10 D. BAO, S.S. CHERN AND Z. SHEN
It is desirable to have a viable formulation of Seiberg-Witten theory from a
Finslerian perspective. Independent trends of thought in this direction are be-
ing pursued by Flaherty (see this volume), Lackey, and no doubt many other re-
searchers.
The study of submanifolds involves analysis. Recently, Burago and Ivanov
[BI]
proved that every compact
cr
(r
~
2) manifold with a symmetric C
2
Finsler
metric admits a
cr
isometric embedding into a finite dimensional Banach space.
Thus subma1lifolds in Minkowskian spaces deserve special attention. See Rund
[Rl]
for some basic formulas.
9. The role of
SM
In §2, we began with the vector bundle
1r*TM
over
SM
as a natural setting
in which to formulate Finsler geometry. Throughout our excursion, we appear to
strive towards producing geometrical constructs directly on
M,
with
1r*TM
and
SM
serving as the means rather than the end. In this concluding section, let us
dispel this impression and make a case for putting S M back on the pedestal.
The 2n - 1 dimensional manifold S M has a natural Riemannian metric of
Sasaki type. Namely
g
:=
Yii dxi
0
dxi
+
Yii oyi
0
oyi,
where
oyi
:=
-j;;
(dyi
+
F Nii dxi)
and the
yi
's are regarded as homogeneous coordinates. It was precisely
this Riemannian structure of
SM
which provided the key insight for formulating
Hodge theory intrinsically on M.
The Laplacian on
SM
has a remarkable property which may prove to be useful
for studying the integral geometry of Finsler manifolds. Consider the differential
ideal generated by the Hilbert form
w. It
starts with the pairs {
w, dw }, { w
1\
dw,
(dw)
2
}, •••
of odd-even degree forms, and terminates with the contact structure
wl\(dw)n-
1
.
In [BLl], it was shown that the above pairs are respectively eigenforms
with eigenvalues
(n-
1), 2(n- 2), ... etc, while the contact form is harmonic.
SM
also figures prominently in the generalization of the Gauss-Bonnet-Chern
theorem
([Ch4], [Ch5])
to the Finsler setting. Lichnerowicz
[L]
carried out the
requisite transgression for Carlan-Berwald spaces. The extension to all Finsler
spaces with constant Vol function (see §4) was given by Bao and Chern ([BC2],
[Ch6]),
after which Shen [S3] discussed an alternate approach that Lichnerowicz
could have taken, had he not focussed on Cartan-Berwald spaces. See also [BCS2].
There are other attempts (in two dimensions), most notably Busemann's
[Bu3]
and Rund's
[R2],
which do not involve Chern's method of transgression; a more
complete list of references can be found in [Ml]
or
[BC2].
Finally, S M is the right space on which to carry out the deformation of given
Finsler structures
F,
or to ascertain the existence of those which have special ge-
ometrical properties. Here, it is assumed that one works with the fundamental
tensors instead of the
F
's directly, as the latter live on
TM \
{0} but not on
SM.
For either purpose, it is useful to know that the Ricci tensor, together with the
notion of Einstein manifolds, have recently been extended by Akbar-Zadeh [AZ2]
to the Finsler setting. In his definition, one multiplies the Ricci scalar Ric (see
§5) by
~F
2
and then take they-Hessian. The resulting components Ricij live on
SM,
while defining a quadratic form on the vector bundle
1r*TM.
This invites the
generalization of Ricci flows (see Hamilton
[H]
and references therein) and other
geometric evolution equations to the Finslerian case. The fact that
SM
is compact
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