相关论文: Spectral uniqueness of bi-invariant metrics on sym…

We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such…

We show that within the class of left-invariant naturally reductive metrics $\mathcal{M}_{\operatorname{Nat}}(G)$ on a compact simple Lie group $G$, every metric is spectrally isolated. We also observe that any collection of isospectral…

We prove that the irreducible symmetric space of complex structures on $\mathbb R^{2n}$ (resp.\ quaternionic structures on $\mathbb C^{2n}$) is spectrally unique within a $2$-parameter (resp.\ $3$-parameter) family of homogeneous metrics on…

The spectral metric, defined by Schwarz and Oh using Floer-theoretical method, is a bi-invariant metric on the Hamiltonian diffeomorphism group. We show in this note that for certain symplectic manifolds, this metric can not be extended to…

Let $G$ be a compact connected Lie group of dimension $m$. Once a bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in correspondence with $m\times m$ positive definite symmetric matrices. We estimate the diameter and…

We consider a strongly damped wave equation on compact manifolds, both with and without boundaries, and formulate the corresponding inverse problems. For closed manifolds, we prove that the metric can be uniquely determined, up to an…

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…

We identify a class of left-invariant pseudo-Riemannian metrics on Lie groups for which the Laplace-Beltrami equation reduces to a first-order PDE and admits exact solutions. The defining condition is the existence of a commutative ideal…

Motivated by considerations of euclidean quantum gravity, we investigate a central question of spectral geometry, namely the question of reconstructability of compact Riemannian manifolds from the spectra of their Laplace operators. To this…

$k$-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant $k$-symplectic…

The Oscillator Groups,$\G_\lambda,$ are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric…

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_\gamma$, the Riemann moduli space of surfaces of genus $\gamma > 1$. This space has…

The action of the subgroup $\operatorname{G}_2$ of $\operatorname{SO}(7)$ (resp.\ $\operatorname{Spin}(7)$ of $\operatorname{SO}(8)$) on the Grassmannian space $M=\frac{\operatorname{SO}(7)}{\operatorname{SO}(5)\times\operatorname{SO}(2)}$…

We propose simple conditions equivalent to the discreteness of the spectrum of the Laplace-Beltrami operator on a class of Riemannian manifolds close to warped products. For this class of manifolds we establish a relationship between…

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of…

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In…

A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the…