# Inaudibility of k-D’Atri Properties

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Main**

**Result.**

**Corollary**

**1.**

## 2. About k-D’Atri Properties

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**2**

**.**M is an n-dimensional k-D’Atri space for all $k=1,\cdots ,n-1$ if and only if for any small real $r>0$ and any unit vector $v\in {T}_{m}M$, the eigenvalues of ${S}_{v}\left(r\right)$ are preserved by the geodesic symmetries ${s}_{m}$ for all $m\in M$, that is

## 3. The Riemannian Manifolds ${\mathit{N}}^{\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}}$

- A two step nilpotent Lie algebra $\mathfrak{g}\left(j\right)=\mathfrak{v}\oplus \mathfrak{z}$ with an inner product for which $\mathfrak{v}$ and $\mathfrak{z}$ are orthogonal, where $\mathfrak{z}$ is central, $j:\mathfrak{z}\to \mathfrak{so}\left(\mathfrak{v}\right)$ is a linear map and the Lie bracket $[\xb7,\xb7]:\mathfrak{v}\times \mathfrak{v}\to \mathfrak{z}$ is given by the equation$$\langle [X,Y],Z\rangle =\langle {j}_{Z}X,Y\rangle ,\phantom{\rule{1.em}{0ex}}X,Y\in \mathfrak{v},\phantom{\rule{4pt}{0ex}}Z\in \mathfrak{z}.$$The Lie algebra $\mathfrak{g}\left(j\right)$ has an associated two-step simply connected nilpotent Lie group $\tilde{G}\left(j\right)$ defined by the exponential map, $exp:\mathfrak{v}\oplus \mathfrak{z}\to \tilde{G}\left(j\right)$ by $exp(X,Z)=(X+Z)$. Its Lie group multiplication is given by the Campbell-Baker-Hausdorff formula as follows$$exp(X,Z)\xb7exp(Y,W)=exp\left(X+Y,Z+W+{\displaystyle \frac{1}{2}}[X,Y]\right).$$Please note that the inner product on the Lie algebra $\mathfrak{g}\left(j\right)$ defines a left-invariant metric on the Lie group $\tilde{G}\left(j\right)$, that is a metric for which the left translations by group elements are isometries.
- We consider the submanifold of $\tilde{G}\left(j\right)$ without boundary$$\tilde{N}\left(j\right)=\left\{exp(X,\tilde{Z})\in \tilde{G}\left(j\right):\phantom{\rule{4pt}{0ex}}X\in {S}^{dim\mathfrak{v}-1}\phantom{\rule{4pt}{0ex}}\mathsf{and}\phantom{\rule{4pt}{0ex}}\tilde{Z}\in \mathfrak{z}\right\}\cong {S}^{dim\mathfrak{v}-1}\times \mathfrak{z}.$$
- Now, to obtain a closed manifold, we take a lattice $\mathcal{L}$ of full rank in $\mathfrak{z}$ and we consider $G\left(j\right)=\tilde{G}\left(j\right)/exp\left(\mathcal{L}\right)$.
- Finally, we obtain the closed submanifold$$N\left(j\right)=\left\{exp(X,Z)\in G\left(j\right):\phantom{\rule{4pt}{0ex}}X\in {S}^{dim\mathfrak{v}-1}\phantom{\rule{4pt}{0ex}}\mathsf{and}\phantom{\rule{4pt}{0ex}}Z\in \mathfrak{z}/\mathcal{L}\right\}\cong {S}^{dim\mathfrak{v}-1}\times {T}^{dim\mathfrak{z}}.$$

**Proposition**

**3**

**.**If two linear maps $j,{j}^{\prime}:\mathfrak{z}\to \mathfrak{so}\left(\mathfrak{v}\right)$ have the same eigenvalues counting multiplicities in $\mathbb{C}$, then the closed Riemannian manifolds $N\left(j\right)$ and $N\left({j}^{\prime}\right)$ are isospectral for the Laplace operator on functions.

**Corollary**

**2.**

**Proposition**

**4**

**.**${N}^{(a+b,0)}$ are locally homogeneous while ${N}^{(a,b)},\phantom{\rule{4pt}{0ex}}b\ge 0$ are not.

## 4. Proof of Main Results

**Lemma**

**1**

**.**Let M be a complete, simply connected, weakly-locally symmetric Riemannian manifold. Then M is weakly symmetric. In particular, the universal Riemannian covering of any complete, weakly-locally symmetric Riemannian manifold is weakly symmetric.

## 5. Conclusions and Applications

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Arias-Marco, T.; Fernández-Barroso, J.M.
Inaudibility of *k*-D’Atri Properties. *Symmetry* **2019**, *11*, 1316.
https://doi.org/10.3390/sym11101316

**AMA Style**

Arias-Marco T, Fernández-Barroso JM.
Inaudibility of *k*-D’Atri Properties. *Symmetry*. 2019; 11(10):1316.
https://doi.org/10.3390/sym11101316

**Chicago/Turabian Style**

Arias-Marco, Teresa, and José Manuel Fernández-Barroso.
2019. "Inaudibility of *k*-D’Atri Properties" *Symmetry* 11, no. 10: 1316.
https://doi.org/10.3390/sym11101316