Abstract
The theory of curvature describes the essential differences between the underlying manifold and its tangent spaces. This cannot be done solely in terms of the connection coefficients, since these do not possess suitable transformation properties. We are therefore compelled to seek a set of tensors for this purpose. The simplest analytical approach to this problem is indicated by the fact that covariant differentiation is not in general commutative. Thus, in the first section of this chapter, we shall study the relevant commutation formulae which give rise to the required curvature tensors. It will be seen that the latter satisfy a large number of identities, the most important of which we shall derive. Unfortunately the analysis is occasionally apt to prove somewhat complicated. We have therefore endeavoured to break the monotony of mere analysis as soon as possible, namely by inserting a section on geodesic deviation, by means of which some of the most striking geometrical properties of the curvature and related tensors may be illustrated. We shall also discuss the second variation of the length integral, thus reestablishing the fundamental link with the calculus of variations, and in particular with the accessory problem.
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References
Cartan [1], Ch. XIII.
[5], p. 91.
This construction is described in more detail by Rund [5], p. 99, where with respect to each direction ξ i at the point x k under consideration such a — in general— non-integrable vector field is defined geometrically by means of a normal coordinate system.
Similar tensors related to the geometry of paths are derived by Douglas [1], Knebelman [1], Bompiani [1]. See also Ancochea [1]. The tensor (1.7) appears also in the work of Davies [1], p. 263, where it is denoted by R*j.
Similar relations hold for Berwald’s covariant derivatives as defined in Ch. III, § 3 (Berwald [2], p. 53) and in the general geometry of paths (Knebelman [1], p. 532). It is to be noted, however, that while formula (1.4) may be used to derive the correct variation of a vector when transported by δ-parallelism about an infinitesimal closed circuit, this is not true for formula (1.10) unless special conditions concerning the element of support prevail, which we shall discuss at a later stage.
Similar relationships have been derived by Berwald [2], p. 53, and Knebelman [1], p. 532. However, the commutation formulae discussed by these authors do not, of course, involve the Fi but only the Gi, the latter always replacing the former wherever they occur in the analysis. Hence a term of the type (1.18) would be zero identically, and as a consequence those formulae of Berwald and Knebelman which correspond to (1.17) do not contain a term analogous to our first term on the right-hand side of (1.17). See § 6.
Cartan [1], p. 34, formula (XVI).
A most elegant geometrical treatment of this tensor is given by Varga [8]: we shall, however, defer an account of this treatment until the next chapter.
Cartan [1], p. 35, formula (XVII)’.
Cartan [1], p. 36, formula (XIX)’. See also Davies [1], p. 363.
For the general theory of exterior differential forms the reader is referred to Cartan [7, 8], Kähler [1], Ch. I; Schouten [1], Ch. II, § 12.
Cartan [7], Ch. VII—VIII.
Cartan [7], p. 34 and pp. 179–181.
Compare Cartan [7], p. 51, where the notation Xi|k|k instead of Xi|kk is used.
For instance, in Riemannian geometry equation (1.43) gives, where denotes the curvature tensor of the Riemannian space (Cartan [7], p. 182). There is a discrepancy in sign between corresponding equations in the works [1] and [7] of Cartan: this maybe traced to a slight difference in notation regarding the curvature tensors.
Rund [5], p. 95. Here the writer erroneously assumed that the skew-symmetry of the K ijhk in i, j would follow from (1.15) in the transition from the “relative” to the well-defined curvature tensor. The correct equation is, of course, our equation (2.10).
Cartan [1], p. 36, formula XIX.
This identity is stated by Cartan [1], p. 37. Equation (2.17) is given by Rund [5], p. 96. Clearly (2.17) is satisfied also by the relative curvature tensor
Cartan [1], p. 37, formula XXIV. The form (3.2a) was given by Rund [5], p. 97. Between the formula (3.3) and formula XXIV of Cartan there is a discrepancy in sign for which the writer cannot give an explanation, as Cartan did not explicitly indicate the steps taken in the course of his calculation.
Cartan [7], pp. 210–211.
Rund [7], p. 6. A direct proof of this statement may also be established on the basis of the fact that the tangent vectors to the geodesics v = const. form a unit vector field. We shall omit this discussion since no further use will be made of this simplification.
