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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 Lη is positive definite. The Levi form is defined by Lη(X, Y ) = −dη(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, dη(ξ, X) = 0 for any X ∈ X(M). The Webster metric is defined by gη(X, Y ) = Lη(X, Y ), gη(X, ξ) = 0, gη(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, gη) 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 structure1 given by HM = Im(ϕ), J = ϕ|HM. If the almost CR structure is integrable, M is a pseudohermitian manifold, whose Webster metric gη coincides with g.

1 J2 = −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)-space2 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

2Lξϕ.

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 ¯ η = aη, ¯ ξ = 1 aξ, ¯ 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

• f ξ) are local isometries.

A Sasakian manifold is said to be a globally ϕ-symmetric4 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, dxσ|HxM⊕D∞(x)⊥ = −Id, D∞(x) = {Xx | X ∈ D∞}. A connected Hermitian almost CR manifold M is called (globally) symmetric5 if for each point x ∈ M there exists a symmetry σx 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 σx defined by σx = expx ◦Lx ◦ exp−1

x ,

where Lx = −Id + 2ηx ⊗ ξx, 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 σx 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 = gη is Sasakian. The following conditions are equivalent: a) (M, HM, J, gη) 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− × Cn × N+, where N−, Cn 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: S2n+1 → CPn, H2n+1 → Cn, Bn × R → CHn.

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 gη is not Sasakian. The following conditions are equivalent: a) (M, HM, J, gη) 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 T1Hn+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 change8 η′ = e2µη, ξ′ = e−2µ(ξ + Q), ϕ′ = ϕ + η ⊗ P, g′(X, Y ) = e2µg(X, Y ) ∀ X, Y ∈ D where µ is a C∞-function, P ∈ D is defined by g(P, X) = dµ(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. Takemura9 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 s(X, Y ) = tr(V → ˜ R(V , X)Y ), k(X, Y ) = 1 2tr(ϕ˜ R(X, ϕY )). The Webster scalar curvature is ρ = tr(s). Consider the tensor fields l, m, L and M defined by g(LX, Y ) = l(X, Y ) = −

1 2(n+2)k(X, Y ) + 1 8(n+1)(n+2)ρg(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 = B0 + B1, where, for any X, Y , Z ∈ D, B0(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 , B1(X, Y )Z = 1 2

• ˜

R(JX, JY )Z − ˜ R(X, Y )Z

• .

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 14 / 24

Let (M, HM, J, η) be a pseudohermitian manifold of CR-dimension n ≥ 2. Let σ ⊂ HxM be a holomorphic 2-plane at a point x ∈ M and {X, JX} an orthonormal basis of σ. The pseudoholomorphic sectional curvature of σ is ˜ K(σ) = ˜ Rx(X, JX, X, JX) which depends only on σ. A pseudohermitian space form is a pseudohermitian manifold for which ˜ K(σ) does not depend on σ and on the point x.

Theorem

A pseudohermitian space form of CR-dimension n ≥ 2 is a spherical CR manifold.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 15 / 24

Theorem

Let (M, HM, J, η) be a non Sasakian locally symmetric pseudohermitian manifold having CR-dimension n ≥ 2. Let (ϕ, ξ, η, g) be the underlying (κ, µ)-contact metric structure. Then, the following conditions are equivalent: i) B = 0, ii) the Boeckx invariant I = 0, iii) µ = 2, iv) the Webster scalar curvature ρ vanishes, v) M has constant pseudoholomorphic curvature. If any of the above conditions holds, then ˜ K = 0, but ˜ R = 0.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 16 / 24

• 4. Classification result.

Let (M, HM, J, η) be a simply connected pseudohermitian manifold

• f CR-dimension n ≥ 2 which is a spherical CR manifold and such

that gη is symmetric. If gη is not Sasakian, then M is a (k, µ)-space with vanishing Boeckx

• invariant. Hence, M is homothetic to T1Hn+1.

