(α,β)-Metrics Of Certain Important Non-Riemann Curvature Properties

In Finsler geometry, (α,β)-metrics of certain important non-Riemann curvature have been a hot topic of great concern of geometrists. It studies a special class of (α,β)-metrics that Matsumoto metric is of relatively istropic mean Landsberg curvature in the third part of this paper. Later, this paper studies a class of (α,β)-metrics such that F =α+ (?)β+ 2(β²/α)-(1/3)(β⁴/α³) and exponential metric F =αe^{λβ/α} in the forth section, whereα=(a_ij(x)yⁱy^j)^1/2 is a Riemann metric,β=b_iyⁱ is a 1-form, (?) is a constant and A is a non-zero constant. It studies the properties of such two class of (α,β) metrics when F is a Douglas metric and that is of isotropic S-curvatrue respectively in the forth section. It gets the following main results:Theorem 3.1 Let F =α²/(α-β) is a Matsumoto metric on manifold M of dimension n(≥3). Then there is a scalar function c(x) on manifold M such that F is of relatively isotropic mean Landsberg curvature (J_i + cFI_i = 0) if and only ifβis parallel with respect toα.Proposition 3.1 Let F =α²/(α-β) is a Matsumoto metric that is of scalar flag curvatureK = K(x,y) on manifold M of dimension n(≥3). If there is a scalar function c(x) on manifold M such that F is of relatively isotropic mean Landsberg curvature (J_i + cFI_i = 0), then K = 0, F is a local Minkowskian metric.Theorem 4.1 Let F =α+ (?)β+ 2(β²/α) - (1/3)(β⁴/α³) is an (α,β)-metric on a manifold M of dimension n(≥3), where (?) is a constant. Then F is Douglas metric and that is of istropic Scurvatrue if and only if F is a Berwald metric. Corollary 4.1 Let F =α+ (?)β+ 2β²/α- (1/3)(β⁴/α³) is an(α,β)-metric on a manifold M of dimension n(≥3), where e is constant. If F is Douglas metric and that is of istropic S- curvatrue, then F is a weak Berwald metric.Theorem 4.2 Let F =α+(?)β+2β²/α-(1/3)(β⁴/α³) is an (α,β)-metric on a manifold M of dimension n, where (?) is constant. Then F is of istropic S- curvatrue if and only ifβis Killing1-from of constant length with respect toα, in this case, F is a weak Berwald metric.Theorem 4.3 LetF=αe^{λβ/α} is an (α,β)-metric on a manifold M of dimension n(≥3), where A is a non-zero constant. Then F is Douglas metric and that is of istropic S- curvatrue if and only if F is a Berwald metric.