6
D. BAO, S.S. CHERN AND Z. SHEN
One feature of Landsberg spaces is perhaps worth mentioning. This concerns
the volume Vol(x) of the indicatrix
{y
E
TxM: F(x,y)
=
1} in each punctured
tangent space
TxM \
{0}. The metric in question is
9iJ(X, y) dyi
®
dyJ,
which has
a surprising rigidity property discovered by Brickell
[Bri]
(see also Schneider [Sc]
for a totally different proof).
In the Riemannian case, the function Vol(x) is constant and has value equal to
the volume Vol(sn-
1
)
of the unit sphere in Euclidean !Rn. In the general Finsler
setting, Vol(x) is generically nonconstant and, even for the occasions when it is
constant, its value is typically different from Vol(sn-
1
).
Furthermore, when F
happens to be symmetric [that is,
F(x, -y)
=
F(x, y)],
it is known that Vol(x) :::;
Vol(sn-
1
),
and equality holds only if
Fx
is the norm induced by an inner product
(see Schneider [Sc] and references therein). In
[BS],
the constancy of Vol(x) was
characterized by a simple criterion in terms of
A.
In particular, Vol(x) was found
to be constant on Landsberg spaces.
For a detailed survey of Berwald and Landsberg spaces, as well as further ref-
erences, see the book
[Ml]
by Matsumoto. Examples of Berwald spaces are also
adequately treated in Rund's book
[Rl]
and in the monograph
[AIM].
Unfortu-
nately, as of this writing there is no known explicit example of a Landsberg space
which is not already of Berwald type. Quite the contrary, there are criteria galore
which readily reduce a Landsberg space to a Berwald one. For a succinct discussion
of such matters, see the article
[M2]
by Matsumoto in this volume. A treatment of
Landsberg surfaces from the vantage point of exterior differential systems has been
given by Bryant
[Br2].
He informed us that genuine (that is, non-Berwald)
local
examples of Landsberg surfaces can be constructed with this approach. Perhaps
global examples are also within reach.
5. The flag curvature
From
Aijk
and
Rijkl
one can construct numerical invariants. The most im-
portant one is the flag curvature, which reduces to the sectional curvature in the
Riemannian case. We would like to argue that when suitably presented, the concept
of flag curvature
([Ch2], [AZl])
actually makes good intuitive sense on M.
To this end, consider erecting a flag at the point
x
E
M. The flagpole y
is a nonzero element of
TxM.
It singles out a particular inner product, namely
gij ( x, y) dxi
®
dxj,
which we shall use here to measure lengths and angles. For sim-
plicity, let the actual flag itself be specified by the unit vector£
:= }
along the flag-
pole and another unit vector V (the transverse edge) which is perpendicular to the
flagpole. The flag curvature is then defined as
K(x,y,£
A
V) :=Vi(£] RJikt£1) Vk.
It is a function of the flags on M. In the Riemannian case, one can check that there
is no dependence on the flagpole y.
The quantity we just defined is typically insensitive to the choice of connec-
tion implicit in
Rjikl·
In other words, in place of the hh-curvature of the Chern
connection, one could have used that of the Berwald or Cartan connections, and
the resulting flag curvatures will all agree with the one above. The same answer
can even be obtained through a Jacobi endomorphism in Foulon's
[Fl]
dynamical
systems approach to Finsler geometry.
Rotating the transverse edge about a fixed flagpole generates n -1 orthonormal
V 's. Summing up the corresponding flag curvatures (essentially an averaging
procedure) gives a function
K(x, y)
of locations and flagpoles, called the Ricci
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