Some applications of prolonged covariant differentiation in Weyl spaces

Abstract

The notion of an \(n\)-dimensional Weyl space was formalized by \textit{J. A. Schouten} in his seminal monograph [``Der Ricci Kalkül'' (Grundlehren 10, Springer-Verlag, Berlin) (1924; see Zbl 0403.53007, Zbl 0057.37803)], as a generalized space which upon suitable specialization reduces to an \(n\)-dimensional Riemannian space, and much of the literature on Weyl spaces consists of an explication of its general properties. Essentially, the present study is concerned with the reverse idea of investigating how various concepts/notions of Riemannian geometry can be generalized to a more general context in a Weyl geometry. In particular, the authors investigate how the geometric properties of a recurrent Riemannian space can be generalized to a Weyl space. The discussion is motivated by physical considerations, e.g., quantum and twistor theory, although no explicit physical applications are indicated. Contents include: an introduction; generalized geodesic coordinates and the Bianchi identities for Weyl spaces; recurrent Weyl spaces; and hypersurfaces of recurrent Weyl spaces.