Properties of a scalar curvature invariant depending on two planes

Abstract

Based on Schouten’s interpretation of the Riemann–Christoffel curvature tensor R, a geometrical meaning for the tensor R·R is presented. It follows that the condition of semi-symmetry, i.e. R·R = 0, can be interpreted as the invariance of the sectional curvature of every plane after parallel transport around an infinitesimal parallelogram. Using the tensor R· R, and in analogy with the definition of the sectional curvature K(p,π) of a plane π, a scalar curvature invariant L(p,π, ${\overline{\pi}}$) is constructed which in general depends on two planes π and ${\overline{\pi}}$ at the same point p. This invariant can be geometrically interpreted in terms of the parallelogramoïds of Levi–Civita and it is shown that it completely determines the tensor R· R. Further it is demonstrated that the isotropy of this new scalar curvature invariant L(p,π, ${\overline{\pi}}$) with respect to both the planes π and ${\overline{\pi}}$ amounts to the Riemannian manifold to be pseudo-symmetric in the sense of Deszcz.