| \documentclass[12pt,a4paper]{article} |
|
|
| |
| \usepackage[utf8]{inputenc} |
| \usepackage[english]{babel} |
| \usepackage{geometry} |
| \geometry{a4paper} |
| \usepackage{titlesec} |
| \usepackage{hyperref} |
| |
| \usepackage{yfonts} |
| \usepackage{amssymb} |
| \usepackage{amsmath} |
| \usepackage{amsthm} |
| \usepackage[all]{xy} |
| \usepackage{comment} |
| |
| |
| \usepackage{xcolor} |
|
|
| |
|
|
| |
| \newcommand{\metrica}[1]{g\left(#1\right)} |
| \newcommand{\tr}[0]{\textup{tr}} |
| \newcommand{\ric}[0]{\textup{ric}} |
| \newcommand{\ii}{\textbf{i}} |
| \renewcommand{\j}{\textbf{j}} |
| \renewcommand{\k}{\textbf{k}} |
| \newcommand{\im}{\textup{Im}} |
| \newcommand{\A}{$\mathcal{A}$ } |
|
|
| \title{Non-compact inaudibility of Naturally Reductive property} |
| \author{Teresa Arias-Marco and José-Manuel Fernández-Barroso} |
| \date{\today} |
|
|
|
|
| \begin{document} |
| \newtheorem{theorem}{Theorem}[section] |
| \newtheorem{proposition}[theorem]{Proposition} |
| \newtheorem{corollary}[theorem]{Corollary} |
| \newtheorem{lemma}[theorem]{Lemma} |
|
|
| \newtheorem*{main}{Main Theorem} |
| \newtheorem*{maincor1}{Main corollary 1} |
| \newtheorem*{maincor2}{Main corollary 2} |
|
|
| \theoremstyle{definition} |
| \newtheorem{definition}[theorem]{Definition} |
|
|
| \theoremstyle{definition} |
| \newtheorem{remark}[theorem]{Remark} |
|
|
| \theoremstyle{definition} |
| \newtheorem{example}[theorem]{Example} |
|
|
| \theoremstyle{definition} |
| \newtheorem*{notation}{Notation} |
|
|
| \author{Teresa Arias-Marco\footnote{ORCID: 0000-0003-0984-0367;\ email: ariasmarco@unex.es} \ and Jos\'e-Manuel Fern\'andez-Barroso\footnote{ORCID: 0000-0003-3864-9967;\ email: ferbar@unex.es}} |
| \date{Universidad de Extremadura, Departamento de Matemáticas, Badajoz, Spain.} |
|
|
| \maketitle |
|
|
| \begin{abstract} |
| |
| Naturally reductive manifolds are an important class of Riemannian manifolds because they provide examples that generalize the locally symmetric ones. A property is said to be inaudible if there exists a unitary operator which intertwines the Laplace-Beltrami operator of two Riemannian manifolds such that one of them satisfies the property and the other does not. |
|
|
| In this paper, we study the relation between 2-step nilpotent Lie groups and the naturally reductive property to prove that this property is inaudible, using a pair of non-compact 11-dimensional generalized Heisenberg groups. |
| |
| \end{abstract} |
|
|
| \textbf{Keywords:} Laplace-Beltrami operator; Isospectral Riemannian manifolds; Naturally reductive manifold; 2-step nilpotent Lie group. |
|
|
| \textbf{MSC2020:} |
| 58J53; 53C25; 58J50. |
|
|
| \section*{Introduction} |
|
|
| Two Riemannian manifolds $M$ and $M'$ are said to be \textit{isospectral} if there exists a unitary operator $T:L^2(M')\to L^2(M)$ which intertwines their Laplacians, that is such that $T\circ\Delta'=\Delta\circ T$. |
