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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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[1] Fiedler, B, Determination of the structure of algebraic curvature tensors by means of Young symmetrizers, Sém. lothar. combin., 48, (2002), Art. B48d, 20 p. (Electronic) · Zbl 1043.53016
[2] B. Fiedler, Short formulas for algebraic covariant derivative curvature tensors via algebraic combinatorics, in press
[3] Gilkey, P, Geometric properties of natural operators defined by the Riemann curvature tensor, (2001), World Scientific Publishing Co., Inc River Edge, NJ · Zbl 1007.53001
[4] Gilkey, P; Ivanova, R, The Jordan normal form of Osserman algebraic curvature tensors, Comment. math. univ. carolin., 43, 2, 231-242, (2002) · Zbl 1090.53022
[5] Greene, R.E, Isometric embeddings, Bull. amer. math. soc., 75, 1308-1310, (1969) · Zbl 0187.19201
[6] Nash, J, The imbedding problem for Riemannian manifolds, Ann. of math., 63, 2, 20-63, (1956) · Zbl 0070.38603
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