Number field

2.3 Number field

Complex fields may also be introduced on a real manifold. Such fields have also been used for the construction of unified field theories, although mostly after the period dealt with here (cf. Part II, in preparation). In particular, manifolds with a complex fundamental form were studied, e.g., with $gik = sik + iaik$ , where $√ --- i = − 1$ [97 ]. Also, geometries based on Hermitian forms were studied [313

]. In later periods, hypercomplex numbers, quaternions, and octonions also were used as basic number fields for gravitational or unified theories (cf. Part II, forthcoming).

In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive, e.g., at complex manifolds and so on. In this part of the article we do not need to take into account this generalisation.