| If $M$ and $M'$ are compact, this definition is equivalent to the condition that their Laplacians have the same spectrum. This compact setting is widely studied in the literature (see \cite{AL.97,LMP.23}). In \cite{Sz.99}, Szabó constructed an operator intertwining the Laplacians of two generalized Heisenberg groups with 3-dimensional center. Similarly, the authors founded in \cite{AF.24} an operator for the 7-dimensional center case. |
|
|
| The well-known locally symmetric manifolds are those whose local geodesic symmetries are isometries (see \cite{H.62} for more details). The locally symmetric manifolds are also weakly symmetric, commutative, and g.o. manifolds. See, for example, the survey \cite{BTV.95} about these properties, or the comprehensive reference \cite{W.07} which provides additional background on related geometric structures. However, it is an open question whether there exists a pair of isospectral Riemannian manifolds such that one of them is locally symmetric while the other is not. |
|
|
| A geometric property is said to be \textit{inaudible}, or it cannot be heard, when one can find isospectral Riemannian manifolds such that one of them satisfies that property and the other does not. Gordon in \cite{G.96} noted the inaudibility of being a g.o. manifold using a pair of non-compact isospectral 23-dimensional generalized Heisenberg groups. Moreover, the authors in \cite{AF.24} used the same pair to prove that weakly symmetry and commutativity are inaudible properties on non-compact Riemannian manifolds. |
|
|
| Naturally reductive Riemannian manifolds $M$ are those whose geodesics in $M$ are the orbit of a one-parameter subgroup of the group of isometries, generated by a vector in the subspace $\mathfrak{m}$ of a reductive decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ of the Lie algebra $\mathfrak{g}$ of the isometry group $G$, where $\mathfrak{h}$ denotes the Lie algebra of the isotropy group $H$ of $G$. Every locally symmetric manifold is also naturally reductive. Moreover, naturally reductive manifolds are g.o. manifolds. |
| The classification of naturally reductive Riemannian manifolds is known up to dimension eight: in dimension three the main authors who studied this property were Tricerri and Vanhecke in \cite{TV.83}; for dimensions four and five, Kowalski and Vanhecke gave important results of their classification in \cite{KV.83} and in \cite{KV.85}, for the dimensions four and five, respectively; then, Agricola, Ferreira and Friedrich classified the six-dimensional naturally reductive spaces in \cite{AFF.15}, and more recently, Storm developed a new method in \cite{S.20} to classify naturally reductive spaces and used it to classify the seven and eight dimensional ones. |
|
|
| In this paper, we study the audibility of the naturally reductive property using a pair of 11-dimensional non-compact generalized Heisenberg groups. In Section \ref{sec:discussionNR-gHg} we recall the definition of generalized Heisenberg groups and we discuss when they are naturally reductive. Then, in Section \ref{sec:inaud.NR.noncompact}, we use a result of Szabó concerning isospectral non-compact generalized Heisenberg groups to set the inaudibility of being a naturally reductive manifold. |
