Math @ Duke

Publications [#49065] of Robert Bryant
search arxiv.org.Papers Published
 On the geometry of almost complex 6manifolds,
The Asian Journal of Mathematics, vol. 10 no. 3
(September, 2006),
pp. 561606 [math.DG/0508428]
(last updated on 2006/09/17)
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 6manifolds. In particular, I define a 2
parameter family of intrinsic firstorder functionals on
almost complex structures on 6manifolds and compute
their EulerLagrange 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 6sphere admits such nontrivial bundles.
This class of almost complex manifolds in dimension 6
will be referred to as quasiintegrable. Some of the
properties of quasiintegrable structures (both almost
complex and unitary) are developed and some examples
are given. However, it turns out that quasiintegrability is
not an involutive condition, so the full generality of these
structures in Cartan's sense is not wellunderstood.
Keywords: almost complex manifolds • quasiintegrable • Nijenhuis tensor


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