On minimal Endo hypersurfaces in nearly Kahlerian manifolds
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BANARU, 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.
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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


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].