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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.: Year: Volume: Issue: Page: Find

 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2018, Volume 146, Pages 17–47 (Mi into288)

On Almost Complex Structures on Six-Dimensional Products of Spheres

N. A. Daurtseva, N. K. Smolentsev

Kemerovo State University

Abstract: In this paper, we discuss almost complex structures on the sphere $S^6$ and on the products of spheres $S^3\times S^3$, $S^1\times S^5$, and $S^2\times S^4$. We prove that all almost complex Cayley structures that naturally appear from their embeddings into the Cayley octave algebra $\mathbb{C}\mathrm{a}$ are nonintegrable. We obtain expressions for the Nijenhuis tensor and the fundamental form $\omega$ for each gauge of the space $\mathbb{C}\mathrm{a}$ and prove the nondegeneracy of the form $d\omega$. We show that through each point of a fiber of the twistor bundle over $S^6$, a one-parameter family of Cayley structures passes. We describe the set of $U(2)\times U(2)$-invariant Hermitian metrics on $S^3\times S^3$ and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on $S^3\times S^3=SU(2)\times SU(2)$ and prove the properties of left-invariant structures that yield the maximal value of the norm of the Nijenhuis tensor on the set of left-invariant, orthogonal, almost complex structures.

Keywords: product of spheres, complex structure, almost complex Cayley structure, octave algebra

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Bibliographic databases:
UDC: 514.76
MSC: 51M15

Citation: N. A. Daurtseva, N. K. Smolentsev, “On Almost Complex Structures on Six-Dimensional Products of Spheres”, Geometry, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 146, VINITI, Moscow, 2018, 17–47

Citation in format AMSBIB
\Bibitem{DauSmo18} \by N.~A.~Daurtseva, N.~K.~Smolentsev \paper On Almost Complex Structures on Six-Dimensional Products of Spheres \inbook Geometry \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2018 \vol 146 \pages 17--47 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into288} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3824398}