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Linear algebra of curvature tensors and their covariant derivatives. (English) Zbl 0652.53012
Let V be a finite-dimensional vector space with inner product g, and let Curv and $$\nabla Curv$$ denote the spaces of tensors having all the symmetries of a Riemann curvature tensor and its first covariant derivative, respectively, if V were the tangent space to a Riemannian manifold at a point, with metric g at that point. The orthogonal group O(g) acts naturally on Curv and $$\nabla Curv$$. The decompositions of Curv (previously known) and $$\nabla Curv$$ into irreducible representations of O(g) and the corresponding projection operators are given explicitly, and the relationships between them under the map $$\nabla$$ are found. These results hold for semi-Riemannian metrics with minor changes.
Analogous results for curvature tensors of symmetric (torsion-free) connections with respect to the general linear group are also obtained, and one of the projection operators is interpreted as the Weyl projective curvature tensor. Applications of the decomposition of Curv involving orthogonal Radon transforms are given. Finally, some preliminary ideas are introduced to tackle the more difficult problem of describing the orbit structure of Curv under O(g), or equivalently, of finding a canonical form for curvature tensors.
Reviewer: R.S.Strichartz

##### MSC:
 53B05 Linear and affine connections 53A45 Differential geometric aspects in vector and tensor analysis
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