## 14 Additional Contributions from Japan

We already met Japanese theoreticians with their contributions to non-local field theory in Section 3.3.2, to wave geometry as presented in Section 4.3, and to many exact solutions in Section 9.6.1. The unfortunate T. Hosokawa showed, in the paper mentioned in Section 4.3, that “a group of motions of a Finsler space has at most 10 parameters” [287]. Related to the discussion about exact solutions is a paper by M. Ikeda on boundary conditions [299]. He took up Wyman’s discussion of boundary conditions at spacelike infinity and tried to formulate such conditions covariantly. Thus, both spacelike infinity and the approach to it were to be defined properly. He expressed the (asymmetric) metric by referring it to an orthormal tetrad tetrad , where are the tetrad vectors, orthonormalized with regard to . The boundary condition then was for , where and the integral is taken over a path from to on a spacelike hypersurface with parameters and metric .Much of the further research in UFT from Japan to be discussed, is concerned with structural features of the theory. For example, S. Abe and M. Ikeda generalized the concept of motions expressed by Killing’s equations to a non-symmetric fundamental metric [1]. Although the Killing equations (43*) remain formally the same, for the irreducible parts of , they read as:

In (497*), . All index-movements are done with . The authors derive integrability conditions for (497*); it turns out that for space-time the maximal group of motions is a 6-parameter group.^{267}

Possibly, in order to prepare a shorter way for solving (30*), S. Abe and M. Ikeda engaged in a systematic study of the concomitants of a non-symmetric tensor , i.e., tensors which are functionals of [301*, 300]. A not unexpected result are theorems 7 and 10 in ([301], p. 66) showing that any concomitant which is a tensor of valence 2 can be expressed by and scalar functions of as factors. In the second paper, pseudo-tensors (e.g., tensor densities) are considered.

A different mathematical interpretation of Hoffmann’s meson field theory as a “unitary field theory” in the framework of what he called “sphere-geometry” was given by T. Takasu [598]. It is based on the re-interpretation of space-time as a 3-dimensional Laguerre geometry. The line element is re-written in the form

It can be viewed as “the common tangential segment of the oriented sphere with center and radius …” ([599].