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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 $$2$$.

##### MSC:
 53B20 Local Riemannian geometry
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##### References:
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