**Historical Remarks on Finsler Geometry**

**Hanno Rund**

**University of Natal**

**(1959)**

The fundamental idea of a Finsler space may be traced back
to the famous lecture of **Riemann**:"*Uber
die Hypothesen, welche der Geometrie zugrnde liegen*." In this memoir of 1854 **Riemann**
discusses various possibilities by means of which an n-dimensional manifold may be endowed
with a metric, and pays particular attention to a metric defined by the positive square
root of a positive definite quadratic differential form. Thus the foundations of
Riemannian geometry are laid; nevertheless, it is also suggested that the positive fourth
root of a fourth order differential form might serve as a metric function. These functions
have three properties in common: they are positive, homogeneous of the first degree in the
differentials, and are also convex in the latter. It would seem natural, therefore, to
introduce a further generalisation to the effect that the distance *ds* between two
neighbouring points represented by the coordinates *x* and *x +dx* be
defined by some functions *F(x, dx):*

*ds=F(x, dx)*,

where this function satisfies these three properties.

It is remarkable that the first systematic study of
manifolds endowed with such a metric was delayed by more than 60 years. It was an
investigation of this kind which formed the subject matter of the thesis of **Finsler**
in 1918, after whom such spaces were eventually named. It would appear that this new
impulse was derived almost directly from the calculus of variations, with particular
reference to the new geometrical background which was introduced by **Caratheodory**
in connection with problems in parametric form. The kernel of these methods is the
so-called indicatrix, while the property of convexity is of fundamental importance with
regard to the necessary conditions for a minimum in the calculus of variations. In fact,
the remarkable affinity between some aspects of differential geometry and the calculus of
variations had been noticed some years prior to the publication of Finsler's thesis, in
particular by **Bliss**, **Landsberg** and **Blaschke**.
Both **Bliss** and **Landsberg** introduced (distinct)
definitions of angle in terms of invariants of a parametric problem in the calculus of
variations, while an analytic study of such invariants had been made by **E. Noether**
and **A. Underhill**. Yet the geometrical theories of **Bliss**
and **Landsberg** were developed against an Euclidean background and cannot,
therefore, be regarded as fulfilling the true objectives of the generalisation of **Riemann**'s
proposal. Clearly, **Finsler**'s thesis must be regared as the first step in
this direction.

A few years later, however, the general development took a
curious turn away from the basic aspects and methods of the theory as developed by **Finsler**.
The latter did not make use of the tensor calculus, being guided in principle by the
notions of the calculus of variations; and in 1925 the methods of the tensor calculus were
applied to the theory independently but almost simultaneously by **Synge**, **Taylor**
and **Berwald**. It was found that the second derivatives of the half of the
squre of *F(x,dx)* with respect to the differentials, *dx*, served admirably
as components of a metric tensor in analogy with Riemannian geometry, and from the
differential equations of the geodesics connection coefficients could be derived by means
of which a generalisation of Levi-Civita's parallel displacement could be defined. While
the corresponding covariant derivatives as introduced by **Synge** and **Taylor**
coincide, the theory of **Berwald** shows a marked difference, in the sense
that in his geometry the lemma of **Ricci** (which in Riemannian geometry
implies the vanishing of the covariant derivative of the metric tensor) is no longer
valid. Nevertheless, **Berwald** continued to develop his theory with
particular reference to the theory of curvature as well as to two-dimensional spaces. The
significance of his work was enhanced by the advent of the general geometry of paths (a
generalisation of the so-called Non-Riemannian geometry) due to **Douglas**
and **Knebelman**, for the initial approach of **Berwald** was
such as to establish a close affinity between these branches of metric and non-metric
differential geometry.

Again, the theory took a new and unexpected turn in 1934
when **E. Cartan** published his tract on Finsler spaces. He showed that it
was indeed possible to define connection coefficients and hence a covariant derivative
such that the preservation of **Ricci**'s lemma was ensured. On this basis **Cartan**
developed a theory of curvature, and practically all subsequent investigations concerning
the geometry of Finsler spaces were dominated by this approach. Several mathematicians
expressed the opinion that the theory had thus attained its final form. To a certain
extent this was correct, but not altogether so, as we shall now indicate.

The above-mentioned theories make use of a certain device
which basically involves the consideration of a space whose elements are not the points of
the underlying manifold, but the line-elements of the later, which form a
(2n-1)-dimensional variety. This facilitates the introduction of what **Cartan**
calls the "Euclidean connection", which , by means of certain postulates, may be
derived uniquely from the fundamental metric function *F(x, dx)*. The method also
depends on the introduction of a so-called "element of support", namely, that at
each point a previously assigned direction must be given, which then serves asdirectional
argument in all functions depending on direction as well as position. Thus, for instance,
the length of a vector and the vector obtained from it by an infinitesimal parallel
displacement depend on the arbitrary choice of the element of support. It is this device
which led to the development of Finsler geometry in terms of direct generalisations of the
methods of Riemannian geometry.

It was felt, however, that the introduction of the element
of support was undesirable from a geometrical point of view, while the natural link with
the calculus of variations was seriously weakened. This view was expressed independently
by several authors, in particular by **Vagner**, **Busemann**
and the present writer. It was emphasised that the natural local metric of a Finsler space
is a Minkowskian one, and that the arbitrary imposition of a Euclidean metric would to
some extent obscure some of the most interesting characteristics of the Finsler space.
Thus at the beginning of the present decade further theories were put forward. The
rejection of the use of the element of support, however desirable from a geometrical point
of view, led to new difficulties: for instance, the natural orthogonality between two
vectors is not in general symmetric, while the analytical difficulties are certainly
enhanced, particularly since **Ricci**'s lemma cannot be generalised as
before. Fortunately, from the point of view of differential invariants, there exist marked
similarities between all these theories, which is a perfectly natural phenomenon and could
have been expected. It is in the application and in the interpretation of these invariants
that the two points of view appear to be irreconcilable.