|
|
|
|
| \section{Naturally reductive generalized Heisenberg groups}\label{sec:discussionNR-gHg} |
|
|
| Let $\mathfrak{n}=\mathfrak{v}\oplus\mathfrak{z}$, where $\mathfrak{v}$ and $\mathfrak{z}$ are orthogonal real vector spaces with respect to an inner product $g$, and $j:\mathfrak{z}\to\mathfrak{so(v)}$ is a linear map. Then, a Lie bracket is defined on $\mathfrak{n}$ by |
| \begin{equation}\label{eq:relacion-j-corchete} |
| \metrica{[X^\mathfrak{v},Y^\mathfrak{v}]^j,Z^\mathfrak{z}}=\metrica{j_{Z^\mathfrak{z}}X^\mathfrak{v},Y^\mathfrak{v}}, |
| \end{equation} |
| for $X^\mathfrak{v},Y^\mathfrak{v}\in\mathfrak{v}$ and $Z^\mathfrak{z}\in\mathfrak{z}$, such that $(\mathfrak{n},[\cdot,\cdot]^j)$ forms a 2-step nilpotent Lie algebra (i.e. $[\mathfrak{n},\mathfrak{n}]^j\subseteq\mathfrak{z}$ and $[\mathfrak{n},\mathfrak{z}]^j=0$). We denote $(\mathfrak{n},j)$ by $\mathfrak{n}(j)$, and $(N(j),g)$ be the 2-step nilpotent Lie group whose Lie algebra is $\mathfrak{n}(j)$ with the left-invariant Riemannian metric induced by $g$, which is also denoted by $g$. The exponential map $\exp:\mathfrak{n}(j)\to N(j)$ is a diffeomorphism since $N(j)$ is simply connected and nilpotent. |
|
|
| When $[\mathfrak{n}(j),\mathfrak{n}(j)]^j\neq\mathfrak{z}$, the Lie group $N(j)$ is diffeomorphic to $N_1\times \mathbb{R}^k$, where $N_1=\exp(\mathfrak{v}\oplus[\mathfrak{n}(j),\mathfrak{n}(j)]^j)$ and $\mathbb{R}^k=\exp(([\mathfrak{n}(j),\mathfrak{n}(j)]^j)^\perp\cap\mathfrak{z})$. Therefore, we say that a 2-step nilpotent Lie group, $(N(j),g)$, has \textit{Euclidean factor} if it is isometric to some $(N_1,g_{|\mathfrak{n}_1\times\mathfrak{n}_1})\times\mathbb{R}^k$. Gordon proved in \cite{G.85} that a 2-step nilpotent Lie group $(N(j),g)$ has no Euclidean factor if and only if $\ker(j)=\{0\}$. |
|
|
| \begin{example} |
| Let $\mathfrak{v}=\mathbb{R}^2$, $\mathfrak{z}=\mathbb{R}^2$ and $\{e_1,e_2,e_3,e_4\}$ an orthonormal basis of $\mathfrak{n}=\mathfrak{v}\oplus\mathfrak{z}$ with respect to an inner product $g$. For each $Z=z_3e_3+z_4e_4\in\mathfrak{z}$, consider the linear map $j:\mathfrak{z}\to\mathfrak{so}(\mathfrak{v})$ given by |
| $$ |
| j_Z=\begin{pmatrix} |
| 0&z_3-z_4\\ |
| -z_3+z_4&0 |
| \end{pmatrix}. |
| $$ |
| By \eqref{eq:relacion-j-corchete}, the only non-zero Lie bracket on $\mathfrak{n}$ is |
| $$ |
| [e_1,e_2]^j=e_3-e_4. |
| $$ |
| Thus, $(N(j),g)$ has Euclidean factor because $\ker(j)=\textup{span}\{(1,1)\}\neq\{0\}$. In this case, $(N(j),g)$ is isometric to the 3-dimensional Heisenberg group times a 1-dimensional Euclidean factor, $(H_3,g_{|\mathfrak{h}_3\times \mathfrak{h}_3})\times\mathbb{R}$. |
| \end{example} |
|
|
| Naturally reductive 2-step nilpotent Lie groups without Euclidean factor were characterized by Gordon in \cite{G.85}. Lauret in \cite{L.99} provided an alternative proof of this characterization using different techniques. |
|
|
