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].

Chen ideal submanifolds M n in Euclidean ambient spaces E n+m (of arbitrary dimensions n ≥ 2 and codimensions m ≥ 1) at each of their points do realise an optimal equality between their squared mean curvature, which is their main extrinsic scalar valued curvature invariant, and their δ-(= δ(2)-) curvature of Chen, which is one of their main intrinsic scalar valued curvature invariants. From a geometric point of view, the pseudo-symmetric Riemannian manifolds can be seen as the most natural symmetric spaces after the real space forms, i.e. the spaces of constant Riemannian sectional curvature. From an algebraic point of view, the Roter manifolds can be seen as the Riemannian manifolds whose Riemann-Christoffel curvature tensor R has the most simple expression after the real space forms, the latter ones being characterisable as the Riemannian spaces (M n , g) for which the (0, 4) tensor R is proportional to the Nomizu-Kulkarni square of their (0, 2) metric tensor g. In the present article, for the class of the Chen ideal submanifolds M n of Euclidean spaces E n+m , we study the relationship between these geometric and algebraic generalisations of the real space forms.Mathematics Subject Classification (2010). Primary 53B20; Secondary 53B25, 53C42.

We investigate semi-Riemannian manifolds satisfying some curvature conditions. Those conditions are strongly related to pseudosymmetry.

In this paper, we study Chen ideal submanifolds M n of dimension n in Euclidean spaces E n+m (n ≥ 4, m ≥ 1) satisfying curvature conditions of pseudosymmetry type of the form: the difference tensor R · C − C · R is expressed by some Tachibana tensors. Precisely, we consider one of the following three conditions: R ·C −C · R is expressed as a linear combination of Q(g, R) and Q(S, R), R ·C −C · R is expressed as a linear combination of Q(g, C) and Q(S, C) and R · C − C · R is expressed as a linear combination of Q(g, g∧S) and Q(S, g∧S). We then characterize Dedicated to the memory of Professor Franki Dillen. Communicated by Young Jin Suh. B Miroslava Petrović-Torgasev mirapt@kg.ac.rs Ryszard Deszcz Ryszard.Deszcz@up.wroc.pl

In this paper it is shown that all Wintgen ideal submanifolds in ambient real space forms are Chen submanifolds. It is also shown that the Wintgen ideal submanifolds of dimension $ >3 $ in real space forms do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann--Christoffel curvature tensor, of their Ricci curvature tensor and of their Weyl conformal curvature tensor.