Abstract
In the first five chapters of this monograph we have endeavoured to present a general outline of the most basic aspects of Finsler geometry. There are, however, a large number of topics which we have left virtually untouched, and it is the purpose of the present chapter to fill in these gaps. The subjects to be treated are mostly unrelated to each other, and for the purpose of convenient orientation on the part of the reader the various sections of this chapter have been written in such a manner that they may be read independently, although, of course, reference will be made to earlier chapters. We shall concern ourselves briefly with the theory of groups of motions, conformal geometry, certain aspects of the equivalence problem, imbedding theories and the geometry of two-dimensional Finsler spaces. At the end of this chapter we have listed bibliographical references which are relevant to topics which could not conveniently be treated within the framework of this monograph.
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References
In fact, this has been done by Soós [1].
Cf. Eisenhart [1], § 27.
The form (1.2 a) of the Killing equation is given by Hokari [1] and Soós [1]. See also Akbar-Zadeh [2], where an ingenious decomposition of the Lie derivative is used.
Knebelman [1], p. 557.
Eisenhart [1], p. 223; T. Y. Thomas [1], p. 35.
This result is due to Knebelman ([1], p. 557), who stipulates in addition that. But this assumption is already contained in condition B of § 1, Ch. I; for if were to vanish, we would have as a result of (1.6). Cf. also Soós [1], p. 78.
In Riemannian geometry translations are defined as motions for which each point moves through the same distance, and the definition given here is proved as a theorem. But by means of a construction similar to that of Ch. II, § 1, it is easy to prove for Finsler spaces that if the paths of a motion are geodesics, each point moves through the same distance.
Knebelman [1], p. 559.
Eisenhart [1], p. 239; Knebelman, loc. cit.
Knebelman [1], p. 561.
Knebelman [1], p. 549 and p. 561.
Wang [1], p. 5.
Homothetic transformations were defined in Riemannian geometry by Shanks [1].
This result is due to Knebelman [2], p. 376.
Using the form (1.7.12) of Landsberg’s angle, Golab [4] shows that if a Finsler space F n is conformai to a euclidean space, then F n is necessarily Riemannian. It was pointed out by Knebelman that this theorem is a special case of the one enunciated above, i. e. a direct consequence of (2.2), but in Golab [5] it is explained that while definition (2.1) implies proportionality of lengths, the equality of the (Landsberg) angles was the point of departure of Golab’s definition of conformality. The equivalence of the two properties is established in [5]. Also, in the proof of (2.2) it has been assumed that the g ij g ̄ ij have continuous directional derivatives. Golab [6] proves the validity of (2.2) for n = 2 under weaker assumptions, namely that the g ij be merely continuous. This result is extended by Szmydt [1], where it is proved that this result holds for any n, with special reference to particular metric functions. Cartan [4] approaches the same problem in a slightly different manner and shows that for n > 2 the space must be Riemannian in order that the directions at every point of a Finsler space be conformai to the directions at a fixed point of a euclidean space.
This is the generalisation of a theorem of Riemannian geometry due to Weyl [1] which is of great importance in the theory of relativity. A different approach to the same problem is given by Laugwitz [4]. The projective relationships between two conformai metrics are studied also by Hosokawa [1].
Knebelman [2], p. 377. The corresponding relation in Riemannian geometry is given by Eisenhart [1], p. 89, eqn. (28.5).
Eisenhart [1], p. 90 et seq.
T. Y. Thomas [1], p. 66 et seq.
Knebelman [2], p. 378.
A process of this kind is sketched very briefly by Knebelman [2], who also indicates how conformai tensors may be obtained if further assumptions are made. Hombu [1] derives conformai scalar invariants for two-dimensional Finsler spaces. A further discussion, indicating the construction of a complete set of conformai invariants, is given by Hombu [3].
Chern [1,2]; Varga [11].
Compare, for instance, T. Y. Thomas [1], p. 206.
References to this method are listed under 3°, § 1 (Ch. IV).
Throughout this section Latin indices run from 1 to n, Greek indices from 1 to n - 1.
Chern [2], p. 98.
Chern [1], p. 34.
Chern [1], p. 35.
Chern [2], p. 105; Cartan [5, 6].
Chern [2], p. 107.
See Ch. IV, § 1, 3°; also Ch. III, § 1.
For the somewhat complicated analysis involving the equations of structure on which these statements are based, the reader is referred to Chern [2], pp. 107 to 120.
Vagner [13], especially §§ 3–4; Kawaguchi [5]; Barthel [3, 4, 6]. From the point of view of the general geometry of paths Friesecke [1], Bortolotti [3], Kawaguchi [3] should be mentioned.
Vagner [11, 15], also [13], pp. 65–67.
Essentially this represents the point of view taken ab initio by Mikami [1] and Vagner ([13], p. 108). Kawaguchi [5], starting with a less general set of connection coefficients uses equations corresponding to (4.2) to define the from the, noting, of course, that these equations do not uniquely define the in terms of the. In fact, in order to be able to derive the equivalent of Ricci’s lemma, Kawaguchi used connection coefficients which do not correspond to equations (4.3) directly ([5], p. 197).
The relation (4.11) is given by Kawaguchi ([5], p. 188) for his special class of metric connections, but this relation is derived from the condition that, which is not generally satisfied in the present case, at least not until further assumptions are made.
