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The axiom of Sasakian hypersurfaces and six-dimensional Hermitian submanifolds of the octonion algebra. (English. Russian original) Zbl 1348.53061
Math. Notes 99, No. 1, 155-159 (2016); translation from Mat. Zametki 99, No. 1, 140-144 (2016).
From the introduction: This paper is part of a research that the author began in the 1990s under the supervision of Kirichenko and continues up to now. In particular, the paper [the author, Sb. Math. 194, No. 8, 1125–1136 (2003; Zbl 1079.53103); translation from Mat. Sb. 194, No. 8, 13–24 (2003)] contains the following result on six-dimensional Hermitian submanifolds of the octonion algebra: if a six-dimensional Hermitian submanifold \(M^6\) of the Cayley algebra satisfies the axiom of \(u\)-Sasakian hypersurfaces (i.e., a totally umbilical Sasakian manifold. hypersurface passes through each point of \(M^6\)), then \(M^6\) is a Kähler manifold. In this paper, we remove the requirement that the Sasakian hypersurface be totally umbilical. Our purpose is to determine properties of a six-dimensional Hermitian submanifold satisfying the axiom of Sasakian hypersurfaces in the octonion algebra.

MSC:
53C40 Global submanifolds
11R52 Quaternion and other division algebras: arithmetic, zeta functions
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
16S35 Twisted and skew group rings, crossed products
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