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

Noboru TANAKA

doi:10.2969/jmsj/01440397

Noboru TANAKA

Author information

JOURNAL FREE ACCESS

1962 Volume 14 Issue 4 Pages 397-429

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

Details

Published: 1962 Received: May 09, 1962 Available on J-STAGE: September 26, 2006 Accepted: - Advance online publication: - Revised: -
Correction information
Date of correction: September 26, 2006 Reason for correction: - Correction: CITATION Details: 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.

Date of correction: September 26, 2006 Reason for correction: - Correction: PDF FILE Details: -

Corresponding author

Correction information

Register with J-STAGE for free!