A classification of spherical symmetric CR manifolds Giulia Dileo - PowerPoint PPT Presentation

Workshop on CR and Sasakian Geometry University of Luxembourg, 24-26 March 2009 A classification of spherical symmetric CR manifolds Giulia Dileo joint work with Antonio Lotta University of Bari Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 1 / 24

1. Pseudohermitian manifolds and contact metric spaces. A pseudohermitian manifold ( M , HM , J , η ) is a strongly pseudoconvex CR manifold of hypersurface type endowed with a pseudohermitian structure , i.e. a nowhere zero 1-form η such that Ker ( η ) = HM and the Levi form L η is positive definite. The Levi form is defined by L η ( X , Y ) = − d η ( X , JY ) X , Y ∈ D where D denotes the module of all smooth sections of HM . Let ξ be the unique nowhere vanishing globally defined vector field transverse to HM such that η ( ξ ) = 1 , d η ( ξ, X ) = 0 for any X ∈ X ( M ) . The Webster metric is defined by g η ( X , Y ) = L η ( X , Y ) , g η ( X , ξ ) = 0 , g η ( ξ, ξ ) = 1 , X , Y ∈ D . Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 2 / 24

Let ϕ be the tensor field of type (1 , 1) such that ϕ ( ξ ) = 0 ϕ ( X ) = JX for any X ∈ D . Then ( ϕ, ξ, η, g η ) is a contact metric structure on M . Conversely, if ( ϕ, ξ, η, g ) is a contact metric structure on M , then M admits a strongly pseudoconvex almost CR structure 1 given by HM = Im ( ϕ ) , J = ϕ | HM . If the almost CR structure is integrable, M is a pseudohermitian manifold, whose Webster metric g η coincides with g . 1 J 2 = − Id and [ X , Y ] − [ JX , JY ] ∈ D for any X , Y ∈ D . Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 3 / 24

A ( κ, µ )-space 2 is a contact metric manifold ( M , ϕ, ξ, η, g ) such that the Riemannian curvature satisfies R ( X , Y ) ξ = κ ( η ( Y ) X − η ( X ) Y ) + µ ( η ( Y ) hX − η ( X ) hY ) for any X , Y ∈ X ( M ), some κ, µ ∈ R , with h = 1 2 L ξ ϕ . The real number κ satisfies κ ≤ 1. If κ = 1 then h = 0 and M is a Sasakian manifold. If κ < 1 then the Riemannian curvature is completely determined. In any case the underlying almost CR -structure is integrable: ( κ, µ )-spaces are pseudohermitian manifolds. 2 D.E. Blair, T. Koufogiorgos, B.J. Papantoniou, Contact metric manifolds satisfying a nullity condition , Israel J. Math., 1995. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 4 / 24

The ( κ, µ )-nullity condition is preserved under a D -homothetic deformation of the structure, defined for any real number a > 0 by ξ = 1 ¯ η = a η, ¯ a ξ, ϕ = ϕ, ¯ g = ag + a ( a − 1) η ⊗ η. ¯ Non Sasakian ( κ, µ )-spaces are locally ϕ -symmetric spaces, 3 i.e. the characteristic reflections (reflections with respect to the integral curve of ξ ) are local isometries. A Sasakian manifold is said to be a globally ϕ -symmetric 4 space if the characteristic reflections are global automorphisms of M , ξ generates a global one-parameter group of automorphisms of the contact structure. We investigate ( κ, µ )-spaces and the notion of ϕ -symmetry in the context of CR geometry. 3 E. Boeckx, A class of locally ϕ -symmetric contact metric spaces , Arch. Math., 1999. 4 T. Takahashi, Sasakian ϕ -symmetric spaces , Tohoku Math. J., 1977. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 5 / 24

2. Symmetric Webster metrics. Let ( M , HM , J , g ) be a Hermitian almost CR manifold, i.e. an almost CR manifold of CR -codimension k ≥ 1, endowed with a Riemannian metric g whose restriction to HM is Hermitian with respect to J . Denote by D ∞ ⊂ X ( M ) the Lie algebra generated by D . Let σ : M → M be an isometric CR -diffeomorphism. Then σ is called a symmetry at the point x ∈ M if σ ( x ) = x , d x σ | H x M ⊕D ∞ ( x ) ⊥ = − Id , D ∞ ( x ) = { X x | X ∈ D ∞ } . A connected Hermitian almost CR manifold M is called (globally) symmetric 5 if for each point x ∈ M there exists a symmetry σ x at x . 5 W. Kaup, D. Zaitsev, On symmetric Cauchy-Riemann manifolds , Adv. Math., 2000. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 6 / 24

Since the symmetry at x in uniquely determined, one can define Hermitian locally symmetric almost CR spaces in a natural manner. Hermitian locally symmetric spaces are CR manifolds. Let ( M , HM , J , η ) be a pseudohermitian manifold with associated contact metric structure ( ϕ, ξ, η, g ). Then M is a Hermitian locally symmetric CR space if and only if at each point x ∈ M the local symmetry σ x defined by σ x = exp x ◦ L x ◦ exp − 1 x , where L x = − Id + 2 η x ⊗ ξ x , is a local isometric CR -diffeomorphism. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 7 / 24

