Abstract
Having described the more essential metric properties of the local tangent spaces, we now proceed to study the underlying manifold X n , which we shall henceforth denote by F n in order to stress the fact that a Finsler metric has been imposed upon it. Some of the most fundamental properties of F n are described by the extremals of the problem in the calculus of variations which provides us with our metric. Thus we shall first derive the differential equations satisfied by the extremals — or ‘‘geodesies” — of F n ; this will be done not by means of the customary method involving the first variation of the length integral, but by means of a method specially adapted to illustrate clearly the geometrical background underlying this derivation. The basic problem in a geometry such as that of F n is the investigation of the mutual relationship between tangent spaces attached to neighbouring points of F n : more precisely, one seeks to establish geometrically meaningful mappings of one such tangent space onto another. In more elementary terms this problem may be formulated by posing the question as to the type of conditions which must be satisfied such that two vectors belonging to distinct but “neighbouring” tangent spaces may be described as being parallel. This question is by no means trivial: analytically it presents itself through the phenomenon that the ordinary derivative of a tensor is not in general also a tensor. It is from this point of view that we shall carry out a preliminary analysis of this problem in the present chapter.
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References
Carathéodory [1], Chapter XII.
Frank and von Mises [1], Ch. V.
The Hamilton- Jacobi equation is not usually expressed in the form (1.15) which involves the unique Hamiltonian function H. For the standard treatment the reader should consult Carathéodory [1], Ch. XIII, or Bolza [1], Ch. V. Further properties of families of geodesics with special reference to contact transformations and Lagrange brackets are described by Maurin [1], Douglas [2], Rund [13]. The analogy of the above construction with geometrical optics (as well as with mechanics) should be immediately obvious to the reader. This analogy is discussed in some detail by Carathéodory [5]. The approach of Synge [4] to geometrical optics may also be interpreted to some extent from this point of view.
Carathéodory [1], pp. 240–245. See also Ch. III, § 6.
Bolza [1], p. 258.
The reader may verify that the expressions on the left-hand side of (2.1) transform like the components of a covariant vector under the transformation (1.1.1). Thus equations (2.1) and hence also (2.5) are invariant (which is also obvious from our construction). However, the tensor character of (2.5) may also be established by direct transformation. It is to be noted that although the Christoffel symbols (2.3) by themselves do not possess the same transformation properties as in Riemannian geometry, the combination (2.4) is such as to cause the left-hand side of (2.5) to represent the components of a covariant vector [see equation (3.11)]. This question will be fully dealt with in the next section.
Note that this is a property enjoyed by straight lines in euclidean geometry.
This is immediately obvious if (3.1) is differentiated, while the variation of the directional arguments is being taken into account.
We shall amplify this remark in Ch. III in connection with the соvariant derivative of E. Cartan.
Rund [5, 6]. The reader is referred to a further discussion of this question in chapter III, § 2.
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© 1959 Springer-Verlag OHG., Berlin · Göttingen · Heidelberg
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Rund, H. (1959). Geodesics: Covariant Differentiation. In: The Differential Geometry of Finsler Spaces. Die Grundlehren der Mathematischen Wissenschaften, vol 101. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51610-8_2
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DOI: https://doi.org/10.1007/978-3-642-51610-8_2
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