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

Abstract : In this paper, we study Chen ideal submanifolds Mn of dimension n in Euclidean spaces En+m (n≥4, m≥1) satisfying curvature conditions of pseudo-symmetry 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 Chen ideal submanifolds Mn of dimension n in Euclidean spaces En+m (n≥4, m≥1) which satisfy one of the following six conditions of pseudo-symmetry type: R⋅C−C⋅R and Q(g , R) are linearly dependent, R⋅C−C⋅R and Q(S , R) are linearly dependent, R⋅C−C⋅R and Q(g , C) are linearly dependent, R⋅C−C⋅R and Q(S , C) are linearly dependent, R⋅C−C⋅R and Q(g,g∧S) are linearly dependent and R⋅C−C⋅R and Q(S,g∧S) are linearly dependent. We also prove that the tensors R⋅R−Q(S,R) and Q(g, C) are linearly dependent at every point of Mn at which its Weyl tensor C is non-zero.
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Submitted on : Monday, February 22, 2021 - 11:10:41 AM
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### Citation

Ryszard Deszcz, Miroslava Petrović-Torgašev, Leopold Verstraelen, Georges Zafindratafa. On Chen Ideal Submanifolds Satisfying Some Conditions of Pseudo-symmetry Type. Bulletin of the Malaysian Mathematical Sciences Society, Springer Singapore, 2016, 39 (1), pp.103-131. ⟨10.1007/s40840-015-0164-7⟩. ⟨hal-03148398⟩

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