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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 \(u\).
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)