K型D 'atri空间及与Jacobi算子有关的几何类
D'atri spaces of type k and related classes of geometries concerning jacobi operators
摘要：
在这篇文章中，我们继续研究几何的$k$-D 'Atri空间，$% 1\leq k$ $\leq n-1$（$n$表示维数的流形）$,$开始的第二作者。众所周知，$k$ -D 'Atri空间，$k\geq 1,$与Jacobi算子$R_{v}$沿测地线的性质有关，因为她已经证明了${\operatorname{tr}}R_{v}$，${\operatorname{tr}}R_{v}^{2}$在测地线流下对任何单位切向量$v$是不变的。在这里，假设黎曼流形是一个D 'Atri空间，我们证明了在我们的主要结果，${\operatorname{tr}}R_{v}^{3}$也是不变的测地线流下，如果$ k\geq 3$。此外，还得到了Jacobi算子与Ledger条件有关的其它性质，并将它们应用于Iwasawa型空间。在Iwasawa型的D 'Atri空间类中，我们给出了非紧型对称空间的两种不同的刻画：一方面它们是$\frak{C}$ -空间，另一方面对于某些$k\geq 3.$它们是$k$-D' Atri空间，最后一种情形对于所有$k=1,...,n-1$它们也是$k$-D 'Atri空间.特别地，对于某些$k\geq 3$为$k$-D 'Atri的Damek-Ricci空间是对称的。最后，我们刻画了所有$k=1,...,n-1$的$k$-D 'Atri空间为$% \frak{SC}$ -空间（测地对称保持小测地球面的主曲率）。此外，应用这个结果在4%维齐性空间的情况下，我们证明了是一个D 'Atri（1-D' Atri）空间，或3-D 'Atri空间的性质，是一个$k$-D' Atri空间的性质等价于所有$k=1,2,3$。
英文摘要：
In this article we continue the study of the geometry of $k$-D'Atri spaces, $% 1\leq k$ $\leq n-1$ ($n$ 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 $% \frak{SC}$-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 $k=1,2,3$.
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