Contact hypersurfaces and CR-symmetry
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Contact hypersurfaces and CR‑symmetry Jong Taek Cho1 Received: 11 September 2019 / Accepted: 18 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that the contact (k, 𝜇)-spaces whose Boeckx invariant is ≠ 1 are realized as real hypersurfaces of the complex quadric Qn and its non-compact dual Qn∗ . Then, we classify simply connected, complete, non-K-contact, CR-symmetric contact metric spaces by real hypersurfaces in Hermitian symmetric spaces. Keywords Contact hypersurface · Complex quadric · Non-compact dual of complex quadric · CR-symmetry Mathematics Subject Classification 53C40 · 53C25 · 53C35
1 Introduction In contact Riemannian geometry, as a generalization of Sasakian manifolds, the notion of contact (k, 𝜇)-spaces was introduced by Blair, Koufogiorgos, and Papantoniou [2]. For (k, 𝜇) ∈ ℝ2 , a contact (k, 𝜇)-space is a contact metric manifold (M;𝜂, 𝜙, 𝜉, g) whose curvature tensor R satisfies
R(X, Y)𝜉 = (kI + 𝜇h)(𝜂(Y)X − 𝜂(X)Y) for all vector fields X, Y on M, where I denotes the identity transformation and 2h ∶= L𝜉 𝜙 is the Lie derivative of 𝜙 along 𝜉 . Contact (k, 𝜇)-spaces provide a large class of strictly pseudo-convex CR manifolds including Sasakian manifolds when k = 1 (and h = 0 ). 𝜇)-space is locally homogeneous. Boeckx [4] showed that every non-Sasakian contact (k, √ Moreover, in [5], he defined a number I = (1 − 𝜇∕2)∕ 1 − k , which is invariant under pseudo-homothetic deformations, and then by using this, he proved a fundamental rigidity theorem. Then, up to equivalence and pseudo-homothetic deformations, the family of simply connected, complete contact (k, 𝜇)-spaces of fixed dimension is parametrized by This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2019R1F1A1040829). * Jong Taek Cho [email protected] 1
Department of Mathematics, Chonnam National University, Gwangju 61186, Korea
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the real numbers. The unit tangent sphere bundles T1 B(c) of a space of constant curvature c ≠ 1 provide examples which exhaust all possible values I > −1 . For the case I ≤ −1 , Boeckx himself gave examples of contact (k, 𝜇)-spaces by a two-parameter family of solvable Lie groups which admit a left-invariant contact metric structure. However, he gave neither geometric description nor explicit realization of such abstract examples. In this context, another construction of models of I ≤ −1 has been given recently by Loiudice and Lotta [20, 21] and Lotta [23]. Namely, they are obtained as tangent hyperquadric bundles T−1 B(c) = {(p, v) ∈ TB(c)|gp (v, v) = −1} of a Lorentzian space form B(c) with c ≠ −1 . On the other hand, Dileo and Lotta [17] showed that for non-Sasakian contact metric manifolds, the (k, 𝜇)-condition is equivalent to the local CR-symmetry, which was introduced by Kaup and Zaitsev [18] as an analogue of the (local) Hermitian symmetry. Boeckx an
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