FINSLER GEOMETRY OVER THE REALS
5
The Chern connection, as do many other connections, solves the equivalence
problem for Finsler structures. Namely, it gives rise to a list of criteria which decides
when two such structures differ only by a change of coordinates. For a treatment
of this connection using moving frames, see Chern's article
[Ch3]
in this volume.
An explicit formula in terms of natural coordinates can also be written down. To
this end, let
{ijk}
denote the formal Christoffel symbols of the fundamental tensor
and introduce the quantities Nij
:=
{ijk}£k- Aijk {\s}fr £8

Then the Chern
connection coefficients are riJk = {iJd - gir (ArJs Nsk - AJks Nsr
+
Akrs N 8J).
Relationships among the Berwald-, Cartan-, Chern-, and other connections can be
found in
[S4]
and
[BCSl].
Since the Chern connection is torsion-free, its curvature consists of a hh-part
R/kt and a hv-part P/kt· Unlike that for the Cartan connection (which has torsion
but is entirely metric compatible), there is no vv-part. The tensor
P/kl
can be
expressed in terms of Aijk and its horizontal covariant derivatives. Thus the main
local invariants of Finsler geometry are Aij k and R/ kt· Riemannian geometry
remains an important special case, with Aijk
=
0 and Rjikl the Riemann-Christoffel
tensor.
4. The Cartan tensor and name-brand Finsler spaces
There is a close relationship between the Cartan tensor Aijk on SxM and the
Minkowski norm Fx on TxM. More precisely, the horizontal covariant derivative
AiJkll of Aijk describes the infinitesimal change of Fx as one varies
x.
One can show that AiJkll
=
0 if and only if PJ ikl
=
0. Thus for Finsler spaces
with Pj ikl = 0, the Minkowski norms Fx are in some sense independent of
x.
It was L. Berwald who first undertook a systematic study of such spaces, aptly
named Berwald spaces. It is a remarkable fact that on a Berwald space (M, F), not
only do the Berwald and Chern connections agree, but they also induce a linear
connection on the underlying manifold M. Thanks to a work of Szabo's
[Sz],
this
linear connection is actually the Christoffel (Levi-Civita) connection of a certain
Riemannian metric. Viewed in this light, Berwald manifolds might arguably be
regarded as natural generalizations of Riemannian manifolds.
In his study, Berwald settled the 2-dimensional case by explicitly determining
all Berwald surfaces. This program of investigation has been continued all the
way to the present by Matsumoto
[Ml],
Hashiguchi
[H],
Ichijyo
[11],
Wagner
[W],
SzabO
[Sz],
and many other dedicated scholars. Most notably, lchijyo proved that
all the (Minkowski) normed tangent spaces (TxM, Fx) of a Berwald manifold are
linearly isometric to each other through parallel translation, while Szabo gave a
complete list of locally irreducible globally symmetric Berwald manifolds which are
non-Riemannian.
Some recent results suggest that Landsberg spaces constitute just as important
a name-brand as the Berwald ones. These are defined by Aijk
:=
AiJkll £1
=
0 or,
equivalently, fJ P1ikl
=
0; they encompass the Riemannian, locally Minkowskian,
and Berwald categories. An interesting geometric characterization was given by
Aikou
[Ak].
Landsberg spaces have been studied by many geometers. In particular,
it was observed by Ichijyo
[12]
that, as in Berwald spaces, the (TxM, Fx) 's are
mutually isometric in a natural way; but unlike the Berwald setting, this natural
isometry is not necessarily linear.
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