Characterization of three- and four-dimensional Riemannian manifolds by pointwise conditions on trace and determinant of the Jacobi operator (Q5954178)

Let \((M,g)\) be a finite dimensional Riemannian manifold with curvature tensor \(R\) and let \(R_{X}\) be the Jacobi operator defined for the unit tangent vector \(X\) in the tangent space \(M_{p}\) at a point \(p\in M\). The characteristic equation of \(R_{X}\) can be represented in the form \(\sum_{k=0}^{n}(-1)^{k}J_{k}c^{n-k}=0\), where \(n=\dim M,\) \(J_{0}=0,\) \(J_{i}=J_{i}(p,X)\), \(i=1,2,\dots ,n.\) Since \(X\) is an eigenvector of \(R_{X}\) corresponding to the eigenvalue \(0,\) it results \(J_{n}=0\). The author proves that a three-dimensional Riemannian manifold \((M,g)\) is a space of non-zero constant sectional curvature if and only if at any point \(p\in M\) and for any unit tangent vector \(X\in S_{p}M\) one of the characteristic coefficients \(J_{1}(p,X),\) \(J_{2}(p,X)\neq 0\) of the Jacobi operator \(R_{X}\) is a pointwise constant. The coefficient \(J_{2}(p,X)\) is equal to zero at any point of the three-dimensional Riemannian manifold \((M,g)\) if and only if either \((M,g)\) is both foliated and non-flat or it is flat. The author proves for a four-dimensional manifold \(M\) that the following three assertions are equivalent: 1. \(M\) is pointwise Osserman; 2. \(M\) is 2-Stein, 3. locally there is a choice of orientation of \(M\) for which the metric is self-dual and Einstein. The four-dimensional pointwise Osserman manifolds are characterized by means of the characteristic coefficients \(J_{1}(p,X),\) \(J_{3}(p,X)\neq 0\) of the Jacobi operator \(R_{X}.\) The non-flat four-dimensional Einstein Riemannian manifolds for which the Jacobi operators \(R_{X}\) are degenerated are locally classified. They are locally isometric to one of the following two types of manifolds: 1. a pointwise Osserman manifold for which the Jacobi operator \(R_{X}\) has a double eigenvalue \(0\); 2. a reducible space.