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A homogeneous submanifold with nonzero parallel mean curvature vector in a Euclidean sphere. (English) Zbl 1244.53062

Summary: We show that every sufficiently high dimensional Euclidean sphere admits an odd dimensional Riemannian submanifold \(M\) having the following properties: (1) \(M\) is a homogeneous submanifold with nonzero parallel mean curvature vector in the ambient sphere; (2) \(M\) is a Berger sphere; (3) \(M\) is a Sasakian space form of constant \({\phi}\)-sectional curvature. Note that our manifold \(M\) is diffeomorphic but not isometric to a Euclidean sphere.

MSC:

53C40 Global submanifolds
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[1] Adachi T., Maeda S., Yamagishi M.: Length spectrum of geodesic spheres in a non-flat complex space form. J. Math. Soc. Jpn. 54, 373–408 (2002) · Zbl 1037.53019 · doi:10.2969/jmsj/05420373
[2] Blair D.E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Mathematics, vol. 509. Springer, Berlin (1976) · Zbl 0319.53026
[3] do Carmo M.P., Wallach N.R.: Minimal immersions of spheres into spheres. Ann. Math. 93, 43–62 (1971) · Zbl 0218.53069 · doi:10.2307/1970752
[4] Ferus D.: Immersions with parallel second fundamental form. Math. Z. 140, 87–93 (1974) · Zbl 0287.53037 · doi:10.1007/BF01218650
[5] Niebergall R., Ryan P.J. et al.: Real hypersurfaces in complex space forms. In: Cecil, T.E. (eds) Tight and Taut Submanifolds, pp. 233–305. Cambridge University Press, Cambridge (1998) · Zbl 0904.53005
[6] O’Neill B.: Isotropic and Kaehler immersions. Can. J. Math. 17, 907–915 (1965) · Zbl 0171.20503 · doi:10.4153/CJM-1965-086-7
[7] Takahashi T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Jpn. 18, 380–385 (1966) · Zbl 0145.18601 · doi:10.2969/jmsj/01840380
[8] Weinstein A.: Distance spheres in complex projective spaces. Proc. Am. Math. Soc. 39, 649–650 (1973) · Zbl 0243.53040 · doi:10.1090/S0002-9939-1973-0315631-0
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