2 1 0 D O C . 2 1 6 R I E M A N N I A N G E O M E T R Y & P A R A L L E L I S M 216. “Riemannian Geometry Retaining the Concept of Distant Parallelism” [Einstein 1928n] Presented 7 June 1928[1] Published 10 July 1928 In: Preußische Akademie der Wissenschaften (Berlin). Physikalisch-mathematische Klasse. Sitzungsberichte (1928): 217–221. Riemannian geometry led to a physical description of the gravitational field within general relativity theory, but it provides no concepts that can be associated with the electromagnetic field. For this reason, theoretical efforts have been directed toward finding a natural generalization or extension of Riemannian geometry that will provide more terms, in the hope of arriving at a logical structure that will permit us to unify all the physical field descriptions under a single point of view. Such efforts have led me to a theory that I shall disclose here without at- tempting to give it any physical interpretation, because the newly introduced con- cepts themselves are worthy of a certain interest, owing to the natural manner of their formulation. Riemannian geometry is characterized by the fact that the infinitesimal neigh- borhood of each point P exhibits a Euclidean metric and also that the magnitudes of two line elements that belong to the infinitesimal neighborhoods of two points P and Q at a finite distance from each other are comparable. On the other hand, the property of parallelism is lacking for two such line elements a concept of direction does not exist for finite regions. The theory discussed in the following is character- ized by the definition of a “direction,” or of equivalent directions, that is, “parallelism”[2] for finite regions, in addition to the Riemannian metric. This cor- responds to the fact that alongside the invariants and tensors of Riemannian geom- etry, new invariants and tensors will occur. § 1. The n-Bein Orthogonal-Frame Field and Metric. We imagine that at an arbitrary point P in the n-dimensional continuum, a local n-Bein orthogonal frame of n unit vectors has been attached it represents a local coordinate system. Let A a be the components of a line element or of some vector [p. 217]
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