Papers Published
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.
Keywords:
almost complex manifolds • quasi-integrable • Nijenhuis tensor