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.