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Geometry of SU(3) manifolds. Feng Xu

Geometry of SU(3) manifolds.

Feng Xu

Published
ISBN : 9780549665205
NOOK Study eTextbook
118 pages
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 About the Book 

I study differential geometry of 6-manifolds endowed with various SU(3) structures from three perspectives. The first is special Lagrangian geometry- The second is pseudo-Hermitian-Yang-Mills connections or more generally, o-anti-self dualMoreI study differential geometry of 6-manifolds endowed with various SU(3) structures from three perspectives. The first is special Lagrangian geometry- The second is pseudo-Hermitian-Yang-Mills connections or more generally, o-anti-self dual instantons- The third is pseudo-holomorphic curves.-For the first perspective, I am interested in the interplay between SU(3)-structures and their special Lagrangian submanifolds. More precisely, I study SU(3)-structures which locally support as nice special Lagrangian geometry as Calabi-Yau 3-folds do. Roughly speaking, this means that there should be a local special Lagrangian submanifold tangent to any special Lagrangian 3-plane. I call these SU(3)-structures it admissible. By employing Cartan-Kahler machinery, I show that locally such admissible SU(3)-structures are abundant and much more general than local Calabi-Yau structures. However, the moduli space of the compact special Lagrangian submanifolds is not so well-behaved in an admissible SU(3)-manifold as in the Calabi-Yau case. For this reason, I narrow attention to nearly Calabi-Yau manifolds, for which the special Lagrangian moduli space is smooth. I compute the local generality of nearly Calabi-Yau structures and find that they are still much more general than Calabi-Yau structures. I also discuss the relationship between nearly Calabi-Yau and half flat SU(3)-structures. To construct complete or compact admissible examples, I study the twistor spaces of Riemannian 4-manifolds. It turns out that twistor spaces over self-dual Einstein 4-manifolds provide admissible and nearly Calabi-Yau manifolds. I also construct some explicit special Lagrangian examples in nearly Kahler CP3 and the twistor space of H 4.-For the second perspective, we are mainly interested in pseudo-Hermitian-Yang-Mills connections on nearly Kahler six manifolds. Pseudo-Hermitian-Yang-Mills connections were introduced by R. Bryant in [4] to generalize Hermitian-Yang-Mills concept in Kahler geometry to almost complex geometry. If the SU(3)-structure is nearly Kahler, I show that pseudo-Hermitian-Yang-Mills connections (or, more generally, o-anti-self-dual instantons) enjoy many nice properties. For example, they satisfy the Yang-Mills equation and thus removable singularity results hold for such connections. Moreover, they are critical points of a Chern-Simons functional. I derive a Weitzenbock formula for the deformation and discuss some of its application. I construct some explicit examples which display interesting singularities.-For the third perspective, I study pseudo-holomorphic curves in nearly Kahler CP3. I construct a one-to-one correspondence between null torsion curves in the nearly Kahler CP3 and contact curves in the Kahler CP3 (considered as a complex contact manifold). From this, I derive a Weierstrass formula for all null torsion curves by employing a result of R. Bryant in [9]. In this way, I classify all pseudo-holomorphic curves of genus 0.