Lie contact manifolds. II
type: Item
instance of: scholarly article
title: Lie contact manifolds. II
zbMATH Open document ID: 0789.53021
DOI: 10.1007/BF01459487
author: Hajime Sato
author: Keizo Yamaguchi
published in: Mathematische Annalen
publication date: 1994-06-13T00:00:00Z
full work available at URL: https://eudml.org/doc/165117
review text: This is the continuation of our previous paper [Lie contact manifolds, Geometry of manifolds, Coll. Pap. 35th Symp. Differ. Geom., Matsumoto/Japan 1988, Perspect. Math. 8, 191-238 (1989; Zbl 0705.53019)]. In that paper, the notion of Lie contact manifolds has been introduced. The Lie contact structure is a geometric structure on contact manifolds, which is modelled after the classical Lie sphere geometry of oriented hyperspheres in the unit sphere \(S^ n\). The flat model of this geometry is the unit tangent bundle \(T_ 1(S^ n) = G/G'\) of \(S^ n\), where \(G = PO(n+1,2)\) is the Lie sphere transformation group. Especially it is shown that the tangent sphere bundle \(S(M)\) of a conformal manifold \(M\) has a Lie contact structure. In this paper, we further investigate the interplay of these structures and show that the canonical lift to \(S(M)\) of the normal conformal connection on \(M\) gives the normal Lie contact connection of the induced Lie contact structure on \(S(M)\). As a corollary, we obtain the following results due to \textit{R. Miyaoka} [Kodai Math. J. 14, No. 1, 42-71 (1991; Zbl 0732.53024)]; \(S(M)\), endowed with the induced Lie contact structure, is Lie flat if and only if \(M\) is conformally flat.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 537198
zbMATH Keywords: Lie contact structure
zbMATH Keywords: Lie sphere geometry
zbMATH Keywords: unit tangent bundle
zbMATH Keywords: normal Lie contact connection
zbMATH Keywords: Lie flat
zbMATH Keywords: conformally flat
reviewed by: Hiroshi Sato
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 537198