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SM ISO690:2012 BANARU, Galina, BANARU, Mihail. Hermitian geometry of six-dimensional planar submanifolds of Cayley algebra. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Casa Editorial-Poligrafică „Bons Offices”, 2022, Ediţia a 29, pp. 132-134. ISBN 978-9975-81-074-6. |
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Conference on Applied and Industrial Mathematics Ediţia a 29, 2022 |
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Conferința "Conference on Applied and Industrial Mathematics" 29, Chişinău, Moldova, 25-27 august 2022 | |||||
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Pag. 132-134 | |||||
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1. The almost Hermitian structures belong to the most important and meaningful differential-geometrical structures. The existence of Gray-Brown 3-vector cross products [1] in Cayley algebra gives a set of substantive examples of almost Hermitian manifolds. As it is well known, every 3-vector cross product on Cayley algebra induces a 1vector cross product (or, what is the same in this case, an almost Hermitian structure) on its six-dimensional 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 2n-dimensional 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 six-dimensional Hermitian planar submanifolds of Cayley algebra. We present the following results. Theorem 1. If a six-dimensional Hermitian planar submanifold of Cayley algebra satisfies the U-Kenmotsu hypersurfaces axiom, then it is K¨ahlerian. Theorem 2. A symmetric non-K¨ahlerian Hermitian six-dimensional submanifold of Ricci type does not admit totally umbilical Kenmotsu hypersurfaces. Theorem 3. If a six-dimensional Hermitian planar submanifold of general type of Cayley algebra satisfies the 1-cosymplectic hypersurfaces axiom, then it is K¨ahlerian. Theorem 4. The Hermitian structure on a 6-dimensional planar submanifold of Cayley algebra is stable if and only if such submanifold is totally geodesic. Theorem 5. The quasi-Sasakian structure on a totally umbilical hypersurface of a six-dimensional Hermitian planar submanifold of Cayley algebra is Sasakian. |
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<?xml version='1.0' encoding='utf-8'?> <oai_dc:dc xmlns:dc='http://purl.org/dc/elements/1.1/' xmlns:oai_dc='http://www.openarchives.org/OAI/2.0/oai_dc/' xmlns:xsi='http://www.w3.org/2001/XMLSchema-instance' xsi:schemaLocation='http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd'> <dc:creator>Banaru, G.A.</dc:creator> <dc:creator>Banaru, M.B.</dc:creator> <dc:date>2022</dc:date> <dc:description xml:lang='en'><p>1. The almost Hermitian structures belong to the most important and meaningful differential-geometrical structures. The existence of Gray-Brown 3-vector cross products [1] in Cayley algebra gives a set of substantive examples of almost Hermitian manifolds. As it is well known, every 3-vector cross product on Cayley algebra induces a 1vector cross product (or, what is the same in this case, an almost Hermitian structure) on its six-dimensional 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 2n-dimensional 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 six-dimensional Hermitian planar submanifolds of Cayley algebra. We present the following results. Theorem 1. If a six-dimensional Hermitian planar submanifold of Cayley algebra satisfies the U-Kenmotsu hypersurfaces axiom, then it is K¨ahlerian. Theorem 2. A symmetric non-K¨ahlerian Hermitian six-dimensional submanifold of Ricci type does not admit totally umbilical Kenmotsu hypersurfaces. Theorem 3. If a six-dimensional Hermitian planar submanifold of general type of Cayley algebra satisfies the 1-cosymplectic hypersurfaces axiom, then it is K¨ahlerian. Theorem 4. The Hermitian structure on a 6-dimensional planar submanifold of Cayley algebra is stable if and only if such submanifold is totally geodesic. Theorem 5. The quasi-Sasakian structure on a totally umbilical hypersurface of a six-dimensional Hermitian planar submanifold of Cayley algebra is Sasakian.</p></dc:description> <dc:source>Conference on Applied and Industrial Mathematics (Ediţia a 29) 132-134</dc:source> <dc:title>Hermitian geometry of six-dimensional planar submanifolds of Cayley algebra</dc:title> <dc:type>info:eu-repo/semantics/article</dc:type> </oai_dc:dc>