A de Rham decomposition type theorem for contact sub-Riemannian manifolds
type: Item
instance of: scholarly article
title: A de Rham decomposition type theorem for contact sub-Riemannian manifolds
zbMATH Open document ID: 1483.53054
author: Marek Grochowski
published in: Analysis and Mathematical Physics
publication date: 2021-12-02T00:00:00Z
review text: The author proves a result which can be regarded as a sub-Riemannian version of de Rham decomposition theorem. More precisely, suppose that \((M, H, g)\) is a contact and oriented sub-Riemannian manifold such that the Reeb vector field \(\xi\) is an infinitesimal isometry. Under such assumptions there exists a unique metric and torsion-free connection on \(H\). Suppose that there exists a point \(q\in M \) such that the holonomy group \(\Psi(q)\) acts reducibly on \(H(q)\) yielding a decomposition \(H(q)=H_1(q)\oplus\cdots\oplus H_m(q) \) into \(\Psi(q)\)-irreducible factors. Using parallel transport the authors obtain the decomposition \(H = H_1 \oplus\cdots\oplus H_m\) of \(H\) into sub-distributions \(H_i\). Unlike the Riemannian case, the distributions \(H_i\) are not integrable, however they induce integrable distributions \(\delta_i\) on \(M/\xi\), which is locally a smooth manifold. As a result, every point in \(M\) has a neighborhood \(U\) such that \(T (U/\xi) = \delta_1\oplus\cdots\oplus\delta_m\), and the latter decomposition of \(T(U/\xi)\) induces the decomposition of \(U/\xi\) into the product of Riemannian manifolds. One can restate this as follows: every contact sub-Riemannian manifold whose holonomy group acts reducibly has, at least locally, the structure of a fiber bundle over a product of Riemannian manifolds. The authors also give a version of the theorem for indefinite metrics.
reviewed by: Peibiao Zhao
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 58A12
zbMATH DE Number: 7438140
zbMATH Keywords: contact distributions
zbMATH Keywords: connections
zbMATH Keywords: sub-Riemannian geometry
zbMATH Keywords: de Rham decomposition theorem
MaRDI profile type: Publication
Wikidata QID: Q115376016
full work available at URL: s13324-021-00624-y
OpenAlex ID: W3215298410
cites work: Exponential mappings for contact sub-Riemannian structures
cites work: Sub-Riemannian structures on 3D Lie groups
cites work: Q2973950
cites work: Constant curvature models in sub-Riemannian geometry
cites work: The tangent space in sub-Riemannian geometry
cites work: Riemannian geometry of contact and symplectic manifolds
cites work: Horizontal holonomy and foliated manifolds
cites work: Smoothness of subRiemannian isometries
cites work: Holonomy of Sub-Riemannian Manifolds
cites work: The de Rham decomposition theorem for metric spaces
cites work: On contact sub-pseudo-Riemannian isometries
cites work: Q4583913
cites work: Connections on bundles of horizontal frames associated with contact sub-pseudo-Riemannian manifolds
cites work: Sub-Riemannian geometry on infinite-dimensional manifolds
cites work: On the de Rham decomposition theorem
DOI: 10.1007/S13324-021-00624-Y
description: scientific article; zbMATH DE number 7438140