Vagner [13], p. 113. In this context the term “metric” implies that the length of a (contravariant) vector remains unchanged under parallel displacement. In the theory of non-linear connections this does not necessarily imply that the covariant derivative of the metric tensor vanishes identically (which state of affairs we had previously described by means of the term “metric connection”). It should be noted also that in view of (4.11) condition (4.12) implies that the length of covariant vectors remains unchanged under parallel displacement. This result follows immediately from the fact that. Condition (4.12) may also be derived from a consideration of the covariant derivatives of scalars (Rund [15]). Furthermore, Vagner (loc. cit.) also introduces the more general notion of a “semi-metric” connection, which implies that under parallel displacement of two vectors the ratio of their lengths remains constant. In this respect the theory of non-linear connections is closely connected with the conformai geometry discussed in §2.
This conclusion does not apply to the covariant derivative as defined by Kawaguchi [5], p. 196, for the method of the latter is formally similar to that of Cartan (Ch. III, §§ 1–2). For instance, Kawaguchi (loc. cit.) defines
Rund [15].
Ch. III, § 3.
This result seems to be the essence of the fundamental uniqueness theorem due to Vagner ([13], p. 116). A detailed treatment of the geodesics, including such aspects as the second variation and the Jacobi equation, from the point of view of the theory of non-linear connections, is given by Vagner [13], § 4.
Vagner [13], p. 124.
See Ch. IV; Vagner [13], p. 126.
Vagner [11, 15]; [13], §§ 5–6.
Kawaguchi [5], p. 186.
The idea of a “generalised” homogeneous function is also due to Kawaguchi and is described in detail in [4].
Kawaguchi [5], p. 191.
It should be mentioned, however, that Rachevski [1] had previously shown how a two-dimensional Finsler metric can be obtained by a contact transformation from a metric involving the line element.
Results similar to those given by Yano and Davies [1] were found independently and almost simultaneously by Takano [2], while a slightly more general discussion is given by Suguri and Nakayama [1]. For the literature concerning non-holonomic subspaces of Riemannian or affinely connected spaces see Yano and Davies [1].
Davies and Yano [1], p. 5.
These coefficients must not be confused with the parameters (3.1.30).
In fact, the reader may verify that these conditions result from the requirement that δg AB = 0. See Yano and Davies [1], p. 13.
Equations (5.16) and (5.17) exhibit the special form of the frame as used by Yano and Davies [2], p. 412. In Yano and Davies [1] (p. 14) the special form is taken to be The case considered by Davies [5] is more special, namely: while Doyle [1] stipulates
Yano and Davies [2], p. 414.
This theorem is due to Yano and Davies [2], p. 416. Closely related to this is the work of Deicke [2, 3], who proved that it is in general not possible to imbed a Finsler space in a Riemannian space without torsion, but that it is always possible to determine the metric and torsion tensors of a (2n-1)-dimensional space V 2n -1 such that a given F n may be regarded as a non-holonomic subspace of V 2n -1 In this process no conditions such as (5.33) are imposed; instead it is stipulated with respect to the surrounding space that the autoparallel curves shall be geodesics.
Galvani [1–6]. A similar method for two-dimensional Finsler spaces is suggested by Rachevsky [2].
Galvani [1]; [4], p. 133.
Galvani [4], p. 137.
Galvani [4], p. 141.
Galvani [4], p. 143
Galvani [2,3]; [6], p. 431.
Galvani [6], p. 444 et seq.
Berwald [5], Cartan [5]. A report on earlier results is given by Berwald [6]. We should also draw the reader’s attention to the fact that a number of results concerning the curvature of two-dimensional Finsler spaces has been given in §§ 4–5 of Ch. IV. For the sake of completeness, we should mention that Grüss [1] made a detailed study of systems of meshes in an F 2 . The concept of “geodesic conics” in an F2 is discussed by Grüss [2] and Nakajima [1]. A treatment of Cartan’s parallelism in two-dimensional Finsler spaces is given by Galvani [6].
Note that this is essentially the construction used in § 6 of Ch. V.
This scalar appears for the first time in Berwald [5], p. 204, where it differs by the factor 1/2 from the scalar used in the present text.
Cartan [5], p. 121.
Berwald [5], p. 206. In this connection the work of Moór [2] concerning two-dimensional Finsler spaces of constant curvature should be mentioned. By means of (6.23) the commutation formulae for a scalar function 0 may be written as In particular, it follows from (6.2 a) and (6.4) that we have (Berwald [5], p. 203) If we assume that the space is such that K |1= K|2= 0, K ≠ 0, and apply this formula to Φ= K, we find. But by homogeneity,, so that, and hence, from the definition of K|1, K | 2 it also follows that K x 1=K x 2= 0. Hence the conditions K |1 = 0 and K = const. are equivalent (Moor [2], p. 4). From (6.23) we deduce that these conditions are also equivalent to. The Bianchi identities become A detailed discussion of this equation is given by Moór [2].
Berwald [10, III], p. 97.
“Quasi-geodesic” systems are systems of differential equations in which is a homogeneous second order polynomial in. Cf. Blaschke and Bol [1], §29; Berwald [10, III], p. 98.
This, incidentally, is an alternative proof of the latter part of the second theorem stated on p. 144.
This may also be proved directly from considerations concerning the integrability of the conditions that the geodesics be rectilinear (Berwald [10, III], p. 96). See also Funk [2].
Berwald [5], p. 215 et seq.
Berwald [10, III], p. 104.
Berwald [10, III], p. 107 et seq.
Such spaces were first considered by Landsberg [2], p. 334 et seq.; Cartan [5], p. 33 et seq.; Berwald [5], p. 208; Berwald [10, III], p. 109 et seq.
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Rund, H. (1959). Miscellaneous Topics. 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_6
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