| \begin{theorem}[\cite{G.85,L.99}]\label{theo:caracterizacionNR} |
| Let $(N(j),g)$ be a 2-step nilpotent Lie group without Euclidean factor. Then, $(N(j),g)$ is naturally reductive if and only if |
| \begin{enumerate} |
| \item $j_\mathfrak{z}=\{j_Z\}_{Z\in\mathfrak{z}}$ is a Lie subalgebra of $\mathfrak{so(v)}$. |
| \item $\tau_X\in\mathfrak{so(z)}$ for any $X\in\mathfrak{z}$, where $\tau_X$ is given by $j_{X^\mathfrak{z}}j_{Y^\mathfrak{z}}-j_{Y^\mathfrak{z}}j_{X^\mathfrak{z}}=j_{\tau_{X^\mathfrak{z}}Y^\mathfrak{z}}$. |
| \end{enumerate} |
| \end{theorem} |
|
|
| Kaplan introduced the generalized Heisenberg groups in \cite{K.81} as special cases of 2-step nilpotent Lie groups. These manifolds are also commonly known as \textit{H-type groups} in the literature. |
| \begin{definition} |
| A \textit{generalized Heisenberg algebra} is a 2-step nilpotent Lie algebra $\mathfrak{n}(j)$ satisfying |
| \begin{equation} |
| j_{Z^\mathfrak{z}}^2=-\|Z^\mathfrak{z}\|^2\textup{Id}_\mathfrak{v}, |
| \end{equation} |
| for every $Z^\mathfrak{z}\in\mathfrak{z}$. The attached simply connected Lie group $(N(j),g)$ is the \textit{generalized Heisenberg group}. And such $j$ map is called a \textit{map of Heisenberg type}. |
| \end{definition} |
|
|
| The geometric information of a generalized Heisenberg group is encoded in its 2-step nilpotent Lie algebra. According to \cite{ABS.64}, the number of irreducible representations of $\mathfrak{v}$ viewed as Clifford modules together with $\dim{\mathfrak{z}}$ classifies generalized Heisenberg algebras as follows: |
| \begin{itemize} |
| \item[i)] If $\dim\mathfrak{z}\not\equiv3\mod4$, the Clifford module $Cl(\mathfrak{z})$ has a unique irreducible module $\mathfrak{v}_0$. Then, $\mathfrak{v}=(\mathfrak{v}_0)^p$ with $p\geq1$. That is, the generalized Heisenberg algebra is obtained by taking the direct sum of $p$ times the irreducible module. |
| \item[ii)] If $\dim\mathfrak{z}\equiv3\mod4$, the Clifford module $Cl(\mathfrak{z})$ has two non-equivalent irreducible modules, $\mathfrak{v}_1$ and $\mathfrak{v}_2$. Thus, the generalized Heisenberg algebra is obtained by taking $\mathfrak{v}=(\mathfrak{v}_1)^p\oplus(\mathfrak{v}_2)^q$, with $p\geq0,q\geq0,p+q\geq1$. We name the generalized Heisenberg algebra by $\mathfrak{n}(p,q)$ and by $N(p,q)$ its associated generalized Heisenberg group. Note that $\mathfrak{n}(p,q)$ is isomorphic to $\mathfrak{n}(q,p)$. |
| \end{itemize} |
| With this notation, if the Clifford module structure is irreducible then $\mathfrak{v}$ is said to be \textit{isotypic}. Thus, $\mathfrak{v}$ is trivially isotypic when $\dim\mathfrak{z}\not\equiv3\mod4$. If $\dim\mathfrak{z}\equiv3\mod4$, $\mathfrak{v}$ is isotypic if either $p=0$ or $q=0$. |
|
|
|
|
| Kaplan in \cite{K.83} classified naturally reductive generalized Heisenberg groups according to their dimension. |
