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Locally symmetric contact metric manifolds. (English) Zbl 1103.53047

In 1962, M. Okumura proved that a locally symmetric Sasakian manifold is of constant curvature \(+1\) [see Tohoku Math. J. 14, 135–145 (1962; Zbl 0119.37701)]. S. Tanno proved the same result for a locally symmetric \(K\)-contact manifold [see Proc. Japan Acad. 43, 581–583 (1967; Zbl 0155.49802)]. D. Blair considered unit tangent sphere bundles with their standard contact metric structure, he showed that these are locally symmetric if and only if the base space is either the sphere \(S^2(1)\) or some Euclidean space [see Geometry and topology, Singapore, World Scientific 1989, 15–30]). Concerning the classification of all locally symmetric contact metric \((2n+1)\)-manifolds, Blair conjectured that “the only two possibilities are contact metric manifolds which are locally isometric to \(S^{2n+1}(1)\) or \(E^{n+1}\times S^n(4)\)” [see D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics 203. Boston, MA: Birkhäuser (2002; Zbl 1011.53001)].
In the present paper the authors prove this conjecture. Moreover, they remark that their method, which is a computational one, does not offer any geometric insight into the question why local symmetry and the presence of a contact metric structure are “\(incompatible\)”. They raise the question to find a more conceptual proof of their result. The paper is well written, the topic is well explained and motivated.
Reviewer: D. Perrone (Lecce)

MSC:

53D10 Contact manifolds (general theory)
53C35 Differential geometry of symmetric spaces
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

[1] Blair DE (1989) When is the tangent sphere bundle locally symmetric? In: Geometry and Topology, pp 15–30. Singapore: World Scientific
[2] Blair DE (2001) Riemannian Geometry of Contact and Symplectic Manifolds. Birkhäuser
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