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Geometria del calcolo delle variazioni pp 173–226Cite as

Geometrical Methods in the Differential Geometry of Finsler Spaces

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Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 23))

Abstract

The treatment of metric spaces (Finsler spaces) by the methods of differential geometry involves a lot of geometric objects (tensors, objects of connection etc.), the geometrical background of Which is in most cases not obvious. In some cases, these objects are more or less formal generalizations of corresponding objects of Riemannian geometry; in other instances, corresponding Riemannian objects do not exist. In Riemannian geometry, all of the objects needed have a significant geometrical background, either from elementary surface theory or from intrinsic properties. The situation in the more general metric spaces is apparently far more complicated. For instance, geometrical reasons for the choice of the tensors \( {\text{C}}_{{\text{j}}\,\,{\text{k}}}^{\text{i}} \,{\text{,}}\,{\text{A}}_{{\text{j}}\,\,{\text{k}}}^{\text{i}} \) in the theory of Finsler spaces (CARTAN [1] ) and for the choice of the different connection coefficients introduced by various authors are far from being evident. In fact, there are lots of connections for Finsler spaces 1) which are essentially different from each other, each of them having its special advantages for certain problems and all of them generalizing Levi-Civita's parallelism for Riemannian geometry. The method for introducing a connection in Finsler geometry has in general been the setting up of a number of postulates which lead to a certain object of connection. All of these sets of postulates take a few of the properties of Levi-Civita's connection, which are said to be the essential geometrical properties of this connection; looked at without any bias, any choice of such “fundamental” properties will seem arbitrary. This state of basing Finsler geometry on objects which are apparently derived by only formal deductions, may look very unsatisfactory. One may ask, if any geometrical procedures might be found to furnish the fundamental geometric objects of Finsler geometry with a geometrical background. Such procedures should, moreover, decide for or against one or the other formal possibility to generalize an object of Riemannian geometry to Finsler spaces. The treatment of such geometrical procedures will be the subject of these lectures.

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  1. D. Laugwitz
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E. Bompiani

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Laugwitz, D. (2011). Geometrical Methods in the Differential Geometry of Finsler Spaces. In: Bompiani, E. (eds) Geometria del calcolo delle variazioni. C.I.M.E. Summer Schools, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10959-1_3

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