A survey on Lagrangian submanifolds of nearly Kaehler six-sphere
type: Item
instance of: scholarly article
title: A survey on Lagrangian submanifolds of nearly Kaehler six-sphere
DOI: 10.1007/978-981-99-9750-3_3
author: Ramesh Sharma
publication date: 2024-10-15T00:00:00Z
review text: As it is known, the Gray-Brown three-fold vector cross-products in the Cayley algebra induce canonical nearly Kähler structures on the six-dimensional sphere \(S^6\). A submanifold \(M\) of the nearly Kähler six-dimensional sphere \(S^6\) is said to be Lagrangian (or totally real) if the operator \(J\) of the almost complex structure maps the tangent bundle of \(M\) onto the normal bundle, and the normal bundle onto the tangent bundle.\N\NThis survey presents the most important results on Lagrangian submanifolds of the nearly Kähler six-sphere obtained by S. Deshmukh, F. Dillen, N. Ejiri, L. Schafer, R. Sharma, K. Smoczyk, L. Vrancken and others. As an example of such a result we mention the following theorem proved by \textit{R. Sharma} [Contemp. Math. 756, 219--228 (2020; Zbl 1459.53031)]:\N\NTheorem. If the almost contact metric structure induced by a unit tangent vector field on a Lagrangian submanifold of the nearly Kählerian six-dimensional sphere is normal, then this almost contact metric structure is Sasakian.\N\NFor the entire collection see [Zbl 1537.53001].
reviewed by: Mihail B. Banaru
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 7926430
zbMATH Keywords: nearly Kähler unit six-sphere
zbMATH Keywords: Lagrangian submanifolds
zbMATH Keywords: Chen's equality
zbMATH Keywords: almost contact structure
zbMATH Keywords: Sasakian
zbMATH Keywords: normal connection
zbMATH Keywords: normal curvature operator
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7926430