4 D. BAO, S.S. CHERN AND Z. SHEN
(resp. cotangent) bundle. Let
SM
be the manifold of rays of
M,
obtained from
T M by identifying the nonzero vectors which differ by a positive multiplicative
factor. There is a natural projection
1r :
SM ____, M,
sending a ray to its origin.
This projection induces vector bundles n*T M and n*T* M (of fibre dimension n)
over the 2n - 1 dimensional manifold
SM.
Denote local coordinates on M by xi, i
=
1, ... , n. Throughout our discussion,
lower case
Latin indices (except n) will range from 1
ton.
The element
(x,
y)
E
TM
with origin x and y
=
yi 8 ~, will then have (xi, yJ) as its local coordinates. The
xi 'sand yJ 'scan also serve as local coordinates on
SM,
provided that the yJ 's
are treated as homogeneous coordinates. This means that some local calculations
on
SM
are more easily done on
TM \
{0} using natural coordinates, but one must
make sure that the formulas obtained at each stage are indeed invariant under
positive rescaling in y (so that they make sense on
SM).
A Finsler structure on M corresponds to a smooth function F : T M ____, [0, oo)
which is strictly positive at ally=/=- 0 and such that F(x,ty)
=
tF(x,y) for all
t 0. We require the fundamental tensor 9ij(x,y)
:= [~F 2 ]y'y1
to be positive-
definite. Here, Fy' means
g;,,
and so on. Note that the 9ij 's are functions on
SM;
they allow us to define inner products on the fibres of the vector bundles n*T M
and n*T* M.
Put another way, for each fixed x we think of g;j(x, y) dxi
®
dxJ as a sphere's
worth (that sphere being the indicatrix) of inner products on the tangent space
TxM.
It is a bit of an irony that Finsler geometry started out to generalize Rie-
mannian geometry by replacing the inner product on each
TxM
by a Minkowski
norm, but ended up regaining a family of inner products on the same
TxM.
From
this perspective, one would expect most Riemannian constructions to carry over to
the Finsler setting, and indeed they do.
Anyway, with respect to the inner products on n*T M and n*T* M, the global
sections f.
:=
~
8
~,
(of n*T
M)
and w
:=
Fy' dxi (of n*T*
M)
are both of unit
length. Remarkably, they are also naturally dual to each other by Euler's theorem.
The section w can be regarded, through an abuse of notation, as a 1-form on
SM.
This is the Hilbert form we mentioned earlier.
Besides the fundamental tensor and the Hilbert form, another important in-
variant of the Finsler structure F is the Cartan tensor A;jk ·- ~ (F 2 )y'y1yk. It
vanishes if and only if the Finsler structure is Riemannian.
3. A choice of connection
After Einstein's formulation of general relativity, Riemannian geometry became
fashionable and one of the connections, namely that due to Christoffel (Levi-Civita),
came to the forefront. This connection is both torsion-free and metric-compatible.
Likewise, connections in Finsler geometry can be prescribed on n*T M and
its tensor products. Examples of such were proposed by J. L. Synge (1925), J.H.
Taylor (1925), L. Berwald (1928)
[Be]
and, most important of all, Elie Cartan
(1934) [C]. There is also one proposed by Chern
[Chl]
in 1948. It is torsion-free
but is not completely compatible with the inner product (on n*T
M)
defined by
the 9ij 's. Incidentally, in the generic Finslerian setting, it is not possible to have a
connection on n*T M which is both torsion-free and compatible with the said inner
product.
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