| Tricerri and Vanhecke in \cite{TV.83} proved the same result using homogeneous structures. In both proofs, a particular map $j:\mathfrak{z}\to\mathfrak{so(v)}$ is considered given by $j_Z(X)=Z\cdot X$, with $Z\in\mathfrak{z},X\in\mathfrak{v}$, where $\cdot$ denotes the usual multiplication in $\mathfrak{v}$. Moreover, if $\mathbb{A}$ denotes the complex $\mathbb{C}$, the quaternion $\mathbb{H}$ or the Cayley (octonion) $\mathbb{O}$ numbers, $\mathfrak{v}$ is the direct sum of some copies of $\mathbb{A}$ and the center $\mathfrak{z}$ is $\mathbb{A}^*$, the non-real elements of $\mathbb{A}$. The generalized Heisenberg groups endowed with the previous $j$ maps are referred to as the Heisenberg group ($\mathfrak{z}=\mathbb{C}^*$), the quaternion analog ($\mathfrak{z}=\mathbb{H}^*$) and the Cayley analog ($\mathfrak{z}=\mathbb{O}^*$). |
|
|
| \begin{theorem}[\cite{K.83,TV.83}]\label{theo:NRKTV} |
| A generalized Heisenberg group is a naturally reductive space if and only if its center has dimension 1 (the Heisenberg group) or 3 (its quaternionic analog). |
| \end{theorem} |
|
|
| This result is previous to the characterization of naturally reductive 2-step nilmanifolds given by Gordon in \cite{G.85}. |
| Thus, it is necessary to clarify and specify that \textit{quaternionic analog} is equivalent to stating that $\mathfrak{v}$ is isotypic. The $j$ map used to prove Theorem \ref{theo:NRKTV} can be generalized in order to understand the isotypic and the non-isotypic generalized Heisenberg algebra at the same time. Suppose that $\dim\mathfrak{z}\equiv3\mod4$ and $\mathbb{A}$ denotes $\mathbb{H}$ or $\mathbb{O}$. Gordon introduced in \cite{G.93} the map $j:\mathfrak{z}\to\mathfrak{so(v)}$ where $\mathfrak{v}=(\mathbb{A})^p\oplus(\mathbb{A})^q$, $p,q\geq0$, $p+q\geq1$, $p,q\in\mathbb{N}$ and $\mathfrak{z}=\mathbb{A}^*$, by |
| $$ |
| j_Z(X_1,\dots,X_p,X_{p+1},\dots,X_{p+q})=(Z\cdot X_1,\dots,Z\cdot X_p,X_{p+1}\cdot Z,\dots,X_{p+q}\cdot Z), |
| $$ |
| where $Z\in\mathfrak{z},X_i\in \mathbb{A}, i=1,\dots p+q$, and $\cdot$ is the usual multiplication in $\mathbb{A}$. In other words, $Z\in\mathfrak{z}$ acts by the left in the first $p$ copies of $\mathbb{A}$, and it acts by the right in the last $q$ copies of $\mathbb{A}$. Note that, in the isotypic case ($p=0$ or $q=0$), this map is the same as the used by Kaplan and by Tricerri and Vanhecke. Moreover, this $j$ map is always of Heisenberg type, for every $Z\in\mathfrak{z}$ and $X\in\mathfrak{v}$, due to |
| $$ |
| \begin{aligned} |
| j^2_Z(X_1,\dots,X_p,X_{p+1},\dots,X_{p+q})&=j_Z(Z\cdot X_1,\dots,Z\cdot X_p,X_{p+1}\cdot Z,\dots,X_{p+q}\cdot Z)\\ |
| &=(Z^2\cdot X_1,\dots,Z^2\cdot X_p,X_{p+1}\cdot Z^2,\dots,X_{p+q}\cdot Z^2)\\ |
| &=-\|Z\|^2\cdot(X_1,\dots,X_p,X_{p+1},\dots,X_{p+q}). |
| \end{aligned} |
| $$ |
|
|
| In addition, $\ker(j)=\{0\}$ and the corresponding generalized Heisenberg groups with $\dim\mathfrak{z}\equiv3\mod4$ do not have Euclidean factor. |
|
|
