On Chen Ideal Submanifolds Satisfying Some Conditions of Pseudo-symmetry Type

Abstract

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

Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

“…We mention that Chen ideal submanifolds in Euclidean spaces satisfying the presented above curvature conditions, as well as other conditions of this kind were studied among others in [10] and [13].…”

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

“…We denote by ∇, R, S, κ and C the Levi-Civita connection, the Riemann-Christoffel curvature tensor, the Ricci tensor, the scalar curvature and the Weyl conformal curvature tensor of (M, g), respectively. We refer to [5,11,14,17] for precise definitions of the symbols used. Let (M , g) and ( N , g), dim M = p, dim N = n − p, 1 p < n, be semi-Riemannian manifolds and F a positive smooth function on M .…”

“…where L R is some function on this set. It seems that the Schwarzschild spacetime, the Kottler spacetime, the Reissner-Nordstrøm spacetime, as well as some Friedmann-Lemaître-Robertson-Walker spacetimes are the "oldest" examples of non-semisymmetric pseudosymmetric manifolds, see, e.g., [11,14]. A semi-Riemannian manifold (M, g), n 3, is called Ricci-pseudosymmetric if the tensors R • S and Q(g, S) are linearly dependent at every point of M , see, e.g., [11,13,19].…”

Curvature properties of some class of minimal hypersurfaces in Euclidean spaces

“…, β 4 are defined by (16). The Roter type manifolds (and in particular, hypersurfaces in space forms) were studied among others in the papers: [19], [20], [22], [25], [26], [32], [39] and [44]. (iii) Let M be a n-dimensional hypersurface in the Euclidean space E n+1 , n ≥ 4.…”

“…We refer to [13], [35], [19], [27] and [32] for results on semi-Riemannian manifolds (M, ), dim M ≥ 4, and in particular,…”