Lagrangian submanifolds of the nearly Kähler full flag manifold F1,2(ℂ3)

2020 ◽  
Vol 158 ◽  
pp. 103844 ◽  
Author(s):  
Reinier Storm
Keyword(s):  
Lagrangian Submanifolds ◽  
Flag Manifold ◽  
Nearly Kähler

scholarly journals Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 from minimal surfaces in 𝕊3

2018 ◽  
Vol 149 (03) ◽  
pp. 655-689 ◽  
Author(s):  
Burcu Bektaş ◽  
Marilena Moruz ◽  
Joeri Van der Veken ◽  
Luc Vrancken
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Lagrangian Submanifolds ◽  
Large Family ◽  
Maximal Rank ◽  
Totally Geodesic ◽  
Nearly Kähler

AbstractWe study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

scholarly journals H-Umbilical Lagrangian Submanifolds of the Nearly Kähler \( {\mathbb{S}^3\times\mathbb{S}^3} \)

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1427
Author(s):  
Miroslava Antić ◽  
Marilena Moruz ◽  
Joeri Van der Veken
Keyword(s):  
Lagrangian Submanifolds ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Nearly Kähler

H-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that, in the homogeneous nearly Kähler S3×S3, also H-umbilical Lagrangian submanifolds are automatically totally geodesic.

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