芬斯拉空間極小超曲面的同態變換

<正> 在芬斯拉-嘉當空間裹,正如J.M.Wegener所指出,極小超曲面的確定是和某一定的超曲面參數族的選擇有關的,並且除了A_i=0的芬斯拉空間而外,在幾何學上很難給它以完備的意義.現時A.Deicke證明了在完全正测度之下具有A~i=0的芬期拉空間恰是黎曼空間.這個驚異的結果使得在這樣特

ON THE ISOMORPHIC TRANSFORMATIONS OF MINIMAL HYPERSURFACES IN A FINSLER SPACE

In a recent paper W. Barthel has investigated the geometry in a Finsler space with new Euclidean connections. The coefficients of the affine connection F_(ikh) there established are different from those utilized by E. Cartan and they obey a new postulate, namely,In such a Finsler space so called "gefaserte" Finsler space we can define a minimal hypersurface by setting the extremal mean curvature of the hypersurface, M, to zero:Consider the infinitesimal deformation x~i = x~i + ξ~i(x) δt, which carries on the point (x) into the point (x) infinitely near (x), δt being an infinitesimal. If a minimal hypersurface S is deformed to a nearby minimal hypersurface S, then we shall call ξ~i(x) the extremal deviation of the isomorphic transformation of the minimal hypersurface S. Just as in a regular Cartan space based on the notion of area we propose to solve the following problem:How depends the deviation of a minimal hypersurface in a Finsler-Barthd space upon the curvature of the space, when the minimal hypersurface is subjected to an isomorphic transformation?The present paper deals with the calculation of the extremal mean curvature M, which corresponds to the formula of the author concerning the extremal deviation in a geometry based on the notion of area. The result runs: δM/δt=E-MG, whence the equation of deviation of a minimal hypersurface turns out as E=O.In rewriting these quantities, especially E, in terms of fundamental tensors, curvature tensors and allied affinors of the space and the minimal hypersurface we arrive at an invariant, analogous to the invariant U of Barthel in the normal form of the second variation of an (n-1)-ple surface-integral.It should be noted that depends upon both the curvature-tensor R_(ρoσo) and the affinor G~(ρσ)≡g~(ρσ) + A~σ A~σ + x_k~ρ A~k ‖~σ.Here we can also generalize the result to the general case where the extermal deviation ξ~i contains not only the position of the point in space, but also the directioncoefficients of the hypersurface. The investigation of such transformations will lead to the normal deviation of a minimal hypersurface.