Sato, Hajime; Yamaguchi, Keizo Lie contact manifolds. II. (English) Zbl 0789.53021 Math. Ann. 297, No. 1, 33-57 (1993). 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) Cited in 3 Documents MSC: 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C05 Connections (general theory) 53C20 Global Riemannian geometry, including pinching Keywords:Lie contact structure; Lie sphere geometry; unit tangent bundle; normal Lie contact connection; Lie flat; conformally flat Citations:Zbl 0705.53019; Zbl 0732.53024 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Blaschke, W.: Vorlesungen ?ber Differentialgeometrie, vol. 3. Berlin Heidelberg New York: Springer 1929 · JFM 55.0422.01 [2] Cecil, T., Chern, S.: Tautness and Lie sphere geometry. Math. Ann.278, 381-399 (1987) · Zbl 0635.53029 · doi:10.1007/BF01458076 [3] Guillemin, V.: The integrability problem forG-structures. Trans. Am. Math. Soc.116, 544-560 (1965) · Zbl 0178.55702 [4] Kobayashi, S.: Transformation groups in differential geometry. Berlin Heidelberg New York: Springer 1972 · Zbl 0246.53031 [5] Miyaoka, R.: Lie contact structures and conformal structures. Kodai Math. J.14, 42-71 (1991) · Zbl 0732.53024 · doi:10.2996/kmj/1138039339 [6] Miyaoka, R.: A note on Lie contact manifolds. In: Recent developments in differential geometry. (Adv. Stud. Pure Math., vol. 22) Tokyo: Kinokuniya 1993 (to appear) · Zbl 0818.53046 [7] Ochiai, T.: Geometry associated with semi-simple flat homogeneous spaces. Trans. Am. Math. Soc.152, 1-33 (1970) · Zbl 0205.26004 · doi:10.1090/S0002-9947-1970-0284936-6 [8] Sternberg, S.: Lectures on differential geometry. New Jersey: Prentice-Hall 1964 · Zbl 0129.13102 [9] Sato, H., Yamaguchi, K.: Lie contact manifolds. In: Shiohama, K. (ed.) Geometry of manifolds, pp. 191-238. Boston: Academic Press 1989 · Zbl 0705.53019 [10] Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan J. Math.2, 131-190 (1976) · Zbl 0346.32010 [11] Tanaka, N.: On the equivalence problems associated with simple graded Lie algebras. Hokkaido Math. J.8, 23-84 (1979) · Zbl 0409.17013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.