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Algebraic curvature tensors which are \(p\)-Osserman. (English) Zbl 1031.53034
Let \(m\geq 4\) and let \(R\in {\otimes}^4(R^m)^*\) be a 4 co-tensor. The curvature tensor of a metric of constant sectional curvature \(+1\) is defined by \(R_0(X,Y,Z,W)=(Y,Z)(X,W)-(X,Z)(Y,W)\) and if \(m\) is even and \(c\) is an almost complex structure on \(R^m\), we define \(R_c(X,Y,Z,W)=(Y,cZ)(cX,W)-(X,cZ)(cY,W)-2(X,cY)(cZ,W)\), where \(X,Y,Z,W\) are unit vectors in \(R^m\). The main result of this paper is given by the following classification theorem.
Let \(2\leq p\leq m-2\).
(1)  Let \(R\) be a \(p\)-Osserman algebraic curvature tensor. If \(m\) is odd, then there exists a constant \(K_0\) so that \(R=K_0R_0\). If \(m\) is even, then either there exists a constant \(K_0\) so that \(R=K_0R_0\) or there exists a constant \(K_c\) and an almost complex structure \(c\) so that \(R=K_cR_c\).
(2)  If \((M,g)\) is a \(p\)-Osserman Riemannian manifold, then \((M,g)\) has constant sectional curvature.

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