On A Class Of Special (α,β)-Metric

Now there are two methods on the resarch of Finsler geometry. One shi tensor method, the other is analytic method. In present papaer, we majorly use the lat tor.In Finsler Geometry, there are many Finsler metrics that we have known. Lots of concrete examples are (α,β)- metrics. And one of fundamental problems in Finsler geometry is to find and study Finsler metrics with constant (flag) curvature. On the basic, we majarly study the following problems in present paper:(A)To the property of a class of (α, β)- metrics in which β is parallel with respect to Riemann metric a and Riemann metric a is of constant curvature, we obtain the followingTheorem4.3 Let F(α,β) be a positive definite metric on the manifold M (dimM> 3). If β is parallel with respect to Riemann metric a and Riemann metric a is of constant curvature n, then(i)F is projective metric with constant curvature K.(ii)(a)If K = 0, then F is flat-parallel metric.(ii)(b)If K ≠ 0, then F = ec(x)α, i.e. F is conformally related to Riemann metric α. where c(x) = (B)To the property of a class of metrics which are conformally related to a special (α,β) - metric, we obtain thefollowingTheorem4.5 Let F be a positive definite metric on the manifold M (dim.M > 3) that is conformally related to (α,β)-metric F(α,β), i.e. T = ec(x)F. If ,/J is parallel with respect to Riemann metric α arid Riemann metric a is of constant curvature u. then(i)When K = 0, F is conformally flat metric.(ii)When K≠ 0,F = ec(x)α, i.e. F is conformally related to Riemann metric α: Where c'(x) = c(x) + 1/2ln(u/K), and K is the flag curvature of F.(C)To the necessary and sufficent condition that projectively flat Randrrs metric F = a + b is of constant curvature, we obtain the followingTheorems. 1 Let Randers metric F = a+B be projectively flat metric- on the manifold M (dimM > 3), then F is of constant curvature if and only if the following holdwhere u is the constant curvature of Riemann metric a, (D)To the necessary and sufficent condition that Shen metric F is projec-tively related to Riemann metric a when B is close, we obtain the followingTheorems. 4 Let Shen metric F metric ou the manifold M (dimM > 3) in which (3 is close, then F is projectively related to Riemann metric a if and only id r00 = 0 or a2bi = yi(yi := aij) holds.