D'atri spaces of type k and related classes of geometries concerning jacobi operators

In this article we continue the study of the geometry of $k$-D'Atri spaces, $$ $\leq n-1$ ($n$ denotes the dimension of the manifold)$,$ began by the second author. It is known that $k$-D'Atri spaces, $k\geq 1,$ are related to properties of Jacobi operators $R_{v}$ along geodesics, since she has shown that ${\operatorname{tr}}R_{v}$, ${\operatorname{tr}}R_{v}^{2}$ are invariant under the geodesic flow for any unit tangent vector $v$. Here, assuming that the Riemannian manifold is a D'Atri space, we prove in our main result that ${\operatorname{tr}}R_{v}^{3}$ is also invariant under the geodesic flow if $k\geq 3$. In addition, other properties of Jacobi operators related to the Ledger conditions are obtained and they are used to give applications to Iwasawa type spaces. In the class of D'Atri spaces of Iwasawa type, we show two different characterizations of the symmetric spaces of noncompact type: they are exactly the $\frak{C}$-spaces and on the other hand they are $k$ -D'Atri spaces for some $k\geq 3.$ In the last case, they are $k$-D'Atri for all $k=1,...,n-1$ as well. In particular, Damek-Ricci spaces that are $k$-D'Atri for some $k\geq 3$ are symmetric. Finally, we characterize $k$-D'Atri spaces for all $k=1,...,n-1$ as the $$-spaces (geodesic symmetries preserve the principal curvatures of small geodesic spheres). Moreover, applying this result in the case of 4% -dimensional homogeneous spaces we prove that the properties of being a D'Atri (1-D'Atri) space, or a 3-D'Atri space, are equivalent to the property of being a $k$-D'Atri space for all $k=1,2,3$.