Math @ Duke

Publications [#243386] of Robert Bryant
search www.ams.org.Papers Published
 Bryant, RL, On the geometry of almost complex 6manifolds,
Asian Journal of Mathematics, vol. 10 no. 3
(January, 2006),
pp. 561605, International Press of Boston [math.DG/0508428], [doi]
(last updated on 2019/08/21)
Abstract: This article discusses some basic geometry of almost complex 6manifolds. A 2parameter family of intrinsic firstorder functionals on almost complex structures on 6manifolds is introduced and their EulerLagrange 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 G2invariant 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 and a corresponding condition for unitary structures is considered. 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. The failure of this involutivity is discussed and some constructions are made to show, at least partially, how general these structures can be. © 2006 Robert L. Bryant.
Keywords: almost complex manifolds • quasiintegrable • Nijenhuis tensor


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