Title: Fractional-linear integrals of geodesic flows on surfaces and Nakai's geodesic 4-webs Show Chinese title

Title: 分数线性积分的曲面上测地线流和Nakai的测地线4-网

Abstract: We prove that if the geodesic flow on a surface has an integral, fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for existence of fractional-linear integrals: such integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves. Show Chinese abstract

Abstract: 我们证明，如果曲面上的测地线流有一个关于动量的分式线性积分，则此类积分的空间维度要么是3，要么是5，后者情况对应于常高斯曲率。 我们还给出了分式线性积分存在的几何准则：当且仅当曲面携带一个具有常交叉比的四重测地线网时，存在这样的积分。