| Finally, it follows that these generalized Heisenberg groups are naturally reductive if their corresponding generalized Heisenberg algebras have $\dim\mathfrak{z}=3$ and $\mathfrak{v}$ is isotypic, for instance $\mathfrak{n}=\mathfrak{n}(p,0),p\geq1$). Consider $\mathfrak{z}=\mathbb{H}^*$ and $\tau:\mathfrak{z}\times\mathfrak{z}\to\mathfrak{z}$ such that $\tau_XY= X\cdot Y-Y\cdot X$ for every orthogonal $X$ and $Y$ in $\mathfrak{z}$. Then, Theorem \ref{theo:caracterizacionNR} is satisfied, as a consequence of the properties of the quaternions and due to |
| $$ |
| \begin{aligned} |
| j_X&j_Y(U_1,\dots,U_p)-j_Yj_X(U_1,\dots,U_p)\\ |
| &=(X\cdot Y\cdot U_1,\dots, X\cdot Y\cdot U_p)-(Y\cdot X\cdot U_1,\dots, Y\cdot X\cdot U_p)\\ |
| &=j_{X\cdot Y-Y\cdot X}(U_1,\dots,U_p)\\ |
| &=j_{\tau_XY}(U_1,\dots,U_p), |
| \end{aligned} |
| $$ |
| for every $U=(U_1,\dots,U_p)\in\mathfrak{v}$. |
|
|
| Now suppose that $\dim\mathfrak{z}=3$ and $\mathfrak{v}$ is not necessarily isotypic, $\mathfrak{n}=\mathfrak{n}(p,q)$, with $p,q\geq0,p+q\geq1, p,q\in\mathbb{N}$. We consider $U=U^{\mathfrak{v}_p}+U^{\mathfrak{v}_q}=(U_1,\dots,U_p,0,\dots,0)+(0,\dots,0,U_{p+1},\dots,U_{p+q})\in\mathfrak{v}$, then |
| $$ |
| \begin{aligned} |
| j_Xj_YU-j_Yj_XU&=X\cdot Y\cdot U^{\mathfrak{v}_p}+U^{\mathfrak{v}_q}\cdot Y\cdot X-Y\cdot X\cdot U^{\mathfrak{v}_p}-U^{\mathfrak{v}_q}\cdot X\cdot Y\\ |
| &=(X\cdot Y-Y\cdot X)\cdot U^{\mathfrak{v}_p}+U^{\mathfrak{v}_q}\cdot(Y\cdot X-X\cdot Y)\\ |
| &=j_{X\cdot Y-Y\cdot X}U^{\mathfrak{v}_p}+j_{Y\cdot X-X\cdot Y}U^{\mathfrak{v}_q}\\ |
| &=j_{X\cdot Y-Y\cdot X}U^{\mathfrak{v}_p}-j_{X\cdot Y-Y\cdot X}U^{\mathfrak{v}_q}\\ |
| &=j_{X\cdot Y-Y\cdot X}(U^{\mathfrak{v}_p}-U^{\mathfrak{v}_q}) |
| \end{aligned} |
| $$ |
| which, in general, cannot be expressed in terms of $j_{\tau_XY}U$. Therefore, these generalized Heisenberg groups with 3-dimensional center and $\mathfrak{v}$ non-isotypic are not naturally reductive. Thus, the theorem proved by Kaplan in \cite{K.83} and by Tricerri and Vanhecke in \cite{TV.83} must be rewritten. |
| \begin{theorem}\label{theo:correccionNRHeis} |
| A generalized Heisenberg group is a naturally reductive space if and only if its center has dimension 1 (the Heisenberg group) or 3 with $\mathfrak{v}$ isotypic (its quaternionic analog). |
| \end{theorem} |
|
|
|
|
| \section{Non-compact inaudibility of the naturally reductivity}\label{sec:inaud.NR.noncompact} |
|
|
| Consider the generalized Heisenberg group $N(p,q), p,q\geq0,p+q\geq1, p,q\in\mathbb{N}$ associated with the generalized Heisenberg algebra $\mathfrak{n}(p,q)$, with 3 or 7 dimensional center. One can construct a lattice $L_{p,q}$, in $\mathfrak{n}(p,q)$, spanned by the standard basis elements. Then, $\Gamma_{p,q}=\exp(L_{p,q})$ is a cocompact discrete subgroup (i.e., it makes the quotient $N/\Gamma$ compact) of $N(p,q)$. We denote by $N^{p,q}$ the 2-step Riemannian nilmanifold $(N(p,q)/\Gamma_{p,q}, g_{p,q})$, where $g_{p,q}$ is the left-invariant Riemannian metric of $N(p,q)$. Gordon proved in \cite{G.93} the following theorem. |
| \begin{theorem} |
| If $p+q=p'+q'$, then the nilmanifolds $N^{p,q}$ and $N^{p',q'}$ are isospectral. They are locally isometric if and only if $(p',q')=\{(p,q),(q,p)\}$. |
| \end{theorem} |
|
|
