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Lie contact manifolds. II. (English) Zbl 0789.53021
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 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.
Reviewer: H.Sato (Nagoya)

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C05 Connections (general theory)
53C20 Global Riemannian geometry, including pinching
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