Abstract
A distinguishing feature of the process of δ-differentiation as developed in the previous chapter is the fact that the covariant derivative of the metric tensor does not in general vanish. Consequently, further developments of the theory of Finsler spaces will differ radically from the established body of theorems of Riemannian geometry, for in the latter the lemma of Ricci (according to which the covariant derivative of the metric tensor vanishes) plays a most decisive role. This divergence cannot be avoided if one continues to regard Finsler spaces as locally Minkowskian spaces, only linear connections being considered. On the other hand, a basically different point of view may be adopted if one introduces the so-called element of support: if this device is accepted, one can indeed construct covariant derivatives for which the analogue of Ricci’s lemma is generally valid. This is achieved by the “euclidean connection” of Cartan, to which the first part of the present chapter is devoted. In view of the great influence which this approach has exerted upon the general development of our subject, the theory of Cartan will be developed ab initio. In the latter half of this chapter we shall introduce a generalisation of Finsler spaces by defining a general space of paths: this will lead to alternative forms of covariant derivatives which will be compared with each other in the light of results of this and the previous chapter.
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References
Cartan [1, 2].
Shorter methods for the determination of Cartan’s coefficients have been given at various times, in particular by Varga [2], Laugwitz [3] and Sulanke [1]. However, since these authors use special devices such as the osculating Riemannian space (see § 4) or the general geometry of paths (in the sense of Douglas [1]), their methods might not exhibit the basic ideas of Cartan as clearly as the original method of the latter. It might, therefore, be more advantageous to follow the historical approach and to postpone the discussion of these more recent constructions. Also, it should be pointed out that Schouten and Haantjes [1] gave a more general theory for the determination of the connection coefficients of spaces in which the fundamental metric function depends on со- and contravariant vector densities. In this connection the reader should also consult Schouten and Hlavatý [1].
In point of fact, Cartan gives a further postulate in addition to conditions (a) to (d) ([1], p. 10); however, this postulate is superfluous since it may be derived from the others (see Cartan [3]). It merely involves the relation between perpendicularity and transversality with respect to the element of support.
It should be stressed, however, that the notion of length as defined in Chapter I is in general not identical to that defined by (1.1’), the identity holding only if the direction of the vector X i coincides with its own element of support.
Cartan [1], p. 15.
Cartan [1], p. 16.
This could be expected also from a purely geometrical point of view, since the element of support defines two osculating (i. e. quadric) indicatrices for the two neighbouring points in analogy with Riemannian geometry.
See the footnote to equation (2.5.8a), where an analogous problem is treated.
See, for instance, T. Y. Thomas [1] or Eisenhart [2].
Douglas [1], also Knebelman [1].
Noether [1].
Berwald [1, 2, 4]. Expository articles dealing with this development are due to Koschmieder [1], Winternitz [1], Berwald [13]. Certain aspects of the metric theory of Berwald in relation to the work of Douglas [1] are discussed by Slebodzinski in an article [1] based on the “kinematic group” (Wundheiler [2]).
Berwald [5, 6]. It should be stressed that Berwald’s theory is also based on the use of the element of support.
С art an [1], formula XV, p. 19. Further comparisons between the two connections are made by Davies [2], p. 262, Hosokawa [3], Laptew [2] and Berwald [7, 9, 10], who also derives relations between the curvature tensors resulting from the two connections. (See Ch. IV). Sasaki [1] investigates the relationship between the coefficients (3.6) associated with special metric functions in connection with his generalisation of projective differential geometry of curved spaces as discussed by Veblen [1].
Berwald [2], equations (25) and (35).
Berwald [2], p. 47. In contrast to this Vagner [1, 4] calls such spaces “Berwald spaces” and shows that the determination of such spaces is related to the problem of determining all hypersurfaces which admit a transitive subgroup of centrally affine transformations. In the paper Vagner [1] special cases are discussed.
Cartan [1], p. 38.
Laugwitz [3], p. 23. The basic idea of this method is similar to one described briefly by Bompiani [1]. A general geometry of paths based on the non-linear displacement (4.4 b) had already been discussed by Miкami [1] in an extension of the work of Bortolotti [1, 3] and Friesecke [1]. See also Schouten [2] and Kawaguchi [3].
Barthel [3,4].
For a more general discussion of non-linear connections see Ch. VI, § 4. A general theory of such connections is developed by Vagner [13].
Compare, for instance, equation (2.6).
Compare Norden [1], p. 135. In fact, (4.7) may be deduced by cyclic interchange of the indices i, k, j in (4.6), which gives two new equations from the sum of which (4.6) may be subtracted. On simplifying the result we find (4.7).
Nazim [1].
Eisenhart [1], p. 62.
[2], p. 450–452.
Varga [2], p. 172. See also Varga [6].
Laugwitz [3], p. 26.
Laugwitz [3], p. 25.
This theorem is due to Douglas [1], p. 163. A similar result has been stated by Busemann [11]. In this context we should remark that the paths (and hence the geodesics of a Finsler space) naturally continue to possess the following property: for any point O of a region there exists a domain R containing O such that any point P of R is joined to О by one and only one path lying in R. In fact, an even more general result is proved by Whitehead [1, 2]: For any point O there exists a domain D containing О such that any two points P and Q of D can be joined by one and only one path lying in the domain D. See also Whitehead [4].
Douglas [1], p. 164 et seq.
Douglas [1], §§ 10–11. For the original definitions of extension and normal tensors see T. Y. Thomas [1], pp. 14 and 102.
Varga [13]. For the general theory of the spaces referred to, the so-called „Affinzusammenhängende Mannigfaltigkeiten“, see Varga [11].
Eisenhart [2], p. 58.
Varga [13], § 1.
Varga [13], p. 155 et seq.; for the classical replacement theorem see T. Y. Thomas [1], § 39. The normal coordinates of Varga are used by Rapcsák [2] to derive an invariant Taylor expansion for tensors of a Finsler space. For a modification of Varga’s definitions see Rapcsák [3].
For the proof of this and of the following statements see Myers [1], § 3. Cf. also Busemann and Mayer [1].
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© 1959 Springer-Verlag OHG., Berlin · Göttingen · Heidelberg
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Rund, H. (1959). The “Euclidean Connection” of E. Cartan. 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_3
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