TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

We consider almost Kenmotsu manifolds with conformal Reeb foliation. We prove that such a foliation produces harmonic morphisms, we study the $k$-nullity distributions and we discuss the isometrical immersion of such a manifold $M$ as hypersurface in a real space form $\widetilde{M}(c)$ of constant curvature $c$ proving that $c \leq -1$ and, if $c<-1$, $M$ is totally umbilical, Kenmotsu and locally isometric to the hyperbolic space of constant curvature $-1$. Finally, the Einstein and $\eta$-Einstein conditions are discussed.

https://projecteuclid.org/download/pdf_1/euclid.bbms/1320763128

Almost Kenmotsu manifolds with conformal Reeb foliation

We classify hypersurfaces of the hyperbolic space ℍn+1(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mn is a complete hypersurfaces with constant scalar curvature n(n − 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n− 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.

Hypersurfaces of the hyperbolic space with constant scalar curvature

Hypersurfaces with constant mean curvature in a hyperbolic space

In this paper, we investigate the nonnegative sectional curvature hypersurfaces in a real space form M
 n+1(c). We obtain some rigidity results of nonnegative sectional curvature hypersurfaces M
 n+1(c) with constant mean curvature or with constant scalar curvature. In particular, we give a certain characterization of the Riemannian product S
 k
 (a) × S
 n-k
 (√1 − a
 2), 1 ≤ k ≤ n − 1, in S
 n+1(1) and the Riemannian product H
 k
 (tanh2
 r − 1) × S
 n-k
 (coth2
 r − 1), 1 ≤ k ≤ n − 1, in H
 n+1(−1).

Nonnegative sectional curvature hypersurfaces in a real space form

Let M n be a complete hypersurface with constant normalized scalar curvature R in a hyperbolic space form H n+1 . We prove that if ¯ R = R + 1 i 0 and the norm square jhj 2 of the second fundamental form of M n satisÞes n ¯ R i supjhj 2 i n

/pdf/hypersurfaces-with-constant-scalar-curvature-in-a-hyperbolic-5cdtvkrwuc.pdf

Hypersurfaces with constant scalar curvature in a hyperbolic space form

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Hypersurfaces with constant scalar curvature in a hyperbolic space form