In the Sasakian case , the local symmetry σ x coincides in a suitable neighborhood of x with the characteristic reflection at x . Proposition Let ( M , HM , J , η ) be a pseudohermitian manifold. Assume that the Webster metric g = g η is Sasakian. The following conditions are equivalent: a) ( M , HM , J , g η ) is a locally (globally) symmetric pseudohermitian manifold. b) ( M , ϕ, ξ, η, g ) is a locally (globally) ϕ -symmetric space. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 8 / 24

Sasakian ϕ -symmetric spaces have been classified. 6 They are principal fibre bundles π : M → N , whose fibres are integral curves of ξ , over a Hermitian symmetric space N = N − × C n × N + , where N − , C n and N + are, respectively, of non-compact, Euclidean, and compact type, with a topological obstruction on N + . Sasakian space forms are ϕ -symmetric spaces and they fibre over K¨ ahler space forms: S 2 n +1 → C P n , H 2 n +1 → C n , B n × R → CH n . 6 J.A. Jim´ enez, O. Kowalski, The Classification of ϕ -symmetric Sasakian Manifolds , Monatsh. Math., 1993. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 9 / 24

In the non Sasakian case we prove the following Theorem Let ( M , HM , J , η ) be a pseudohermitian manifold. Assume that the Webster metric g η is not Sasakian. The following conditions are equivalent: a) ( M , HM , J , g η ) is a locally symmetric pseudohermitian manifold. b) ( M , ϕ, ξ, η, g ) is a ( κ, µ ) -space. Non Sasakian ( k , µ )-spaces are classified, up to D -homothetic deformations, by the following invariant introduced by E. Boeckx: 7 I = 1 − µ/ 2 √ 1 − κ . I = 0 if and only if M is locally homothetic to T 1 H n +1 , the tangent sphere bundle of the Riemannian space form of curvature − 1. 7 E. Boeckx, A full classification of contact metric ( κ, µ ) -spaces , Illinois J. Math., 2000. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 10 / 24

3. The Bochner type tensor of a symmetric CR -manifold. Let ( M , HM , J ) be a strongly pseudoconvex CR manifold of hypersurface type and CR -dimension n ≥ 2. Let η and η ′ be two pseudohermitian structures, with subordinate contact metric structures ( ϕ, ξ, η, g ) and ( ϕ ′ , ξ ′ , η ′ , g ′ ). These structures are related by the pseudoconformal change 8 η ′ = e 2 µ η, ξ ′ = e − 2 µ ( ξ + Q ) , ϕ ′ = ϕ + η ⊗ P , g ′ ( X , Y ) = e 2 µ g ( X , Y ) ∀ X , Y ∈ D where µ is a C ∞ -function, P ∈ D is defined by g ( P , X ) = d µ ( X ) for X ∈ D and Q = JP . 8 K. Sakamoto, Y. Takemura, On almost contact structures belonging to a CR-structure , Kodai Math. J., 1980. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 11 / 24

The Bochner curvature tensor defined by K. Sakamoto and Y. Takemura 9 is a pseudoconformal invariant. It coincides with the Chern-Moser-Tanaka invariant tensor field of type (1 , 3). 10 Hence, M is a spherical CR -manifold if and only if B = 0. The definition of the Bochner curvature tensor B involves the Tanaka-Webster connection ˜ ∇ of the pseudohermitian manifold. 9 K. Sakamoto, Y. Takemura, Curvature invariants of CR-manifolds , Kodai Math. J., 1981. 10 S.S. Chern, J.K. Moser, Real hypersurfaces in complex manifolds , Acta Math., 1974. Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 12 / 24

Let ˜ R be the curvature tensor of ˜ ∇ . Consider the two Ricci-type tensor fields k ( X , Y ) = 1 s ( X , Y ) = tr ( V → ˜ 2 tr ( ϕ ˜ R ( V , X ) Y ) , R ( X , ϕ Y )) . The Webster scalar curvature is ρ = tr ( s ) . Consider the tensor fields l , m , L and M defined by 1 1 g ( LX , Y ) = l ( X , Y ) = − 2( n +2) k ( X , Y ) + 8( n +1)( n +2) ρ g ( X , Y ) , g ( MX , Y ) = m ( X , Y ) = l ( JX , Y ) , for any X , Y ∈ D . Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 13 / 24

The Bochner curvature tensor is defined by B = B 0 + B 1 , where, for any X , Y , Z ∈ D , B 0 ( X , Y ) Z = ˜ R ( X , Y ) Z − 2 { m ( X , Y ) JZ + g ( JX , Y ) MZ } + l ( Y , Z ) X − l ( X , Z ) Y + m ( Y , Z ) JX − m ( X , Z ) JY + g ( Y , Z ) LX − g ( X , Z ) LY + g ( JY , Z ) MX − g ( JX , Z ) MY , B 1 ( X , Y ) Z = 1 � � R ( JX , JY ) Z − ˜ ˜ R ( X , Y ) Z . 2 Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 14 / 24