Symmetries of almost complex structures and pseudoholomorphic foliations
type: Item
instance of: scholarly article
title: Symmetries of almost complex structures and pseudoholomorphic foliations
zbMATH Open document ID: 1304.32016
DOI: 10.1142/S0129167X14500797
published in: International Journal of Mathematics
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1103.4404
Mathematics Subject Classification ID: 32Q60
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53A55
Mathematics Subject Classification ID: 22E65
zbMATH DE Number: 6359572
zbMATH Keywords: almost complex manifold
zbMATH Keywords: pseudogroup of local pseudoholomorphic symmetries
zbMATH Keywords: integrable structure
zbMATH Keywords: Nijenhuis tensor
author: Boris Kruglikov
MaRDI profile type: Publication
arXiv ID: 1103.4404
cites work: A note on mappings and automorphisms of almost complex manifolds
cites work: On the geometry of almost complex 6-manifolds
cites work: Some Remarks on Almost and Stable Almost Complex Manifolds
cites work: On the geometry of model almost complex manifolds with boundary
cites work: Chern numbers of almost complex manifolds
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten. (Plane element fields in 4-dimensional manifolds)
cites work: \(G^*_{2(2)}\)-structures on pseudo-Riemannian manifolds
cites work: Complete prolongation for infinitesimal automorphisms on almost complex manifolds
cites work: Nijenhuis tensors and obstructions to constructing pseudoholomorphic mappings
cites work: Non-existence of higher-dimensional pseudoholomorphic submanifolds
cites work: Tangent and normal bundles in almost complex geometry
cites work: Finite-dimensionality in Tanaka theory
cites work: Q3645910
cites work: Geometry of differential equations
cites work: The tangent bundle of an almost complex manifold
cites work: The local behaviour of holomorphic curves in almost complex 4-manifolds
cites work: Complex analytic coordinates in almost complex manifolds
cites work: Some integration problems in almost-complex and complex manifolds
cites work: Involution. The formal theory of differential equations and its applications in computer algebra
cites work: Overdetermined systems of linear partial differential equations
cites work: Q5337746
cites work: On differential systems, graded Lie algebras and pseudo-groups
cites work: Complex Structures on Real Vector Bundles
cites work: NORMAL FORMS FOR ALMOST COMPLEX STRUCTURES
Wikidata QID: Q126026561
review text: Let \((M,J)\) be a \(2n\)-dimensional almost-complex manifold. This paper is concerned with the pseudogroup of local (pseudoholomorphic) symmetries of \((M,J)\). This contains the group of (global) automorphisms of \((M,J)\), and the main theme of the paper is that a large symmetry pseudogroup implies that \((M,J)\) carries some integrable structure. One of the main results is that if \(J\) is nondegenerate (which here means that the Nijenhuis tensor, viewed as a map \(\Lambda^2 TM\to TM\), is surjective when \(n>2\) and that its image is a non-integrable distribution when \(n=2\)), then the pseudogroup of symmetries is finite-dimensional. In particular, if \(J\) is nondegenerate in at least one point then the automorphism group of \((M,J)\) is a finite-dimensional Lie group.NEWLINENEWLINEFurthermore, if \(n=2\) then the dimension of the symmetries pseudogroup is at most \(4\), with equality holding only for left-invariant almost-complex structures on Lie groups, and if \(n=3\) then the dimension is at most \(14\), with equality holding only for the standard almost-complex structure on \(S^6\) coming from the octonions, and for the similar unit sphere \(S^{2,4}\) in the purely imaginary split octonions.
reviewed by: Valentino Tosatti
description: scientific article; zbMATH DE number 6359572