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A note on the structure of algebraic curvature tensors. (English) Zbl 1056.53014
A \((0,4)\)-tensor on an \(n\)-dimensional vector space \(V^n\) is an algebraic curvature tensor if it satisfies the algebraic identities of a Riemannian curvature tensor. It is known [P. B. Gilkey, Geometric properties of natural operators defined by the Riemann curvature tensor. Singapore: World Scientific (2001; Zbl 1007.53001)] that any algebraic curvature tensor on \(V^n\) can be written as a linear combination of algebraic curvature tensors built from some symmetric bilinear form on \(V\). Using the Nash embedding theorem and the Gauss equation, the authors show that a maximum number of \(n(n+1)/2\) symmetric forms suffices, but this estimate is not optimal. As is shown in the paper, the optimal value for \(n=2\) is \(1\) and for \(n=3\) is \(2\).

53B20 Local Riemannian geometry
Full Text: DOI
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