Remarks on the geometry of almost complex 6-manifolds

dc.contributor.author

Bryant, Robert L

dc.date.accessioned

2016-12-03T19:50:25Z

dc.description.abstract

This article is mostly a writeup of two talks, the first given in the Besse Seminar at the Ecole Polytechnique in 1998 and the second given at the 2000 International Congress on Differential Geometry in memory of Alfred Gray in Bilbao, Spain. It begins with a discussion of basic geometry of almost complex 6-manifolds. In particular, I define a 2-parameter family of intrinsic first-order functionals on almost complex structures on 6-manifolds and compute their Euler-Lagrange equations. It also includes a discussion of a natural generalization of holomorphic bundles over complex manifolds to the almost complex case. The general almost complex manifold will not admit any nontrivial bundles of this type, but there is a large class of nonintegrable almost complex manifolds for which there are such nontrivial bundles. For example, the standard almost complex structure on the 6-sphere admits such nontrivial bundles. This class of almost complex manifolds in dimension 6 will be referred to as quasi-integrable. Some of the properties of quasi-integrable structures (both almost complex and unitary) are developed and some examples are given. However, it turns out that quasi-integrability is not an involutive condition, so the full generality of these structures in Cartan's sense is not well-understood.

dc.format.extent

45 pages, no figures.

dc.identifier

http://arxiv.org/abs/math/0508428v2

dc.identifier.uri

https://hdl.handle.net/10161/13134

dc.relation.ispartof

The Asian Journal of Mathematics

dc.subject

math.DG

dc.subject

math.DG

dc.subject

53C15 (Primary), 53A55 (Secondary)

dc.title

Remarks on the geometry of almost complex 6-manifolds

dc.type

Journal article

pubs.author-url

http://arxiv.org/abs/math/0508428v2

pubs.begin-page

3

pubs.notes

Version 2 contains minor improvements (typos corrected, statements reworded for clarity), a few added remarks, and a dedication

pubs.organisational-group

Duke

pubs.organisational-group

Mathematics

pubs.organisational-group

Trinity College of Arts & Sciences

pubs.volume

10

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