Finsler Spaces With Constant Curvature And L-reducible Finsler Spaces

Finsler spaces with constant curvature are always the focus of Finsler metric study. Recently, D.Bao and Z.Shen have a great progress on the classification of Finsler spaces with constant curvature. They erase some problems that confuse us many years so that we can have a better understanding of Finsler spaces with constant curvature. Their works make the study on Finsler spaces with constant curvature become the hot topic again. In twenty eightieth, H.Akbar-Zadeh, a French mathematician, found that Finsler spaces with constant curvature A must satisfy Now the reverse problem interests some mathematicians, that is, wether Finsler spaces satisfying are of constant curvature. In present paper, it studies the problem first and obtains the following results.Theorem 3.3 Let (M, F) be a Finsler space of dimension n(≥ 3). If F is of scalar curvature K = K(x,y) and satisfies , then there is a function p(x) on M such thatTheorem 3.4 Let (M, F) be a Finsler space of dimension n(≥ 3). If F is of scalar curvature K = K(x,y) and satisfies , then K must be a constant, that is, (M, F) is of constant curvature.Theorem 3.5 (M, F) is a non-Riemannian Finsler space of dimension n(> 3) and it satisfies L:0 -f K(x, y)F2C = 0. (M, F) is of scalar curvature K(x, y) if and only if (M, F) is of constant curvature. In such case, K — K(x, y) is a constant.Theorem 3.6 Let (M,F) be a complete Finsler space of dimension n(> 3), it satisfies L;o + cF2C = 0, where c < 0 is a constant. If Cartan torsion C is bounded, then (M, F) is Reimannian.Theorem 3.7 Let(M, F) be a projectively flat Finsler space of dimension n(> 3). If h-.j-.k, Jj-.k both are sysmetric in j, k, then (M, F) is of constant curvature. Moreover, if (M, F) is of constant curvature c ^ 0, then (M, F) is Reimannian.Corollary 3.8 If Ii-,j± of projectively flat Finsler space (M, F) is sysmetric in i,j, k, then (M, F) is of constant curvature. In such case, Ji:k = 0.Later, this paper studies L-reducible Finsler spaces. M.Matsumoto has defined C-reducible Finsler spaces and has a wonderful classification of them. As we know, C-reducible Finsler spaces must be L-reducible. But, the revere may not be true. By the close relationship between Cartan torsion and Landsberg curvature, it has been realized that L-reducible Finsler spaces turn to be C-reducible Finsler spaces. It gets the following results.Theorem 4.2 (M, F) is Finsler space with isotropic Landsberg curvature. (M, F) is L-reducible if and only if (M, F) is C-reducible. In such case, F is a Randers metric with Douglas curvature D = 0.Proposition 4.3 If L-reducible Finsler space (M, F) is of constant curvature /C,then it must be C-reducible.Proposition 4.4 If L-reducible Finsler space (M, F) satisfies L-oxt + k(x, y)C = 0, where k(x, Xy) = X3k(x, y) ,then it must be C-reducible.Theorem 4.5 If L-reducible Finsler space (M, F) is of scalar curvature K = K(x,y), then it must be C-reducible and mean Cartan torsion is in such form Ik = — ^W{?/fc:o + f(n+l)K.k}.