| Particularly, we have the following situation |
| $$\xymatrix{ N(p,q)\ar[d]& N(p+q,0) \ar[d]\\ N^{p,q}\ar@{~}[r]& N^{p+q,0} }$$ |
| where $N(p,q)$ and $N(p+q,0)$ with $p\geq0,q\geq0,p+q\geq1$, are the Riemannian covering of $N^{p,q}$ and $N^{p+q,0}$, respectively, and $\xymatrix{N^{p,q}\ar@{~}[r]& N^{p+q,0} }$ means that $N^{p,q}$ and $N^{p+q,0}$ are isospectral and not locally isometric in the compact sense. Szabó proved in \cite{Sz.99} the following result. |
|
|
| \begin{proposition}\label{prop:isosp-Szabo} |
| The generalized Heisenberg groups $N(p,q)$ and $N(p+q,0)$, $p\geq0,q\geq0,p+q\geq1$, with 3-dimensional center, are isospectral for the Laplace-Beltrami operator. |
| \end{proposition} |
|
|
| To prove it, Szabó constructed an explicit unitary operator intertwining the Laplacians of both generalized Heisenberg groups. The authors gave the same result as Szabó when $\dim\mathfrak{z}=7$, in \cite{AF.24}. Finally, we can deduce the following proposition. |
|
|
| \begin{theorem} |
| One cannot determine if a non-compact Riemannian manifold is naturally reductive using the Laplace-Beltrami operator. |
| \end{theorem} |
| \begin{proof} |
| Consider the generalized Heisenberg groups $N(1,1)$ and $N(2,0)$ with 3-dimensional center. By Proposition \ref{prop:isosp-Szabo}, these generalized Heisenberg groups are isospectral. Moreover, $\mathfrak{n}(2,0)$ is isotypic while $\mathfrak{n}(1,1)$ is not. Thus, using Theorem \ref{theo:correccionNRHeis}, $N(2,0)$ is naturally reductive while $N(1,1)$ is not. Therefore, the Laplace-Beltrami operator does not determine whether a non-compact Riemannian manifold is naturally reductive or not. |
| \end{proof} |
|
|
|
|
| \textbf{Authors' contributions:} All authors contributed equally to this research and in writing the paper. |
|
|
| \textbf{Funding:} The authors are supported by the grants GR21055 and IB18032 funded by Junta de Extremadura and Fondo Europeo de Desarrollo Regional. |
| The first author is also partially supported by grant PID2019-10519GA-C22 funded by AEI/10.13039/501100011033 and by the grant GR24068 funded by Junta de Extremadura and Fondo Europeo de Desarrollo Regional. |
|
|
| \textbf{Conflicts of Interest:} The authors declare no conflict of interest. The founders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. |
|
|
| \textbf{Remark:} This is a preprint of the Work accepted for publication in Siberian Mathematical Journal, \copyright, copyright 2025, Pleiades Publishing, Ltd. (\url{https://pleiades.online}) |
|
|
| |
|
|
| \begin{thebibliography}{99} |
|
|
| \bibitem{AFF.15}Agricola, I., Ferreira, A. and Friedrich, T. The classification of naturally reductive homogeneous spaces in dimensions $n\leq 6$. {\em Differential Geom. Appl.}. \textbf{39} pp. 59-92 (2015), https://doi.org/10.1016/j.difgeo.2014.11.005 |
| \bibitem{AL.97} Andersson, S., and Lapidus, M. (1997). \textit{Progress in inverse spectral geometry}. Springer Science and Business Media. |
