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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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References:
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