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

Abstract

Let (M, g) be an n-dimensional Riemannian manifold with curvature tensor R and let R X (U) = R U X X = R (U, X, X) be the Jacobi operator defined for a unit tangent vector X in the tangent space M p at a point \( p\in M \). In the present note by a pointwise conditions on the trace and the determinant of R X we characterize locally any classes of a three- and four-dimensional Riemannian manifolds more exactly a spaces of constant sectional curvature, foliated manifolds and a reducible spaces.