Related papers: Spectral uniqueness of bi-invariant metrics on sym…

The reduction of biharmonic maps equation in terms of the Maurer-Cartan form for all smooth map of any compact Riemannian manifolds into a compact Lie group with bi-invariant Riemannian metric is obtained. By this formula, all the…

We explicitely compute the essential spectrum of the Laplace-Beltrami operator for $p$-forms for the class of warped product metrics $d\sigma^2= y^{2a}dy^2 + y^{2b}d\theta_{\partial M}^2$, where $y$ is a boundary defining function on a…

This is a survey on left invariant semi-Riemannian metrics on compact Lie groups.

It is known, that every function on the unit sphere in $\bbr^n$, which is invariant under rotations about some coordinate axis, is completely determined by a function of one variable. Similar results, when invariance of a function reduces…

We prove an inverse spectral result for $S^1$-invariant metrics on $S^2$ based on the so-called asymptotic equivariant spectrum. This is roughly the spectrum together with large weights of the $S^1$ action on the eigenspaces. Our result…

We show that any generalised smooth distribution on a smooth manifold, possibly of non-constant rank, admits a Riemannian metric. Using such a metric, we attach a Laplace operator to any smooth distribution as such. When the underlying…

In this paper the classification of left-invariant Riemannian metrics, up to the action of the automorphism group, on cotangent bundle of (2n+1)-dimensional Heisenberg group is presented. Also, it is proved that the complex structure on…

We prove a formula for the spectrum of the Laplace-Beltrami operator on functions for compact naturally reductive homogeneous spaces in terms of eigenvalues of a generalized Casimir operator and spherical representations. We apply this…

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and B\'{e}rard Bergery, there are two typical types of such metrics, which…

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact…

A real Lie algebra is said to be characteristically solvable if its derivation algebra is solvable. We explicitly determine the moduli space of left-invariant metrics, up to isometric automorphism, for $6$-dimensional nilmanifolds whose…

We first show that a Laplace isospectral family of Riemannian orbifolds, satisfying a lower Ricci curvature bound, contains orbifolds with points of only finitely many isotropy types. If we restrict our attention to orbifolds with only…

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are…

We use Lie-theoretic methods to explicitly compute the full spectrum of the Laplace--Beltrami operator on homogeneous spheres which occur as geodesic distance spheres in (compact or noncompact) symmetric spaces of rank one, and provide a…

For an unbounded operator $S$ on a Banach space the existence of invariant subspaces corresponding to its spectrum in the left and right half-plane is proved. The general assumption on $S$ is the uniform boundedness of the resolvent along…

In this paper such Riemann metrics are established whose Laplace-Beltrami operators are identical to familiar Hamilton operators of elementary particle systems. Such metrics are the natural positive definite invariant metrics defined on…

For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie…

We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant…

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…

We show that any two left-invariant metrics on $S^3\cong\operatorname{SU}(2)$ which are isospectral for the associated classical Dirac operator $D$ must be isometric. In the case of left-invariant metrics of positive scalar curvature, we…