Abstract
We characterize two-point homogeneous spaces, locally symmetric spaces, C and B-spaces via properties of the standard contact metric structure of their unit tangent sphere bundle. Further, under various conditions on a Riemannian manifold, we show that its unit tangent sphere bundle is a (locally) homogeneous contact metric space if and only if the manifold itself is (locally) isometric to a two-point homogeneous space.
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REFERENCES
J. Berndt and L. Vanhecke, Two natural generalizations of locally symmetric spaces, Diff. Geom. Appl. 2 (1992), 57–80.
J. Berndt and L. Vanhecke, Geodesic sprays and C-and P-spaces, Rend. Sem. Mat. Univ. Pol. Torino 50 (1992), 343–358.
D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer-Verlag, Berlin, Heidelberg, New York, 1976.
D. E. Blair, Critical associated metrics on contact manifolds III, J. Austral. Math. Soc. 50 (1991), 189–196.
D. Blair, When is the tangent sphere bundle locally symmetric?, In: Geometry and Topology, World Scientific, Singapore (1989), 15–30.
D. E. Blair, T. Koufogiorgos and B. J. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995), 189–214.
E. Boeckx, A class of locally ϕ-symmetric contact metric spaces, Arch. Math.
E. Boeckx and L. Vanhecke, Geometry of the unit tangent sphere bundle, In: Proc. Workshop on Recent Topics in Differential Geometry, Santiago de Compostela, 1997, Public. Depto. Geometría y Topología, Univ. Santiago de Compostela (Spain), 89 (1998), 5–17.
E. Boeckx and L. Vanhecke, Characteristic reflections on unit tangent sphere bundles, Houston J. Math. 23 (1997), 427–448.
E. Boeckx and L. Vanhecke, Unit tangent sphere bundles with constant scalar curvature, Czechoslovak Math. J.
E. Boeckx and L. Vanhecke, Curvature homogeneous unit tangent sphere bundles, Publ. Math. Debrecen.
E. Cartan, Geometry of Riemannian spaces, Math. Sci. Press, Brooklyn, Mass., 1983.
Q. S. Chi, A curvature characterization of certain locally rank one symmetric spaces, J. Differ. Geom. 28 (1988), 187–202.
I. Dotti and M. J. Druetta, Negatively curved homogeneous Osserman spaces.
P. Gilkey, A. Swann and L. Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator, Quart. J. Math. Oxford 46 (1995), 299–320.
O. Kowalski, Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math. 250 (1971), 124–129.
E. Musso and F. Tricerri, Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl. 150 (1988), 1–20.
D. Perrone, Tangent sphere bundles satisfying ∇ɛτ = 0, J. Geom. 49 (1994), 178–188.
D. Perrone, Torsion tensor and critical metrics on contact (2n + 1)-manifolds, Monatsh. Math. 114 (1992), 245–259.
F. PodestÀ, Isometries of tangent sphere bundles, Boll. Un. Mat. It. 3-A (1991), 207–214.
S. Tanno, The standard CR structure on the unit tangent bundle, Tôhoku Math. J. 44 (1992), 535–543.
Y. Tashiro, On contact structures of tangent sphere bundles, Tôhoku Math. J. 21 (1969), 117–143.
L. Vanhecke, Geometry in normal and tubular neighborhoods, Rend. Sem. Fac. Sci. Univ. Cagliari, supplemento al vol. 58 (1988), 73–176.
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Boeckx, E., Perrone, D. & Vanhecke, L. Unit Tangent Sphere Bundles and Two-Point Homogeneous Spaces. Periodica Mathematica Hungarica 36, 79–95 (1998). https://doi.org/10.1023/A:1004629423529
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DOI: https://doi.org/10.1023/A:1004629423529