The Differential Geometry of Finsler Spaces

The present monograph is motivated by two distinct aims. Firstly, an endeavour has been made to furnish a reasonably comprehensive account of the theory of Finsler spaces based on the methods of classical differential geometry. Secondly, it is hoped that this monograph may serve also as an introduction to a branch of differential geometry which is closely related to various topics in theoretical physics, notably analytical dynamics and geometrical optics. With this second object in mind, an attempt has been made to describe the basic aspects of the theory in some detail - even at the expense of conciseness - while in the more specialised sections of the later chapters, which might be of interest chiefly to the specialist, a more succinct style has been adopted. The fact that there exist several fundamentally different points of view with regard to Finsler geometry has rendered the task of writing a coherent account a rather difficult one. This remark is relevant not only to the development of the subject on the basis of the tensor calculus, but is applicable in an even wider sense. The extensive work of H. BUSEMANN has opened up new avenues of approach to Finsler geometry which are independent of the methods of classical tensor analysis. In the latter sense, therefore, a full description of this approach does not fall within the scope of this treatise, although its fundamental l significance cannot be doubted.

I: Calculus of Variations. Minkowskian Spaces.- § 1. Problems in the calculus of variations in parametric form.- § 2. The tangent space. The indicatrix.- § 3. The metric tensor and the osculating indicatrix.- § 4. The dual tangent space. The figuratrix.- § 5. The Hamiltonian function.- § 6. The trigonometric functions and orthogonality.- § 7. Definitions of angle.- § 8. Area and Volume.- II: Geodesics: Covariant Differentiation.- § 1. The differential equations satisfied by the geodesics.- § 2. The explicit expression for the second derivatives in the differential equations of the geodesies.- § 3. The differential of a vector.- § 4. Partial differentiation of vectors.- § 5. Elementary properties of ?-differentiation.- III: The “Euclidean Connection” of E. Cartan.- § 1. The fundamental postulates of Cartan.- § 2. Properties of the covariant derivative.- § 3. The general geometry of paths: the connection of Berwald.- § 4. Further connections arising from the general geometry of paths.- § 5. The osculating Riemannian space.- § 6. Normal coordinates.- IV: The Theory of Curvature.- § 1. The commutation formulae.- § 2. Identities satisfied by the curvature tensors.- § 3. The Bianchi identities.- §4. Geodesic deviation Ill.- § 5. The first and second variations of the length integral.- § 6. The curvature tensors arising from Berwald’s connection.- § 7. Spaces of constant curvature.- § 8. The projective curvature tensors.- V: The Theory of Subspaces.- § 1. The theory of curves.- § 2. The projection factors.- § 3. The induced connection parameters.;.- § 4. Fundamental aspects of the theory of subspaces based on the euclidean connection.- § 5. The Lie derivative and its application to the theory of subspaces.- § 6. Surfaces imbedded in anF3.- § 7. Fundamental aspects of the theory of subspaces from the point of view of the locally Minkowskian metric.- § 8. The differential geometry of the indicatrix and the geometrical significance of the tensor Sijhk.- § 9. Comparison between the induced and the intrinsic connection parameters.- VI: Miscellaneous Topics.- § 1. Groups of motions.- § 2. Conformai geometry.- § 3. The equivalence problem.- § 4. The theory of non-linear connections.- § 5. The local imbedding theories.- § 6. Two-dimensional Finsler spaces.- Appendix: Bibliographical references to related topics.- Symbols.