In view of (1.34) it is clear that equation (4.16) may also be written in terms of Cartan’s curvature tensor in the alternative form This particular case of (4.11) is obtained by Cartan [1], p. 40 (equation XXV). Similar equations may also be derived in the general geometry of paths (Ch. III, § 3): in fact they may be obtained almost directly from the equations of variation of the differential equations of the paths. This process is discussed by Berwald [10, IV] and Kosambi [1–4]. It should be noted that these equations of variation are intimately connected with the inverse problem in the calculus of variations as is evident in particular from the work of Davis [1, 2]. See also Davies [1], Su [1].
Blaschke [4], p. 218.
Blaschke [4], p. 216.
Eisenhart [1], p. 81.
Alternative forms of equation (4.24) are given by Rund [11], pp. 188–189.
Blaschke [4], p. 216, eqn. (67), where this relation is deduced from the accessory problem with respect to the length integral on a surface. Applications of (4.24 a) are discussed by Mayer [1], Duschek and Mayer [2] for the case of Riemannian spaces.
See for instance, Blaschke [4], p. 153.
Berwald [10, I], p. 54 et seq. For the case of an F 2 of constant curvature (see § 7 of the present chapter) the same formulae are discussed by Moor [2], p. 13 et seq. The proof is quite similar to that of the classical formulae.
Berwald [10, I], p. 47 et seq. See also Berwald [1].
Here is defined with respect to the angle (1.7.12): see Ch. V, § 1.
See Kamke [1], p. 120.
Funk [2], p. 187 et seq.
Busemann [12]; [10], p. 408.
Nazim [1, 2]; Busemann [8]; Rund [3]; also Bliss [2].
This result is due to Lichnerowicz [6, 7].
These theorems are generalisations of corresponding results in Riemannian geometry due to Synge [2]. For the case of a Finsler space the first variation is discussed by Stokes [1], Freeman [1], Rund [11], p. 192.
Davies [3], p. 246. In this paper an elegant geometrical interpretation of the first term on the right-hand side is derived. Similar calculations are given by Auslander [1, 2] and Freeman [1]. The second variation may also be studied by means of the Lie derivative (Ch. V, § 5), as was shown by Laptew [1].
From (5.13), (5.11) and condition C of Ch. I, § 1, we may immediately deduce the following theorem: A curve C being a geodesic, the second variation of its length is positive for all variations with fixed end-points (and its length is therefore a relative minimum with respect to variations of that type), if the function R corresponding to every element (ξ, X) containing the direction of C is zero or negative. For n = 2, see Bolza [1]; for n-dimensional Riemannian spaces, Synge [2], p. 260. Further theorems due to Synge [2] concerning the curvature of the two-dimensional subspace containing C may be generalised similarly.
Synge [2], p. 261; Rund [11], p. 198.
Carathéodory [1], p. 260 et seq. The accessory problem is concerned with the extreme values, in the sense of the calculus of variations, of the integral (5.15).
Bolza [1], p. 62. See also Morse [1], Ch. IV.
This result is a direct generalisation of a well-known theorem of Riemannian geometry: Synge [2], p. 264; [3]; Schoenberg [1]; Myers [3]. A detailed derivation and discussion of related theorems is given by Auslander [3], this work being based on “global” methods.
For a more detailed discussion of conjugate points the reader is referred to treatises on the calculus of variations, especially to Morse [1]. A very complete description of conjugate loci in n-dimensional Finsler spaces is given by Whitehead [4], while Householder [1] investigates the dependence of the position of focal points on the curvature, the latter being based on the definitions of Cartan. See also Rinow [1].
Berwald [10, IV], p. 758. We have permitted ourselves a slight change in notation, since Berwald uses the symbol instead of , but if we were to retain the former notation this would cause confusion with the curvature tensors introduced in § 1.
Berwald calls the tensors (6.4) and (6.6) the „Grundtensor der Krümmung“ and „Krümmungstensor“ respectively.
The definitions (6.2), (6.4) and (6.7) are given in Berwald [2]. The relations (6.5), (6.9a), (6.9b) and (6.9c) between these curvature tensors and those of Cartan are derived (in a slightly different form) in Berwald [9]. It should be noted that the definitions (6.2), (6.4) and (6 7) may be applied equally well to the theory of the general geometry of paths, as is done by Berwald in [10, IV]. More generally, Varga [12] derives formulae for the difference between the curvature tensors resulting from two distinct metric connections defined over the same space of line-elements.
