Local embeddability of real analytic path geometries
type: Item
instance of: scholarly article
title: Local embeddability of real analytic path geometries
zbMATH Open document ID: 1250.32032
author: Thomas Mettler
published in: Differential Geometry and its Applications
publication date: 2012-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.1250
reviewed by: Jiri Lebl
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32V30
Mathematics Subject Classification ID: 58A15
Mathematics Subject Classification ID: 53C10
zbMATH DE Number: 6034647
zbMATH Keywords: path geometries
zbMATH Keywords: CR-structures
zbMATH Keywords: embedding problems
zbMATH Keywords: exterior differential systems
Wikidata QID: Q115356732
MaRDI profile type: Publication
OpenAlex ID: W2064058882
arXiv ID: 1202.1250
cites work: Q5670926
cites work: Q5688871
cites work: Q3997815
cites work: Q5439814
cites work: Q4449111
cites work: Q3040162
cites work: ON A QUESTION OF HANS LEWY
DOI: 10.1016/J.DIFGEO.2012.03.002
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review text: It is known that an almost complex structure \(\mathfrak J\) on a 4-manifold \(X\) can be described using a rank-2 vector bundle \(\Lambda_{\mathfrak J} \subset \Lambda^2TX^*\). A pair of line subbundles \(L_1\) and \(L_2\) is called a splitting if \(\Lambda_{\mathfrak J} = L_1 \oplus L_2\). Then every hypersurface \(M \subset X\) satisfying a nondegeneracy condition inherits a path geometry from the splitting. NEWLINENEWLINENEWLINENEWLINE The main result of the paper is to show that given a real-analytic path geometry on a manifold \(M\), it is locally realizable by a real-analytic embedding into \({\mathbb{C}}^2 \cong {\mathbb{R}}^4\) with the splitting generated by the real and imaginary part of \(dz^1 \wedge dz^2\). That real-analyticity is necessary follows from the smooth nonembeddable example of Nirenberg. A corollary of the main result is the well known fact that every 3 dimensional nongenerate real-analytic CR-structure is induced locally by an embedding into \({\mathbb{C}}^2\).
description: scientific article; zbMATH DE number 6034647