Some curvature conditions of Riemannian manifolds

Nowadays, curvature conditions are one of the powerful tools to study the geometry of Riemannian manifolds. In particular cases, one can obtain geometric properties of a manifold from the curvature operator and its derivatives or vice versa. Now, through this notes we will work in both directions. First, we obtain geometric properties of manifolds working with Ledger's conditions. In particular, we solve the problem of checking if the three-parameter families of six and twelve-dimensional flag manifolds constructed by N. R. Wallach are D'Atri spaces and we obtain the classification of 4-dimensional homogeneous D'Atri spaces. Finally, we introduce the concept of Jacobi osculating rank of a Riemannian g. o. space and, we show how this new concept provide properties of the curvature operator (or, more accurately, of the Jacobi operator) and its derivatives using the geometric properties of a given g. o. space. We also show the known applications and work on explicit examples.