On Wintgen Ideal Submanifolds Satisfying Some Pseudo-symmetry Type Curvature Conditions

Year 2024, Volume: 17 Issue: 1, 72 - 96, 23.04.2024

Miroslava Petrović-torgašev , Ryszard Deszcz , Małgorzata Głogowska , Georges Zafindratafa

https://doi.org/10.36890/iejg.1404576

Abstract

Let M be a Wintgen ideal submanifold of dimension n in a real space form Rn+m(k) of dimension (n + m) and of constant curvaturek, n ≥ 4, m ≥ 1. Let g, R, Ricc, g ∧ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms Rn+m(k), n ≥ 4, m ≥ 1, satisfying the following pseudo-symmetry type curvature conditions:
(i) the tensors R · C and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent;
(ii) the tensors C · R and Q(g, R) (resp., Q(g, C), Q(g, g ∧ Ricc), Q(Ricc, R) or Q(Ricc, g ∧ Ricc)) are linearly dependent;
(iii) the tensors R·C -C ·R and Q(g, R) (resp., Q(g, C), Q(g, g∧Ricc), Q(Ricc, R) or Q(Ricc, g∧Ricc)) are linearly dependent.

Keywords

Tachibana tensor, pseudo-symmetry type curvature condition, generalized Einstein metric condition, submanifold, Wintgen ideal submanifold, DDVV conjecture

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