Some Submanifold Properties And Projective Properties Of (α,β)-space

Finsler geometry is just Riemannian geometry without the quadratic re-striction on the metric([9]). It was the famous mathematician B. Riemann who mentioned the general regular metric spaces in his "Habilitationsvortrag" in 1854. In views of the complicated computations in Finsler geometry, he turned immedi-ately to the geometry of quadratic metric which is called Riemannian geometry. There was no significant progress in the general case until 1918, when P. Finsler studied the variation problem for spaces with a family of norms in his doctor the-sis ([13]) and thereafter the corresponding geometry was called Finsler geometry. In 1900, D.Hilbert formulated 23 problems, in which the 4th and 23rd prob-lems being in Finsler's category ([19]). Thereafter, under the efforts of E.Cartan, S.S.Chern, L.Berwald, J.Douglas and other geometers, Finsler geometry has de-veloped into a fruitful branch in differential geometry.Under the encouragement of S.S.Chern, Finsler geometry has been made a great progress after 1990s. A lot of geometers worked in this field such as D.Bao and Z. Shen, etc. Many classical concepts and results in Reimannian geometry can be extended to Finsler geometry such as the volume comparison theorem ([32]), harmonic map ([16][24]), submanifold geometry ([15],[17],[33],[42],[45]), Einstein metric ([1] [12]), the sphere theorem ([25]) and Gauss-Bonnet theorem([6]), etc.(α,β)-metric F=αφ(β/α) is a class important Finsler metrics constructed by a Riemannian metric (?), a one formβ= bi(x)yi and a smooth positive functionφdefined on an interval (—bo, bo) satisfyingφ(0)= 1 such that F is positive definite. F is obviously reduced to the usual Riemannian metric in the case ofφ(s)= 1. In the caseφ(s)= 1+s, the (α,β)-metric is called Randers metric ([26]). It has many applications for (α,β)-metric in Physics and Biology([2]) which has been extensively studied by a lot of mathematicians ([3] [5] [11] [21] [23] [38]). In this paper, we discuss some submanifold and projective properties in Finsler spaces endowed with (α,β)-metrics. The main content of this paper is divided into there parts:the Berntein theorem in (α,β)-space, the minimal rotational hypersurface in (α,β)-space, and the projective equivalence between an (α,β)-metric and a Randers metric.