## Asian Journal of Mathematics

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

### On the geometry of almost complex 6-manifolds

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 5 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.ajm/1175789046

**Mathematical Reviews number (MathSciNet)**

MR2253159

**Zentralblatt MATH identifier**

1114.53026

**Subjects**

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

**Keywords**

Almost complex manifolds quasi-integrable Nijenhuis tensor

#### Citation

Bryant, Robert L. On the geometry of almost complex 6-manifolds. Asian Journal of Mathematics 10 (2006), no. 3, 561--605. http://projecteuclid.org/euclid.ajm/1175789046.