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On Osserman conjecture by characteristical coefficients. (English) Zbl 0827.53042
Let $$(M,g)$$ be a 4-dimensional Riemannian manifold and let $$R$$ denote the Riemann curvature tensor. For each unit tangent vector $$u \in T_m M$$, $$m \in M$$, the Jacobi operator $$R_u : x \in T_m M \mapsto R_{xu} u$$ is a symmetric operator. Let $$\lambda (\lambda^3 + \alpha_1 (m,u) \lambda^2 + \alpha_2 (m,u)\lambda + \alpha_3(m,u)) = 0$$ be its characteristic equation. $$(M,g)$$ is an Einstein space if and only if $$\alpha_1$$ does not depend on $$u$$, and hence not on $$m$$. Further, $$(M,g)$$ is said to be 2-stein if $$\alpha_1$$ and $$\alpha_2$$ are independent of $$u$$ and finally, $$(M,g)$$ is called a pointwise Osserman space of $$\alpha_1$$, $$\alpha_2$$ and $$\alpha_3$$ or equivalently, if the eigenvalues of $$R_u$$ are independent of $$u$$.
$$(M^4, g)$$ is said to be a globally Osserman space if the eigenvalues are global constants. In this case $$(M^4, g)$$ is isometric to a two- point homogeneous space. Examples of 4-dimensional pointwise Osserman spaces which are not globally Osserman spaces are known [see, for example, P. Gilkey, A. Swann and the reviewer, Q. J. Math., Oxf. II. Ser. 46, 299-320 (1995) also for further references].
In this paper the authors first prove the already known result that the pointwise Osserman condition is equivalent to the 2-stein condition. In their second result they claim that $$(M^4, g)$$ is already a pointwise Osserman space if only $$\alpha_1$$ and $$\alpha_3$$ are independent of $$u$$.
##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
##### Keywords:
2-Stein manifold; Jacobi operator; pointwise Osserman space