Hypersurfaces with two distinct principal curvatures in a real space form

An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms

“…This approach allows us to state a more general version of our Theorem 6 (or, equivalently, of our Theorem 4) in the following terms. Remark 8 It should be remarked that our results, though somewhat related to, are independent and different from a very recent result by Shu and Han (see Main Theorem in [20]). First of all, Shu and Han only consider the case where H 2 + c ≥ 0, and their result says nothing about the case H 2 + c < 0 (when c = −1).…”

“…First of all, Shu and Han only consider the case where H 2 + c ≥ 0, and their result says nothing about the case H 2 + c < 0 (when c = −1). On the other hand, regarding the characterization of the equalities, their result is weaker than ours since they need to assume that both equalities must be satisfied, the one for the infimum of | | and the one for the supremum of | | (see (1.3) and (1.4) in [20]). In particular, for the characterization of the equalities they are assuming that inf | | = sup | |, which means that | | is constant.…”

“…(4) (with c = 0) we find Scal = n 2 H 2 − |A| 2 = −n(n − 1)λ 2 + 2n(n − 1)λH = n(n − 1)λ(2H − λ). (20) Let ⊂ R be the set of non-zero values assumed by λ and μ, and set ± = ∩ R ± . If sup Scal = −τ 2 < 0, then Scal ≤ −τ 2 < 0 and thus, by (20),…”

Curvature properties of some class of hypersurfaces in Euclidean spaces

“…Curvature conditions of pseudosymmetry type on hypersurfaces M in N n+1 s (c), n 4, satisfying on M U H the equation (12) H 2 = ψH + ρg, for some functions ψ and ρ on this set, were investigated among others in [2,7,9,18,27,33,35,45,47,54]. Examples of hypersurfaces in spaces of constant curvature satisfying (1.2) are given among others in [27,35,43,56,57].…”