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Totally Real Minimal Surfaces with Non‐Circular Ellipse of Curvature in the Nearly Kähler S 6
Author(s) -
Bolton John,
Vrancken Luc,
Woodward Lyndon M.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005541
Subject(s) - ellipse , curvature , geodesic , constant curvature , constant (computer programming) , geometry , mathematics , mathematical analysis , computer science , programming language
In [ 2 ] we discussed almost complex curves in the nearly Kähler S 6 . These are surfaces with constant Kähler angle 0 or π and, as a consequence of this, are also minimal and have circular ellipse of curvature. We also considered minimal immersions with constant Kähler angle not equal to 0 or π, but with ellipse of curvature a circle. We showed that these are linearly full in a totally geodesic S 5 in S 6 and that (in the simply connected case) each belongs to the S 1 ‐family of horizontal lifts of a totally real (non‐totally geodesic) minimal surface in C P 2 . Indeed, every element of such an S 1 ‐family has constant Kähler angle and in each family all constant Kähler angles occur. In particular, every minimal immersion with constant Kähler angle and ellipse of curvature a circle is obtained by rotating an almost complex curve which is linearly full in a totally geodesic S 5 .