Sándor Bácsó

On Finsler Spaces of Douglas Type III

The notion of Douglas space was introduced first as a generalization of the notion of Berwald space [3] from the differential equations of geodesies of a Finsler space:

On a Problem of M. Matsumoto and Z. Shen

\begin{gathered} {{\ddot x}^i}{{\dot x}^j} - {{\ddot x}^j}{{\dot x}^i} + 2{D^{ij}}(x,\dot x) = 0, \hfill \\ {D^{ij}}(x,y) = {G^i}(x,y){y^j} - {G^j}(x,y){y^i}. \hfill \\ \end{gathered}

On Finsler spaces of Douglas type

(1.1)

A note on a generalized Douglas space

The paper deals with hyper-quadrics in the real projective 4-space. According to [1] there exist 11 types of hypersurfaces of 2 order, which can be represented by ‘projective normal forms’ with respect to a polar simplex as coordinate frame. By interpreting this frame as a Cartesian frame in the (projectively extended) Euclidean 4-space one will receive sort of Euclidean standard types of hyper-quadrics resp., hypersurfaces of 2 order: the sphere as representative of hyper-ellipsoids, equilateral hyper-hyperboloids, and hyper-cones of revolution. It seems to be worthwhile to visualize the “typical” projective hyper-quadrics by means of descriptive geometry in the (projectively extended) Euclidean 4-space using Maurin’s method [4] or the classical (skew) axonometric mapping of that 4-space into an image plane.

RANDERS SPACES WITH THE H-CURVATURE TENSOR H DEPENDENT ON POSITION ALONE

The purpose of the present paper is to discuss the following two problems: Matsumoto problem: “The most important problem on projectively Berwald spaces is, of course, to find the tensorial characterization of such spaces.” and Shen problem: “Is there any Douglas metric which is not locally projectively Berwald.” Finally we give an example for Finsler spaces which have no common geodesics.