Curvature properties of some class of minimal hypersurfaces in Euclidean spaces

Abstract

We determine curvature properties of pseudosymmetry type of some class of minimal 2-quasiumbilical hypersurfaces in Euclidean spaces E n+1 , n ≥ 4. We present examples of such hypersurfaces. The obtained results are used to determine curvature properties of biharmonic hypersurfaces with three distinct principal curvatures in E 5. Those hypersurfaces were recently investigated by Y. Fu in [38].

Curvature properties of Gödel metric

“…In this way, we obtain examples of quasi-Einstein and 2-quasi-Einstein manifolds [9,10] with pseudosymmetric Weyl conformal curvature tensor. We mention that quasi-Einstein and 2-quasi-Einstein hypersurfaces with pseudosymmetric Weyl conformal curvature tensor immersed isometrically in semi-Riemannian spaces of constant curvature were investigated in the paper [33] (cited in [5]) and [4], respectively.…”

“…We recall that semi-Riemannian manifolds satisfying the condition rank(S − ρg) = 2 are called 2-quasi-Einstein manifolds (see, e.g. [2][3][4]). We present now an application of the last theorem.…”

Professor Mileva Prvanovic: Her contribution to the theory of pseudosymmetry type manifolds

“…There are different extensions of the class of quasi-Einstein manifolds. For instance we have the class of almost quasi-Einstein manifolds, see, e.g., [4], or the class of 2-quasi-Einstein manifolds, see, e.g., [11,12].…”

Curvature properties of some 4-dimensional semi-Riemannian metrics

“…It is clear that the 1-form A as well as the function β are non-zero at every point on U S . The manifold is said to be a 2-quasi-Einstein [29,31,33] manifold if rank(S − αg) ≤ 2 and rank(S − αg) = 2 on some open non-empty subset of U S , where α is some function on U S .…”