J -invariant subspace of T x ( M ). n − p ≤ dim R H x ( M ) ≤ n MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

M is called the Cauchy-Riemann submanifold or briefly CR submanifold if H x has constant dimension for any x ∈ M . R. Nirenberg and R.O. Wells, Jr., Approximation theorems on differentiable submanifolds of a complex manifold , Trans. Amer. Math. Soc. 142 , 15–35, (1965). MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Examples (CR submanifolds of a complex manifold) J-invariant submanifolds. J ı T x ( M ) ⊂ ı T x ( M ) , H x ( M ) = T x ( M ) , dim R H x ( M ) = n . Real hypersurfaces. dim R H x ( M ) = n − 1 . Totally real submanifolds. H x ( M ) = { 0 } holds at every point x ∈ M . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

A submanifold M of M is called a CR submanifold if there exist distributions H and H ⊥ of constant dimension such that H ⊕ H ⊥ = TM , J H = H , J H ⊥ ⊂ T ⊥ M . A. Bejancu, CR-submanifolds of a K¨ ahler manifold I , Proc. Amer. Math. Soc. , 69 , 135–142, (1978). MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Let M n be a CR submanifold of maximal CR dimension dim R ( JT x ( M ) ∩ T x ( M )) = n − 1 at each point x of M Then it follows that M is odd–dimensional and that there exists a unit vector field ξ normal to M such that JT x ( M ) ⊂ T x ( M ) ⊕ span { ξ x } for any x ∈ M MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Examples real hypersurfaces of almost Hermitian manifolds M ; real hypersurfaces M of complex submanifolds M ′ of almost Hermitian manifolds M ; odd-dimensional F ′ -invariant submanifolds M of real hypersurfaces M ′ of almost Hermitian manifolds M , where F ′ is an almost contact metric structure naturally induced by the almost Hermitian structure on M . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

Defining a skew–symmetric (1 , 1)-tensor F from the tangential projection of J by J ı X = ı FX + u ( X ) ξ, for any X ∈ T ( M ), the Hermitian property of ¯ g implies that the subbundle T ⊥ 1 ( M ) = { η ∈ T ⊥ ( M ) | g ( η, ξ ) = 0 } is J -invariant, from which it follows J ξ = − ı U , g ( U , X ) = u ( X ) , U ∈ T ( M ) . Here, U is a tangent vector field, u is one form on M . Also, from now on we denote the orthonormal basis of T ⊥ ( M ) by ξ, ξ 1 , . . . , ξ q , ξ 1 ∗ , . . . , ξ q ∗ , where ξ a ∗ = J ξ a and q = p − 1 2 . MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds

F 2 X = − X + u ( X ) U , FU = 0 , g ( U , X ) = u ( X ) ( F , u , U , g ) defines an almost contact metric structure on M MIRJANA DJORI´ C, University of Belgrade, Faculty of Mathematics, Serbia Geometry of CR submanifolds