On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables

Noboru TANAKA

doi:10.2969/jmsj/01440397

Noboru TANAKA

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1962 年 14 巻 4 号 p. 397-429

DOI https://doi.org/10.2969/jmsj/01440397

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Published: 1962 Received: 1962/05/09 Available on J-STAGE: 2006/09/26 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/09/26 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Right : 1) K. Nomizu, Lie groups and differential geometry, The Math. Soc. of Japan, 1956.
2) We have ∂/∂z_i=∂/∂x_i and √-1·∂/∂z_i=∂/∂y_i (1≤i≤n).
3) As for an exterior differential system, we adopt the definitions and notations given in M. Kuranishi, On E. Cartan's prolongation theorem of exterior differential systems, Amer. J. Math., 1957, Vol. LXXIX, no. 1.
5) and 6) δ_ij=1(i=j),=0(i≠j).∂f/∂z_i=1/2(∂f/∂x_i-√-1∂f/∂y_i) and ∂f/∂z_i=1/2(∂f/∂x_i+√-1∂f/∂y_i). The functions of the right-hand side of (9.1) and of (9.2) should be evaluated at the Point of D with the coordinates: x_i=x_i(p), y_i=y_i(p) (1≤i≤n-1) and t=x_n(p).
4) E. Cartan, Les systemes différentiels extérieurs et leurs applications géométriques,Paris, 1945.
7) We say that a hypersurf ace S satisfies the condition of Levi-Krzoska at a point p of S when we can find a regular local equation F=0 of S at p which satisfies the following condition: ⁿ∑_i,j=1∂²F/∂z_i∂z_j(p)ξ_iξ_j>0 for arbitrary n complex numbers ξ₁, …, ξ_n such that (ξ₁, …, ξ_n)≠0 and ⁿ∑_i=1∂F/∂z_i(p)ξ_i=0.
8) _??__zω_i means the Lie derivative of ω_i with respect to the infinitesimal transformation Z.
9) A hypersurface S is called non-degenerate (resp. of index γ, resp. regular) if it is non-degenerate (resp. of index γ, resp. regular) everywhere.
10) By the trace we always mean the real one.
11) For a proof of this fact, see S. Kobayashi, Theory of connections, Annali. di, Math., 43 (1957), 119-194.
[1] E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, I, Ann. Mat. Pura Appl., 11 (1932), 17-90.
[2] E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, II, Ann. Scuola. Norm. Sup. Pisa, 1 (1932), 333-354.
[3] E. Cartan, Les sous-groupes des groupes continus de transformations, Ann. Sci. Ecole Norm. Sup., 25 (1908), 57-194.
[4] R. S. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc., no. 22, 1957.
[5] N. Tanaka, Conformal connections and conformal transformations, Trans. Amer. Math. Soc., 92 (1959), 168-190.

訂正日: 2006/09/26 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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