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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.

##### MSC:
 53B20 Local Riemannian geometry 53A55 Differential invariants (local theory), geometric objects
##### Keywords:
Rakić duality; algebraic curvature tensor; $$p$$-Osserman
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