正則的嘉當空間極小超曲面的柯什密德不變量和附屬微分方程

<正> 本文是繼作者前篇論文之後的;目的在於詳細研究該文末節所論的關於拓廣的微小變形問題.和芬斯拉空間相類似地有E.Cartan所建立的以面積概念為基礎的

KOSCHMIEDER INVARIANT AND THE ASSOCIATE DIFFERENTIAL EQUATION OF A MINIMAL HYPERSURFACE IN A REGULAR CARTAN SPACE

This note is a sequel to a previous one in which the infinitesimal deformation of a hypersurface in regular Cartan space (so called by L. Berwald) has been generalized so as to depend upon the covariant tangential vector p_m as well as the coordinates x~i of a point on the hypersurface. Besides the notation newly introduced I shall use the same notation.Let us consider that a hypersurface S in a regular Cartan space is subjected to by an infinitesimal deformation of the type x~i=x~i+ξ~i(x,p)δt,(1) where the ξ~i is a function of both x~k and p_m, and is homogeneous of degree zero in p_m, so that ξ~i‖~o = 0.(2)As in the previous paper the equation for the variation of the mean extremal curvature of a general hypersurface in the Cartan space is found to be δα_ρ~ρ/δt=E-Gα_ρ~ρ.(3)Suppose that the original hypersurface S which we have to deform into a near one is minimal,namely, H≡1/n-1α_ρ~ρ=0.(4) In order that the deformed hypersurface S be minimal also, it is necessary and sufficient that the ξ~i shall satisfy the associate differential equation E=0.(5)Now decomposing, as usual, the vector ξ~i into a linear combination of x_α~i and l~i: ξ~i= λ~α x_α~i + V l~i(6) and substituting these components into the expression E, we are finally led to the associate equation of the normal form of Berwald where U_o is Kosehmieder invariant of the hypersurface S.It is quite remarkable that the vanishing of the coefficient of the term λ~α in (5) is due to the generalized Mainardi-Codazzi equations of the hypersurface S: (n-1) H_((k)) = α_k~ρ(ρ) + R_o~ρ_(kρ)+α_ρ~σP_o~ρ_(σk)-α_k~σP_o~ρ_(σρ)+α_ρ~σB_k~ρ_σ(8) and (4). Thus the expression for U_o U_o=(A~μα_ μ~ρA~μB_σ~σ_ρ -A~μA~να_μ~ρα_(ρν)-(n-1)(n-2)K+R_o~ρ_(oρ)-α_ρ~σP_o~ρ_(oσ), as originally given by L. Berwald, is reestablished in quite a different way.(9)