The relationship between D'Atri and $k$-D'Atri spaces

Details The relationship between D'Atri and $k$-D'Atri spaces The relationship between D'Atri and $k$-D'Atri spaces

2011-09-29

arxiv.org/abs/1109.6607v1

Mathematics

Differential Geometry

16 pages

Scientific paper

In this article we continue the study of the geometry of $k$-D'Atri spaces began by the second author. We generalize some results including those related with properties of Jacobi operators and applications to spaces of Iwasawa type. The main result we prove is that every $k$-D'Atri space for some $k$, $1\leq k\leq n-1$ is D'Atri. Moreover, it is known that $k$-D'Atri spaces are related with properties of Jacobi operators as $\tr R_v$, $\tr R_v^2$ be invariant under the geodesic flow. Here we show that $\tr R_v^3$ is also invariant under the geodesic flow. One of the consequences of this fact is that $k$-D'Atri spaces for some $k\geq3$ form a proper subclass of D'Atri spaces. In the case of spaces of Iwasawa type, we show in particular that the condition on $M$ being $k$-D'Atri for some $k\geq3$ characterize the symmetric spaces within this class. Thus, there exit no $k$-D'Atri spaces of Iwasawa type for $k\geq3$ unless $M$ be symmetric, in this case $M$ is $k$-D'Atri for all possible $k\geq1$.

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