On a rigidity condition for Berwald spaces (Q981974)

A connected Finsler space \((M, F)\) is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold \(M\). It is called a Landsberg space if its Landsberg curvature is equal to \(0\). Every Berwald space must be a Landsberg space. It is a long standing problem in Finsler geometry whether there is a Landsberg space which is not Berwald. In this paper, the authors propose a strategy to solve this problem. The main tool is the averaged connection introduced by the first author. They prove, among other things, that, if \((M, F)\) is a Landsberg space such that its averaged connection of the Chern connection does not leave invariant any compact submanifold \(I_x(t)\subset T_x(M)\) of codimension zero, then the space is not Berwald.