Title: Properties of a scalar curvature invariant depending on two planes
Authors: Haesen, Stefan ×
Verstraelen, Leopold #
Issue Date: 2007
Publisher: Springer
Series Title: Manuscripta mathematica vol:122 issue:1 pages:59-72
Abstract: Based on Schouten's interpretation of the Riemann-Christoffel curvature tensor R, a geometrical meaning for the tensor R center dot R is presented. It follows that the condition of semi-symmetry, i.e. R center dot 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 center dot R, and in analogy with the definition of the sectional curvature K(p,pi) of a plane pi, a scalar curvature invariant L(p, pi, (pi) over bar) is constructed which in general depends on two planes pi and (pi) over bar at the same point p. This invariant can be geometrically interpreted in terms of the parallelogramoids of Levi-Civita and it is shown that it completely determines the tensor R center dot R. Further it is demonstrated that the isotropy of this new scalar curvature invariant L(p, pi, (pi) over bar) with respect to both the planes pi and (pi) over bar amounts to the Riemannian manifold to be pseudo-symmetric in the sense of Deszcz.
ISSN: 0025-2611
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Geometry Section
× corresponding author
# (joint) last author

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