Asian Journal of Mathematics

On the geometry of almost complex 6-manifolds

Robert L. Bryant

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This article discusses some basic geometry of almost complex 6-manifolds. A 2-parameter family of intrinsic first-order functionals on almost complex structures on 6-manifolds is introduced and their Euler-Lagrange equations are computed.

A natural generalization of holomorphic bundles over complex manifolds to the almost complex case is introduced. The general almost complex manifold will not admit any nontrivial bundles of this type, but there is a class of nonintegrable almost complex manifolds for which there are such nontrivial bundles. For example, the $G_2$-invariant 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 and a corresponding condition for unitary structures is considered.

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. The failure of this involutivity is discussed and some constructions are made to show, at least partially, how general these structures can be.

Article information

Asian J. Math., Volume 10, Number 3 (2006), 561-605.

First available in Project Euclid: 5 April 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53A55: Differential invariants (local theory), geometric objects

Almost complex manifolds quasi-integrable Nijenhuis tensor


Bryant, Robert L. On the geometry of almost complex 6-manifolds. Asian J. Math. 10 (2006), no. 3, 561--605.

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