We find necessary and sufficient conditions for a Finsler surface $(M,F)$ to be Landsbregian in terms of the Berwald curvature $2$-form. We study Finsler surfaces which satisfy some flag curvature $K$ conditions, viz., $V(K)=0,V(K)=-{ℐ}/F^2$ and $V(K)=-{ℐ}K,$ where ${ℐ}$ is the Cartan scalar. In order to do so, we investigate some geometric objects associated with the global Berwald distribution ${𝒟}:=span\{H,S,V:=JH\}$ of a $2$-dimensional Finsler metrizable nonflat spray $S$. We find out under what conditions these surfaces turn out to be Riemannian. The existence of a first integral for the geodesic flow in each case has some remarkable consequences concerning rigidity results. We prove that a Finsler surface with $V(K)=-{ℐ}/F^2$ and either $S(K)=0$ or $S({𝒥})=0$ is Riemannian. Further, a Finsler surface with $V(K)=-{ℐ}K$ and $S(K)=0$ is Riemannian.

我们发现芬斯勒曲面的充分必要条件$(米,F)$就 Berwald 曲率而言是 Landsbregian$\mathrm{2个}$-形式。我们研究满足某些标志曲率的 Finsler 曲面$钾$条件，即，$V(钾)=0,V(钾)=-{ℐ}/F^{\mathrm{2个}}$和$V(钾)=-{ℐ}钾,$在哪里${ℐ}$是嘉当标量。为此，我们研究了一些与全局 Berwald 分布相关的几何对象${𝒟}:=跨度\{H,\mathrm{小号},V:=杰H\}$的$\mathrm{2个}$-dimensional Finsler 可度量非平面喷涂$\mathrm{小号}$. 我们找出在什么条件下这些表面变成黎曼表面。在每种情况下，测地线流的第一个积分的存在都会对刚度结果产生一些显着影响。我们证明 Finsler 曲面有$V(钾)=-{ℐ}/F^{\mathrm{2个}}$和$\mathrm{小号}(钾)=0$要么$\mathrm{小号}({𝒥})=0$是黎曼。此外，Finsler 曲面具有$V(钾)=-{ℐ}钾$和$\mathrm{小号}(钾)=0$是黎曼。