IX.—On the Line-Geometry of the Riemann Tensor

Abstract

In this paper the curvature tensor Rijkl in a Riemannian Vn is used to define a quadratic complex of lines in an (n – I)-dimensional projective space Sn–1. Work in this direction has been done for a V4 by Struik (1927–28), Lamson (1930), and Churchill (1932). Of these, Struik and Lamson both use 3-dimensional projective geometry, the former for the purpose of defining sets of “principal directions” in V4 by means of the Riemann tensor, and the latter for the purpose of discussing some of the differential and algebraic consequences of the field equations of general relativity. Churchill considers the geometry of the Riemann tensor from the point of view of 4-dimensional Euclidean geometry. In this paper an indication is given of the nature of the general n-dimensional theory, which, by way of elementary illustration, is then applied in moderate detail to a V3. A few general formulae are also obtained for a V4.