If gη is Sasakian, then M is a simply connected Sasakian ϕ-symmetric space and thus it is a principal fibre bundle π : M → N

• ver a simply connected Hermitian globally symmetric space with

fibres tangent to ξ. Since M is spherical, N is Bochner-flat. According to a result of M. Matsumoto and S. Tanno11 we have two possibilities:

11 M. Matsumoto, S. Tanno, K¨

ahlerian spaces with parallel or vanishing Bochner curvature tensor, Tensor (N.S.), 1973.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 17 / 24

• N is a simply connected K¨

ahler space form. In this case, M is a Sasakian space form and hence, it is homothetic to S2n+1, H2n+1, or Bn × R.

• N is isometric to a product Nk(c) × Nn−k(−c), c > 0, of two

simply connected K¨ ahler space forms with holomorphic curvatures respectively c and −c. In this case, up to a homothetic change, we can assume c = 1. Hence, M is homothetic to a principal fibre bunble Pn

k over

CPk × CHn−k, 0 < k < n.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 18 / 24

Theorem

Every simply connected, spherical symmetric pseudohermitian manifold of CR-dimension n ≥ 2 is homothetic to one of the following spaces: S2n+1, H2n+1, Bn × R, T1Hn+1, Pn

1 ,

. . . , Pn

n−1.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 19 / 24

• 5. The CR geometry of tangent sphere bundles.

Let (M, g) be a Riemannian manifold of dimension n ≥ 3. Consider the canonical projection π : TM → M, π(x, u) = x, x ∈ M, u ∈ TxM. For each smooth vector field X ∈ X(M) we denote by X V the vertical lift to TM, X H the horizontal lift with respect to the Levi-Civita connection. Fixed a real number λ = 0, we define the almost complex structure Jλ : TTM → TTM Jλ(X H) = λX V , Jλ(X V ) = − 1 λX H. The tangent sphere bundle TrM of radius r > 0 is the hypersurface

• f TM

TrM = {(x, u) ∈ TM | gx(u, u) = r2}, which inherits an almost CR structure (H(TrM), Jλ) from Jλ.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 20 / 24

Theorem

For each r > 0 and λ > 0, (TrM, H(TrM), Jλ) is a strictly pseudoconvex almost CR manifold. If (M, g) has constant curvature, then TrM is a Hermitian locally symmetric CR space with respect to the Webster metric gηλ, where ηλ is the pseudohermitian structure such that ηλ(ξ) = 1. The vector field ξ is defined by ξ = −2JλU where U is the canonical vertical vector field of TM, locally given by U = vi ∂ ∂vi in a coordinate system (xi, vi) of TM induced by a local chart (U, x1, . . . , xn) of M.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 21 / 24

Theorem

Let (M, g) be a Riemannian manifold with constant curvature K and dimension n ≥ 3. Fix r > 0, λ > 0 and consider the CR manifold (TrM, H(TrM), Jλ). Then a) TrM is spherical if and only if λ2 + Kr2 = 0. b) The metric gηλ is Sasakian if and only if λ2 − Kr2 = 0. c) When (TrM, H(TrM), Jλ, gηλ) is not Sasakian, its Boeckx invariant is I = λ2 + Kr2 |λ2 − Kr2|. Hence, when K = 0, each TrM admits a one-parameter family (H(TrM), Jλ, ηλ) of locally symmetric non homothetic pseudohermitian structures.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 22 / 24

Corollary

Let (M, g) be a Riemannian manifold with constant curvature K and dimension n ≥ 3. Let (H(TrM), J, ηr) be the standard pseudohermitian structure on TrM, corresponding to λ = 1. Then a) K < 0 if and only if there exists r > 0 such that TrM is a spherical CR manifold; r is unique and satisfies K = − 1 r2 . b) K > 0 if and only if there exists r > 0 such that the Webster metric gηr is Sasakian; r is unique and satisfies K = 1 r2 . c) K = 0 if and only if for each r, r′ > 0, (TrM, H(TrM), J, ηr) and (Tr′M, H(Tr′M), J, ηr′) are locally homothetic pseudohermitian manifolds.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 23 / 24

• G. Dileo, A. Lotta,

A classification of spherical symmetric CR manifolds,

• Bull. Aust. Math. Soc. (2009),

doi: 10.1017/S0004972709000252.

Giulia Dileo (University of Bari) A classification of spherical symmetric CR manifolds March 2009 24 / 24