论文标题
在带有某些标志曲率的Finsler表面上
On Finsler surfaces with certain flag curvatures
论文作者
论文摘要
在本文中，我们发现，就Berwald Curvature $ 2 $ -Forms而言，Finsler Surface $（m，f）$的必要条件（m，f）$。我们研究Finsler表面，满足某些标志曲率$ k $条件，即，$ v（k）= 0，\，\，\，\，v（k）= - \ Mathcal {i}/f^2 $ and $ v（k）= - \ Mathcal {i}为此，我们研究了一些与全局Berwald分布相关的几何对象$ \ MATHCAL {D}：= \ operatoTorname {span} \ {s，h，h，v：= jh \} $ a $ 2 $ -2 $ -Dimensional-dimensional-demensional Finsler finsler Metrizable Metrizable Metrizable非flat Spray $ s $。我们获得了此类表面的一些分类，并表明这些表面变为riemannian的假设。在每种情况下，都存在地测量流的第一个积分，这在刚性结果方面都会产生一些显着的后果。我们证明，具有$ v（k）= - \ Mathcal {i}/f^2 $的鳍片表面，以及$ s（k）= 0 $或$ s（\ MATHCAL {J}）= 0 $是riemannian。此外，带有$ v（k）= - \ mathcal {i} \，k $和$ s（k）= 0 $的Finsler表面是Riemannian。
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.
