Die Verallgemeinerung Christoffelscher Zusammenhänge in der nichtlinearen Differentialgeometrie

It was already Riemann who gave fundamental indications to the non-linear differential geometry. At first we outline the different developments of that geometry, which are connected with the names of Finsler, Berwald, E. Cartan, A. Kawaguchi, on the one hand, and of E. Noether (in generalization of Christoffel’s paper), Friesecke, Winternitz, H. Busemann, on the other hand. In the second part we describe the fundamentals of Finsler geometry and areal geometry at the present time. For this we consider two structures in a manifold, a length measurement and a parallel displacement of vectors in the case of Finsler spaces, and an area measurement and a parallel displacement of decomposable p-vectors in the case of areal spaces. In this matter the homogeneous connections in the bundles of Grasmann cones play a particular part generalizing the Christoffel connections of linear differential geometry.