Curvature properties of some class of hypersurfaces in Euclidean spaces

Abstract

We determine curvature properties of pseudosymmetry type of hypersurfaces in Euclidean spaces E n+1 , n 5, having three distinct nonzero principal curvatures λ 1 , λ 2 and λ 3 of multiplicity 1, p and n−p−1, respectively. For some hypersurfaces having this property the sum of λ 1 , λ 2 and λ 3 is equal to the trace of the shape operator of M. We present an example of such hypersurface.

Hypersurfaces in space forms satisfying some generalized Einstein metric condition

“…Hypersurfaces in N n+1 s (c) satisfying pseudosymmetry type conditions were investigated in several papers, see, e.g., [74, Section 1] and references therein. We also refer to [33,36,77] for recent results related to this subject. For instance, if M is a hypersurface in N n+1 s (c), n ≥ 3, satisfying…”

“…where ψ and ρ are some functions on U H . We refer to [31,33,37,73,74,75,76,77] for results on hypersurfaces satisfying (1.17). Evidently, if M is a hypersurface in an 4-dimensional Riemannian space of constant curvature N 4 (c) then (1.17…”

“…where φ, ψ and ρ are some functions on U H . Hypersurfaces M in N n+1 s (c), n ≥ 4, satisfying (1.20) on U H ⊂ M were investigated among others in [8,36,73,77], see also [9,Chapters 3.8.3 and 5.6] and references therein.…”

Curvature properties of some class of minimal hypersurfaces in Euclidean spaces

“…If p n − p − 1 then in view of Theorem 3.2 the Riemann-Christoffel curvature tensor R of the cone M is expressed at every point by a linear combination of the tensors ∧ , ∧ S and S ∧ S, ∧ S 2 , S ∧ S 2 and S 2 ∧ S 2 . We refer to [50] and [52] for further results on hypersurfaces with the curvature tensor having the above presented property. Now (39), by (3.9) of [23], and the conditions κ = 0 and ψ = − εκ n−1 , reduces on U H to…”

“…for some functions φ, ψ and ρ, were investigated in the following papers: [9]- [13], [17]- [18], [21]- [23], [25], [28]- [31], [33], [36], [40], [48]- [52]. The main results of Section 3 are presented in Proposition 3.1 and Theorem 3.2.…”