Commutation formulae involving both Cartan’s and Berwald’s derivatives are given by Davies [1], § 1. Further identities involving the tensor (6.7) are derived by Ispas [1, 2]. Some of these identities are generalisations of the well-known identities of Veblen.
Berwald [2], p. 54.
Finsler [1], p. 105. This result is due to Varga [10], p. 120. The osculating Riemannian metric is used in the course of a fairly simple proof which is based on the scalar form of the equations of geodesic deviation for n = 2. See Ch. V, § 6.
Synge and Schild [1], p. 111. The corresponding definition in Riemannian geometry demands that be independent of both and X i but it is clear that we have to adapt the definition with respect to Finsler spaces as stated above. Berwald [10, IV], p. 774, calls Finsler spaces in which every point is isotropic “spaces of scalar curvature”.
In essence this theorem is due to Berwald [10, IV], p. 778. It was stated by Berwald in terms of his curvature tensors (§ 6) and the scalars resulting from the latter; however, in virtue of the relations (6.2) to (6.6) Berwald’s theorem is equivalent to the one stated above.
Note that the coefficient of , which cannot vanish for all values h, k as a result of condition C (Ch. I, § 1).
Berwald, loc. cit.
Synge and Schild [1], p. 112. For the two-dimensional case in Finsler geometry, see Moór [2], p. 7.
Varga [9], pp. 154–155.
Berwald [9], § 2.
This result could, of course, have been foreseen in the light of § 1, 3°, and in particular as a result of equation (1.44). Nevertheless, the following more detailed discussion may serve to give the reader a deeper insight into the structure of Finsler spaces.
The same result is derived also by Varga [7], p. 375. In fact, it is shown by Varga that if we construct the tensor g ̄ ij (x) = g ij (x, ξ) and regard this tensor as the metric tensor of a non-euclidean space of curvature k, then the relations must be satisfied. These equations together with (7.13) represent the necessary and sufficient conditions that the Finsler space represents a generalised non-euclidean space (in the above sense).
Cartan [1], p. 39, where this result is stated without proof. The construction of the coordinate system (7.16) leading to the equations (7.18) is given by Rund [5], p. 101, but it was erroneously assumed that (7.15) is a sufficient condition for the space to be Minkowskian. See also Varga [1], p. 161, Vagner [13], p. 126.
A glance at the equations (3.3.8a) and (3.3.14) shows that the condition implies. The converse is also true, as may be shown by a short calculation for the expression of the in terms of the and their various derivatives (Berwald [9], equation (11)).
A space is said to have spherical symmetry about a point P if there exist coordinates xi all zero at P, such that the space admits all orthogonal transformations of the xi i. e. if the structure (metric, paths) is transformed into itself by every orthogonal transformation. The term “completely symmetric” implies spherical symmetry about every point.
See also Knebelman [1], p. 534.
the associated tensors were defined in this manner by Berwald [10, IV].
Douglas [1], p. 156.
This is true even in the restricted geometry of paths (cf. Eisenhart [2], p. 101), for the transformation law of the as defined by (8.22) is the same as in the restricted theory. This is immediately obvious if we note that the definition of the projective connection coefficients in the restricted theory is formally the same as (8.22), except that it lacks the last term on the right-hand side, this term being a tensor.
Douglas [1], § 7.
Douglas [1], §§ 5–6.
For n > 2 this theorem is given by Douglas [1], p. 162. The case n = 2 is treated by Berwald [10, III], §§ 9–10. See also Funk [2, 3, 4] and Wirtinger [1].
For the proof of this theorem of the restricted theory we refer the reader to Eisenhart [2], p. 96.
Berwald [10, IV], p. 767. See also Ch. VI, § 6.
Berwald, loc. cit. For n = 2 completely different conditions prevail; see Ch. VI, § 6.
Berwald [10, IV], p. 756. See also Berwald [14], Funk [2, 3, 4].
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© 1959 Springer-Verlag OHG., Berlin · Göttingen · Heidelberg
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Rund, H. (1959). The Theory of Curvature. 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_4
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