﻿﻿ ﻿ ﻿﻿ On minimal Endo hypersurfaces in nearly Kahlerian manifolds
 Articolul precedent Articolul urmator 74 0 SM ISO690:2012BANARU, Mihail. On minimal Endo hypersurfaces in nearly Kahlerian manifolds. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Bons Offices, 2022, Ediţia a 29 (R), pp. 130-132. ISBN 978-9975-81-074-6. EXPORT metadate: Google Scholar Crossref CERIF DataCiteDublin Core
Conference on Applied and Industrial Mathematics
Ediţia a 29 (R), 2022
Conferința "Conference on Applied and Industrial Mathematics"
29, Chişinău, Moldova, 25-27 august 2022

 On minimal Endo hypersurfaces in nearly Kahlerian manifolds

Pag. 130-132

 Banaru Mihail Smolensk State University Disponibil în IBN: 21 decembrie 2022

Rezumat

As it is known, one of the most important examples of almost contact metric structures is the structure induced on an oriented hypersurface in an almost Hermitian manifold [1], [2]. We recall that an almost contact metric structure on a manifold N of odd dimension is defined by a system of tensor fields {Φ, ξ, η, g} on this manifold: here ξ is a vector, η is a covector, Φ is a tensor of the type (1, 1) and g = ⟨·, ·⟩ is the Riemannian metric. Moreover, the following conditions are fulfilled: η(ξ) = 1, Φ(ξ) = 0, η ◦Φ = 0, Φ2 = −id + ξ ⊗ η, ⟨ΦX,ΦY ⟩ = ⟨X, Y ⟩ − η (X) η (Y ) , X, Y ∈ ℵ(N), where ℵ(N) is the module of C∞-smooth vector fields on the manifold N [1]. The almost contact metric structure is called Endo (or nearly cosymplectic), if (∇X Φ) X = 0, X ∈ ℵ(N). In the present note, Endo hypersurfaces in nearly K¨ahlerian manifolds (NK-manifolds, or W1-manifolds, using Gray–Hervella notation [3]) are considered. Such Endo hypersurfaces are characterized in terms of their type number. A simple criterion of the minimality of Endo hypersurfaces of nearly K¨ahlerian manifolds is established. The following Theorem contains the main result of our note. Theorem. If N is an Endo hypersurface in a nearly K¨ahlerian manifold M2n and t is its type number, then the following statements are equivalent: 1) N is a minimal hypersurface in M2n; 2) N is a totally geodesic hypersurface in M2n; 3) t ≡ 0. Taking into account that the six-dimensional sphere is an example of a nearly K¨ahlerian manifold [1], [4], we deduce that there are not minimal non-geodesic Endo hypersurfaces in S6 . We remark that this work is a continuation of the researches of the author, who studied diverse almost contact metric structures on oriented hypersurfaces in nearly K¨ahlerian manifolds [5]–[9].