is) lets us make sense of certain averages that are built into these
Given the Ricci tensor, the criterion for generalized Einstein manifolds ([AZ2])
is then naturally a statement on
which reads
Arguments involv-
ing Euler's theorem show that this equation holds if and only if ).. equals the Ricci
scalar and has no y-dependence (though it can depend on x). In other words, unlike
the Riemannian case, the ).. here is typically a function (rather than a constant)
One can now ask, for example, whether in dimension
3 every manifold
admits a Finsler
structure which satisfies
.(x) 9ij,
or perhaps even one
with constant ).. ? (This is known to be false when posed for Riemannian metrics.)
A.D. Alexandrov, A theorem on triangles in a metric space and some applications, Trudy
Math. Inst. Steklov
(1951), 5-23 (in Russian).
___ , Uber eine Verallgeminerung der Riemannschen Geometri, Forschunginst. Math.
1 (1957), 33-84.
P. Antonelli, R. lngarden and M. Matsumoto, The Theory of Sprays and Finsler Spaces
with Applications in Physics and Biology, Kluwer Academic Publishers, 1993.
T. Aikou, Some remarks on the geometry of tangent bundles of Finsler manifolds, Tensor
(1993), 234-242.
L. Auslander, On curvature in Finsler geometry, Trans. Amer. Math. Soc. 79 (1955),
H. Akbar-Zadeh, Sur les espaces de Finsler
courbures sectionnelles constantes, Bull.
Acad. Roy. Bel. Bull. Cl. Sci. (5) 74 (1988), 281-322.
___ , Generalized Einstein manifolds, J. Geom. Phys. 17(4) (1995), 342-380.
D. Bao and S.S. Chern, On a notable connection in Finsler geometry, Houston J. Math.
19(1) (1993), 135-180.
___ , A note on the Gauss-Bonnet theorem for Finsler spaces, Ann. Math. 143 (1996),
D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Spring-
Verlag, to appear.
___ , On the Gauss-Bonnet integrand for 4-dimensional Landsberg spaces, this pro-
ceedings volume.
L. Berwald, Paralleliibertragung in allgemeinen Riiumen, Atti Congr. Intern. Mat. Bo-
logna 4 (1928), 263-270.
[Ber] M. Berger, La geometrie metrique des variet{s Riemanniennes, Soc. Math. France,
Asterisque hors serie (1985), 9-66.
Burago and S. Ivanov, Isometric embeddings of Finsler manifolds, St. Petersburg
Math. J. 5 (1994), 159-169.
[BK] P. Buser and H. Karcher, Gromov's almost flat manifolds, Soc. Math. France, Asterisque
81 (1981), 1-148.
[BL1] D. Bao and B. Lackey, Special eigenforms on the sphere bundle of a Finsler manifold,
this proceedings volume.
[BL2] ___ , A Hodge decomposition theorem for Finsler spaces, C.R. Acad. Sc. Paris, to
[Br1] R. Bryant, Finsler structures on the 2-sphere satisfying K
1, this proceedings volume.
[Br2] ___ , Finsler surfaces with prescribed curvature conditions, 1995 preprint.
[Bri] F. Brickell, A theorem on homogeneous functions, J. London Math. Soc. 42 (1967),
[BS] D. Bao and Z. Shen, On the volume of unit tangent spheres in a Finsler manifold, Results
in Math. 26 (1994), 1-17.
[Bu1] H. Busemann, The Geometry of Geodesics, Academic Press, 1955.
[Bu2] __ , Intrinsic area, Ann. Math. 48(2) (1947), 234-267.
[Bu3] ___ ,Angular measure and integral curvature, Canadian J. Math. 1 (1949), 279-296.
[Bu4] ___ , On geodesic curvature in two dimensional Finsler spaces, Ann. Mat. Pura Appl.
31 (1950), 281-295.
Previous Page Next Page