Abstract
We investigate the immersed hypersurfaces in space forms ℕn + 1(c), n ≥ 4 with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any immersed hypersurface in space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for the Clifford hypersurfaces in terms of the trace free part of the second fundamental form.
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References
Bao D, Chern S S and Shen Z, An Introduction to Riemannian–Finsler Geometry (Springer-Verlag) (2000)
Chang Y C, Constant mean curvature hypersurfaces with two principal curvatures in a sphere, Monatsh. Math. 158 (2009) 1–22
Hu Z and Zhai S, Hypersurfaces of the hyperbolic space with constant scalar curvature, Results Math. 48 (2005) 65–88
Otsuki T, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970) 145–173
Wu B Y, On hypersurfaces with two distinct principal curvatures in a unit sphere, Diff. Geom. Appl. 27 (2009) 623–634
Wu B Y, On hypersurfaces with two distinct principal curvatures in Euclidean space, Houston J. Math. 36 (2010) 451–467
Wu B Y, On space-like hypersurfaces with two distinct principal curvatures in Lorentzian space forms, J. Math. Anal. Appl. 372 (2010) 244–251
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WU, B.Y. On hypersurfaces with two distinct principal curvatures in space forms. Proc Math Sci 121, 435–446 (2011). https://doi.org/10.1007/s12044-011-0043-6
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DOI: https://doi.org/10.1007/s12044-011-0043-6