Differential Geometry of Spray and Finsler Spaces

by Zhongmin Shen



Local geometric structures of Finsler metrics have been  understood in great depth, due to important contributions by number of geometers after P. Finsler's pioneering work in 1918. Among them are L. Berwald, J. Douglas, E. Cartan, S. S. Chern, H. Rund, M. Matsumoto, R. Miron, etc. Important applications to biology and physics have been given by P. Antonelli, R. Ingarden and G. S. Asanov, etc.

Consider  the following system

wpe1.jpg (1945 bytes)                                      (1)

where wpe2.jpg (980 bytes) satisfy wpe3.jpg (1420 bytes) for t > 0, and  y   stands for wpe5.jpg (1227 bytes).  This is a very common system of non-linear ordinary differential equations. For example, locally minimizing curves in a Finsler manifold can be characterized by (1).

The above system can be described in an index-free form. Let

wpe4.jpg (2091 bytes)

Then G is a vector field on TM-{0}. We call it a  spray. The projection of an integral curve of G satisfies (1). Every spray gives a canonical decomposition wpe6.jpg (1960 bytes),

wpe7.jpg (3719 bytes)                                (2)

where wpe8.jpg (1400 bytes). The decomposition (2) is usually called the  (non-linear) connection of the spray.

Surprisingly, the curvatures of a spray can be defined by very simple formulas. But their geometric meanings have not been completely understood yet. Let

wpeB.jpg (2151 bytes)

wpe12.jpg (3010 bytes)

Then   wpe1D.jpg (2362 bytes) and  wpeA.jpg (1749 bytes)are well-defined quantities. We call B the  Berwald curvature  and R the Riemann curvature.

Taking the mean value of B and R, respectively, we obtain the mean Berwald curvature wpe11.jpg (1668 bytes) and the mean Riemann curvature Ric(y) = (n-1)R(y) (i.e., the Ricci curvature). They are given in a natural way.

wpe12.jpg (1143 bytes)

wpe15.jpg (1733 bytes)

When the spray comes from a Riemann metric, then wpe16.jpg (980 bytes) are quadratic in y. Thus B = 0  and E = 0. Therefore we call B and E the non-Riemannian quantities, while we call R and Ric the Riemannian quantities.

A spray (resp. a Finsler metric) is called a  Berwald spray (resp.  Berwald metric) if the Berwald curvature B = 0. Berwald sprays/metrics are very important special sprays/metrics in applications.

If we are also given a smooth measure wpe17.jpg (1366 bytes) on a spray manifold (M,G), we can define the so-called S-curvature S, which describes the change of the measure along geodesics.

wpe18.jpg (2081 bytes)

If the spray and the measure come from a Riemann metric, then S = 0 . Thus S is another non-Riemannian quantity. This S-curvature  plays an important role in the study of smooth metric measure manifolds or Finsler manifolds.

Given a Finsler metric F, there is an induced inner product wpe1.jpg (799 bytes)   in wpe2.jpg (890 bytes)  for every non-zero tangent vector y at x. We have another important quantity

wpe3.jpg (2288 bytes)

L is called the Landsberg curvature. A Finsler metric is called a Landsberg metric if L = 0. The Landsberg curvature plays an important role in Finsler geometry.

According to a theorem of Numata, if a Landsberg metric  has constant curavature wpe4.jpg (871 bytes) in the following sense

wpe5.jpg (2737 bytes)

then it must be Riemannian. Finsler metrics on wpe8.jpg (782 bytes) with  constant curvature wpeA.jpg (846 bytes) and complete Finsler metrics on wpe9.jpg (832 bytes)with constant curvature wpeB.jpg (964 bytes) have not been completely understood yet. Bryant metrics on wpe6.jpg (780 bytes) has constant curvature = 1, while Hilbert metrics wpe7.jpg (828 bytes) has constant curvature = - 1. But the Akbar-Zadeh theorem says that Finsler metrics of constant curvature = -1 on any closed manifold must be Riemannian.

If one is  interested only  in the geodesics as sets of points, then he needs geometric quantities which are invariant under projective changes

wpe19.jpg (1088 bytes)

where P is a homogeneous function on TM and wpe1A.jpg (1255 bytes) is the canonical vertical vector field on TM.

We have two projective invariant curvatures wpe1B.jpg (2457 bytes) and wpe1C.jpg (1979 bytes). They are given by

wpeC.jpg (3433 bytes)

wpe1E.jpg (4190 bytes)

D and W are called the Douglas curvature and the Berwald-Weyl curvature, respectively. They are two basic projective invariants. In dimension two, the Berwald-Weyl curvature always vanishes. In this case we have and other  projective invariant .

The S-curvature  S associated with an  arbitrary smooth measure is closely related to other non-Riemannian quantities.

wpe10.jpg (1918 bytes)

dbh.jpg (3112 bytes)

It is an important project to study sprays (or Finsler metrics) which projectively related to Berwald sprays (or Berwald metrics).