On Projective Properties Of General Finsler Metric And Three Special Finsler Metrics

In this article, we study protective properties of Finsler metric especially Shen metric, Kropina metric and generalized Randers metric. In the, thirth part we discuss firstly the problem when a kind of protective spray can be induced by protectively flat Finsler metric; Secondly conditions under which a protectively flat Finsler space becomes a locally Minkowski space: Thirdly the special properties of comformally flat and protectively flat Finsler space. In the fourth part we study conditions of Shen metric to be Berwlad, Douglas, protectively flat or to be of zero curvature. In the fifth part we study a condition of Kropina metric to be Dougals and discuss its S-curvature. In the sixth part we dicuss conditions of generalized Randers metric to be Berwald and C-reducible. we obtain mainly the following results.Theorem 3.1 Let F be the Funk metric on a strongly convex domain in Rn and define a protectively flat spray bywhere h(x,y) is O-order positively homogeneous scalar function, then Gh is induced by a projectively Finsler metric F if and only if F satisfies the ODESTheorem 3.2 Let F be the Funk metric on a strongly convex domain in Rn and h is a constant, then if and only if when h = 0,1/2,1, there are Finsler metrics F on which induce the spray Gh Moreover,(i)If h = 0, we may put F = F(y) which is a locally Minkowshi metric.(ii)If h=1.. we may put (iii)if h =1/2, we may put F = F which is the Funk metric.Theorem 3.3 Let {M. F) be a projective flat Finsler space. Then (M. F) is locally Minkowski if and only if the protective factor satisfiesTheorem 3.4 Let (M, F) be a comformally and protectively flat Finsler space. Then (M, F) is Berwald space of constant curvature. Further more, if the conformal factor satisfies then it is locally Minkowshi; Unless (M. F) is Riemanniau space of constant curvature Theorem 4.1 The Shen metric is Berwald if and only if one of the following holds:(a) B is parallel with respect to rv;(b) F is protectively related to a:(c) F has the same geodesies as a. that is Theorem 4.2 (i)If the Shen-metric F = is projectively flat, then a is projectively flat, if and only if B is paralell with respect to a.(ii)If a is projectively flat,, then the Shen-metric F is projectively flat if and only if /3 is paralell with respect to a.Theorem 4.3 If a is projectively flat and the Shen metric F = is projectively flat too, then F must, be of zero curvature.Theorem 5.1 The Kropina metric F= becomes Douglas metric if and only if B is closed.Theorem 5.2 The Kropina metric F = is of constant S-curvature S = (n+1)cF if and only if the constant c satisfieswhereTheorem 6.1 Let be the generalized Randers metric and F is a Berwald metric, then F is Berwald if and only if B is parallel with respect to F.Theorem G.2 The generalized Randers metric F = F + B is C - reducible if and only if F is C - reducible.