Title: Submanifolds of the nearly Kähler manifold S³ x S³
Other Titles: Deelvariëteiten van de bijna-Kählervariëteit S³ x S³
Authors: Dioos, Bart
Issue Date: 29-May-2015
Abstract: Nearly Kähler manifolds are almost Hermitian manifolds for which the Kähler condition is relaxed:instead of being zero, the covariant derivative of the almost complex structure is merely skew-symmetric.The only homogeneous strict nearlyauml;hler manifolds of dimension six are the six-dimensional sphere, the product of two three-spheres S3xS3, the three-dimensional complex projective space and the manifold of full flags in complex three-space. In this thesis we study two types of submanifolds in the nearly Kähler manifold S3xS3, namely almost complex surfaces and Lagrangian submanifolds.Almost complex surfaces are surfaces for which the almost complex structure leaves the tangent spaces invariant. Lagrangian submanifolds are submanifolds for which the almost complex structure gives annbsp;between the tangent spaces and the normal spaces.
Although thesenbsp;types of submanifolds are different, we use a similar approach to study them. In addition to its nearly Kähler structure, the manifold S3xS3 is also endowed withnbsp;almost product structure. The almost product structure is an involutive and symmetric endomorphism that anticommutes with the almost complex structure. Thenbsp;geometry and the shape of a submanifold in S3xS3 are determined by the behaviournbsp;the restriction of the almost product structure on the submanifold.
On an almost complex surface in S3xS3 there exists a holomorphic quadratic differential which describes the behaviour of the almost product structure on thenbsp;There is a correspondence between almost complex surfaces in S3xS3, the so-called H-surfaces innbsp;three-dimensionalnbsp;space and harmonic maps from a surface into the three-sphere. The totally geodesic and flat almost complex surfaces in S3xS3 are classified. Moreover from one almost complex surfacenbsp;S3xS3 with a non-vanishing differential one can construct a whole sequencenbsp;suchnbsp;complexnbsp;innbsp;a Lagrangian submanifold in S3xS3 the behaviour ofnbsp;almost product structure is described in terms of threenbsp;functions. The derivatives of these angle functions give all the components but one of the second fundamentalnbsp;All constant curvature Lagrangian submanifolds in S3xS3 are classified. The techniques fornbsp;Lagrangiannbsp;cannbsp;be used tonbsp;totally real surfaces in S3xS3.
Publication status: published
KU Leuven publication type: TH
Appears in Collections:Geometry Section

Files in This Item:
File Status SizeFormat
BartDioosthesis.pdf Published 1347KbAdobe PDFView/Open Request a copy

These files are only available to some KU Leuven Association staff members


All items in Lirias are protected by copyright, with all rights reserved.