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Properties of a scalar curvature invariant depending on two planes. (English) Zbl 1109.53020
Based on Schouten’s interpretation of the Riemann-Christoffel curvature tensor \(R\) [see L. Verstraelen, Comments on pseudo-symmetry in the sense of Ryszard Deszcz, Dillen, Franki (ed.) et al., Geometry and topology of submanifolds, VI. Singapore: World Scientific, 199–209 (1994; Zbl 0846.53034)], a geometrical meaning for the tensor \(R\cdot R\) is presented. The condition of semi-symmetry, \(R\cdot R=0\), is interpreted as an invariance of the sectional curvature of every plane after parallel transport around an infinitesimal parallelogram. A scalar curvature invariant \(L(p,\pi,\overline\pi)\) for two planes \(\pi\) and \(\overline\pi\) at the point \(p\) is constructed and interpreted in terms of the parallelogramoïds of Levi-Civita. Its isotropy with respect to \(\pi\) and \(\overline\pi\) amounts to the Riemannian manifold to be pseudo-symmetric in the sense of Deszcz.

MSC:
53B20 Local Riemannian geometry
53A55 Differential invariants (local theory), geometric objects
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