On Finsler and Cartan Geometries. III: Two-Dimensional Finsler Spaces with Rectilinear Extremals

Introduction. The principal part of the present paper is devoted to the problem, proposed by P. Funk,2 of characterizing, in an invariant manner, the two-dimensional Finsler spaces the extremals of which can be given, in a suitable coordinate system, by linear equations. We call such spaces Finsler spaces with rectilinear extremals. In an introductory section, we explain briefly, from the beginning, the theory of the two-dimensional Finsler spaces developed especially by the author1 and by E. Cartan.4 The standpoint of this exposition is predominantly formal. Our aim is to develop Cartan's theory of the two-dimensional Finsler spaces independently of the general theory of equivalence, and to connect it with his later theory of the n-dimensional Finsler spaces.5 The single new feature in this section is the connection between the tensor FG11k and the main scalar of a two-dimensional Finsler space, given in ?8. Section II develops two different methods which lead to an invariant characterization of the two-dimensional Finsler spaces with rectilinear extremals. The first is purely analytical, and is based upon the discussion of the conditions of integrability of a certain system of partial differential equations (??9-11). The scope of the second method is first to establish necessary and sufficient conditions in order that a two-dimensional general geometry of paths may have rectilinear paths,6 and then to apply them to a Finsler space. This method has the advantage of showing what is the independent significance of each of the two conditions we obtain (??12, 13). In the third section we determine all two-dimensional Finsler spaces with rectilinear extremals the main scalar of which is a function of position only. First we establish some theorems showing that for such a space the main scalar