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SM ISO690:2012 BANARU, Galina, BANARU, Mihail. Hermitian geometry of sixdimensional planar submanifolds of Cayley algebra. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 2527 august 2022, Chişinău. Chișinău, Republica Moldova: Bons Offices, 2022, Ediţia a 29 (R), pp. 132134. ISBN 9789975810746. 
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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, 2527 august 2022  


Pag. 132134 



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1. The almost Hermitian structures belong to the most important and meaningful differentialgeometrical structures. The existence of GrayBrown 3vector cross products [1] in Cayley algebra gives a set of substantive examples of almost Hermitian manifolds. As it is well known, every 3vector cross product on Cayley algebra induces a 1vector cross product (or, what is the same in this case, an almost Hermitian structure) on its sixdimensional oriented submanifold [2], [3]. Such almost Hermitian structures (in particular, K¨ahlerian, nearly K¨ahlerian, quasi K¨ahlerian, Hermitian, special Hermitian etc) were studied by a number of outstanding geometers: E. Calabi, A. Gray, V. F. Kirichenko, K. Sekigawa and L. Vranchen. We recall that an almost Hermitian manifold is a 2ndimensional manifold M2n with a Riemannian metric g = ⟨·, ·⟩ and an almost complex structure J. Moreover, the following condition must hold ⟨JX, JY ⟩ = ⟨X, Y ⟩ , X, Y ∈ ℵ(M2n), where ℵ(M2n) is the module of smooth vector fields on M2n. An almost Hermitian manifold is Hermitian, if its almost complex structure is integrable [4]. 2. In the present work, we consider sixdimensional Hermitian planar submanifolds of Cayley algebra. We present the following results. Theorem 1. If a sixdimensional Hermitian planar submanifold of Cayley algebra satisfies the UKenmotsu hypersurfaces axiom, then it is K¨ahlerian. Theorem 2. A symmetric nonK¨ahlerian Hermitian sixdimensional submanifold of Ricci type does not admit totally umbilical Kenmotsu hypersurfaces. Theorem 3. If a sixdimensional Hermitian planar submanifold of general type of Cayley algebra satisfies the 1cosymplectic hypersurfaces axiom, then it is K¨ahlerian. Theorem 4. The Hermitian structure on a 6dimensional planar submanifold of Cayley algebra is stable if and only if such submanifold is totally geodesic. Theorem 5. The quasiSasakian structure on a totally umbilical hypersurface of a sixdimensional Hermitian planar submanifold of Cayley algebra is Sasakian. 