| \bibitem{AF.24}Arias-Marco, T. and Fernández-Barroso, J. Non-compact inaudibility of weak symmetry and commutativity via generalized Heisenberg groups. {\em Bulletin Of The Iranian Mathematical Society}. \textbf{50}, 71 (2024) |
| \bibitem{ABS.64}Atiyah, M., Bott, R. and Shapiro, A. Clifford modules. {\em Topology}. \textbf{3}, 3-38 (1964), https://doi.org/10.1016/0040-9383(64)90003-5 |
| \bibitem{BTV.95}Berndt, J., Tricerri, F. and Vanhecke, L. Generalized Heisenberg groups and Damek-Ricci harmonic spaces. (Springer-Verlag, Berlin,1995), https://doi.org/10.1007/BFb0076902 |
| \bibitem{G.96}Gordon, C. Homogeneous Riemannian manifolds whose geodesics are orbits. {\em Topics In Geometry}. \textbf{20} pp. 155-174 (1996) |
| \bibitem{G.93}Gordon, C. Isospectral closed Riemannian manifolds which are not locally isometric. {\em J. Differential Geom.}. \textbf{37}, 639-649 (1993), http://projecteuclid.org/euclid.jdg/1214453902 |
| \bibitem{G.85}Gordon, C. Naturally reductive homogeneous Riemannian manifolds. {\em Canad. J. Math.}. \textbf{37}, 467-487 (1985), https://doi.org/10.4153/CJM-1985-028-2 |
| \bibitem{H.62}Helgason, S. Differential geometry and symmetric spaces. (Academic Press, New York-London,1962) |
| \bibitem{K.83}Kaplan, A. On the geometry of groups of Heisenberg type. {\em Bull. London Math. Soc.}. \textbf{15}, 35-42 (1983), https://doi.org/10.1112/blms/15.1.35 |
| \bibitem{K.81}Kaplan, A. Riemannian nilmanifolds attached to Clifford modules. {\em Geom. Dedicata}. \textbf{11}, 127-136 (1981), https://doi.org/10.1007/BF00147615 |
| \bibitem{KV.85}Kowalski, O. and Vanhecke, L. Classification of five-dimensional naturally reductive spaces. {\em Math. Proc. Cambridge Philos. Soc.}. \textbf{97}, 445-463 (1985), https://doi.org/10.1017/S0305004100063027 |
| \bibitem{KV.83}Kowalski, O. and Vanhecke, L. Four-dimensional naturally reductive homogeneous spaces. {\em Rend. Sem. Mat. Univ. Politec. Torino}. pp. 223-232 (1984) (1983), Conference on differential geometry on homogeneous spaces (Turin, 1983) |
| \bibitem{L.99}Lauret, J. Modified H-type groups and symmetric-like Riemannian spaces. {\em Differential Geom. Appl.}. \textbf{10}, 121-143 (1999), https://doi.org/10.1016/S0926-2245(99)00002-9 |
| \bibitem{LMP.23} Levitin, M., Mangoubi, D., and Polterovich, I. (2023). \textit{Topics in spectral geometry} (Vol. 237). American Mathematical Society. |
| \bibitem{S.20}Storm, R. The classification of 7- and 8-dimensional naturally reductive spaces. {\em Canad. J. Math.}. \textbf{72}, 1246-1274 (2020), https://doi.org/10.4153/s0008414x19000300 |
| \bibitem{Sz.99}Szabó, Z. Locally non-isometric yet super isospectral spaces. {\em Geom. Funct. Anal.}. \textbf{9}, 185-214 (1999), https://doi.org/10.1007/s000390050084 |
| \bibitem{TV.83}Tricerri, F. and Vanhecke, L. (1983) \textit{Homogeneous structures on Riemannian manifolds}. Cambridge University Press. https://doi.org/10.1017/CBO9781107325531 |
| \bibitem{W.07}Wolf, J. (2007). \textit{Harmonic analysis on commutative spaces} (No. 142). American Mathematical Soc.. |
|
|
| \end{thebibliography} |
|
|
| \end{document} |
