On Finsler surfaces with certain flag curvatures

In the present paper, we find out necessary and sufficient conditions for a Finsler surface $(M,F)$ to be Landsbregian in terms of the Berwald curvature $2$-forms. We study Finsler surfaces which satisfy some flag curvature $K$ conditions, viz., $V(K)=0,\,\,V(K)= -\mathcal{I}/F^2$ and $V(K)=-\mathcal{I}\,K,$ where $\mathcal{I}$ is the Cartan scalar. In order to do so, we investigate some geometric objects associated with the global Berwald distribution $\mathcal{D}:= \operatorname{span}\{S, H, V:=JH\}$ of a $2$-dimensional Finsler metrizable nonflat spray $S$. We obtain some classifications of such surfaces and show that under what hypothesis these surfaces turn 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 $V(K)= -\mathcal{I}/F^2$ and either $S(K)=0$ or $S(\mathcal{J})=0$ is Riemannian. Further, a Finsler surface with $V(K)=-\mathcal{I}\,K$ and $S(K)=0$ is Riemannian.