The Some Curvature Properties On Product Finsler Manifolds

There are many very important non-Riemannian geometric quantities^[10,27]in Finsler geometry,and the Cartan torsion C is one of them.Differentiating C along geodesics gives rise to the Landsberg curvature.Landsberg curvature describes the rate of change of the Cartan torsion C,When C=0,the Finsler metric will degenerate into the Riemannian metric.S curvature is also a very important non-Riemannian quantity which was originally introduced for the volume comparison theorem by Shen^[28],which together with the Ricci curvature determines the volume of a Busemann-Hausdorf f ball near a point.The curvature properties of Finsler metric in Finsler geometry has always been a hot topic for scholars,and so is product Finsler metric.but the related properties of Landsberg curvature of warped product of Finsler manifold and the description of the isotropic mean Berward curvature and S curvature properties of doubly warped of Finsler manifold have not been obtained.So we mainly study the curvature properties of these two kinds of important product Finsler manifolds(a manifold M:=I×?M with warped structure F=???and the doubly warped of Finsler manifold(_f2M₁×_f1M₂,F)).The main findings are summarized below:1.First part,in the second section of this paper,we studied the relative Lands-berg curvature properties of warped product of Finsler manifolds M:=I×?M with warped structure F=???.First we described the Finsler warped product met-rics with relatively isotropic Landsberg curvature and obtaining its equations,this conclusion extends the results of the Chen-Song^[9]on spherical symmetric Finsler manifolds with relatively isotropic Landsberg curvature.The necessary and suffi-cient condition equations of the warped product manifold M=I×?M is Landsberg are obtained from the equations.Thus,we simplify the necessary condition for the warped product manifold M=I×?M to be Landsberg.According to the corollary from the Chen-Shen-Zhao^[7]:A spherically symmetric metric is a Finsler warped product metric,we get some new examples of warped product Finsler metrics with relatively isotropic curvature by use the known examples of spherically symmetric metrics with relatively isotropic Landsberg curvature.2.Second part,in the third section of this paper,we studied doubly warped product of Finsler manifolds(_f2M₁×_f1M₂,F).First we discussed the condition of the doubly warped product of Finsler manifolds(_f2M₁×_f1M₂,F)with isotropic mean Berwald curvature,and we show that if the doubly warped product of Finsler manifolds(_f2M₁×_f1M₂,F)has isotropic mean Berwald curvature if and only if it is a weakly Berwald manifold,and this This sufficient condition is equivalent to that F₁and F₂are weak Berwald measures or(g^ik)_yi(f₁²)_xk=(g^??)_v?(f₂²)_u?=0 hold.Then we discussed the properties with S-curvature of doubly warped product of Finsler manifolds.We find that if the doubly warped product of Finsler manifold has weakly isotropic S curvature is equivalent to it is weakly manifold or S=?₁?₂,where?₁,?₂are 1 form on the manifold M₁,M₂respectively.Thus,we derive a corollary that for any doubly warped Finsler manifold(_f2M₁×_f1M₂,F)has isotropic S curvature if and only if S curvature disappears.