**Differential Geometry of Spray and Finsler Spaces**

**by Zhongmin Shen**

08/28/99

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

(1)

where satisfy for t > 0, and *y*
stands for . 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

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 ,

(2)

where . 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

Then and 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 and the mean Riemann curvature **Ric**(y)
= *(n-1)R(y)* (i.e., the Ricci curvature). They are given in a natural way.

When the spray comes from a Riemann metric, then 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 on a spray manifold *(M,***G***)*,
we can define the so-called S-curvature **S**, which describes the change of
the measure along geodesics.

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
in for every non-zero tangent vector *y* at *x*. We have
another important quantity

**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 in the following sense

then it must be Riemannian. Finsler metrics on with constant curvature and complete Finsler metrics on with constant curvature have not been completely understood yet. Bryant metrics on has constant curvature = 1, while Hilbert metrics 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

where *P* is a homogeneous function on *TM* and is the canonical
vertical vector field on *TM*.

We have two projective invariant curvatures and . They are given by

**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.

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