Some Properties Of Finsler Metric With Isotropic Curvature

The non-Riemannian geometric quantities in Finsler geometry describe the difference between Finsler geometry and Riemann geometry. The study of these quantities is benefit for us to make out their distinction and the nature of Finsler geometry. In this article, we mainly discuss the relation between isotropic Berwald curvatures of projectively related Finsler metrics and properties of Finsler metrics with isotropic S-curvature. In the end, we study a class of special metrics. We mainly obtain the following results.Theorem 3. 4 Let (M, F) be a Finsler manifold , F is projectively related to F. If F is of isotropic Berwald curvature c(x), then F is of isotropic Berwald curvature c(x) if and only if c{x) satisfieswhere P = is the projective factor and 0 = bi(x)yi is 1-form.Theorem 3. 7 If F and F are projectively related Finsler metrics with isotropic Berwald curvatures, then F is of S-curvature if and only if F is of isotropic S-curvature.Proposition 3. 8 Let (M, F) be n-dimensional Finsler manifold with scalar curvature K = K(x, y), F is projectively related to F. If F is of almost isotropic S-curvature S = (n + 1){cF + }, then the flag curvature of F satisfieswhere is a scalar function on M.Proposition 4. 1 Let F satisfies Fxk = cFFyk, theni) If c is a constant, then F is projectively flat with constant curvature, constant S-curvature and constant isotropic Berwald curvature.ii) If c = c(x): then F is projectively flat with almost isotropic S-curvature and isotropic Berwald curvature, the flag curvatureiii) If c = c(x,y), then F is non-projectively flat and the flag curvature and the S-curvatureMoreover, F is projectively flat if and only if c = c(x).