Related papers: Diameter and Laplace eigenvalue estimates for left…

Suppose $G$ is a connected noncompact locally compact group, $A,B$ are nonempty and compact subsets of $G$, $\mu$ is a left Haar measure on $G$. Assuming that $G$ is unimodular, and $ \mu(A^2) < K \mu(A) $ with $K>1$ a fixed constant, our…

We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $\mathrm{GL}(n,\mathbb{R})$ action, whose…

In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and…

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $\lambda (M,[g],\si)= \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $\lambda_1^+(\tilde{g})$ is the smallest…

Given a Riemannian submersion $(M,g) \to (B,j)$ each of whose fibers is connected and totally geodesic, we consider a certain 1-parameter family of Riemannian metrics $(g_{t})_{t > 0}$ on $M$, which is called the canonical variation. Let…

In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the…

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…

The space of $G$-invariant metrics on a homogeneous space $G/H$ is in one-to-one correspondence with the set of inner products on the tangent space $\fr{m}\cong T_{{\it o}}(G/H)$, which are invariant under the isotropy representation. When…

The main objective of this dissertation is to analyse thoroughly the construction of self-adjoint extensions of the Laplace-Beltrami operator defined on a compact Riemannian manifold with boundary and the role that quadratic forms play to…

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the…

The goal of this paper is to investigate which one of thenon-isometric left invariant Lorentz metrics on 4-dimensional nilpotent Lie groups $H_3 \times {\Bbb R}$ and $G_4$ satisfy in Ricci Soliton equation. Among the left-invariant…

We consider the infinite-dimensional Lie group $\mathfrak G$ which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold $X$ and the commutative multiplicative group of functions on $X$. The…

Let $(G,h)$ be a nilpotent Lie group endowed with a left invariant Riemannian metric, $\mathfrak{g}$ its Euclidean Lie algebra and $Z(\mathfrak{g})$ the center of $\mathfrak{g}$. By using an orthonormal basis adapted to the splitting…

A study is made of left-invariant $\mathrm{G}_2$-structures with an exact 3-form on a Lie group $G$ whose Lie algebra $\mathfrak{g}$ admits a codimension-one nilpotent ideal $\mathfrak{h}$. It is shown that such a Lie group $G$ cannot admit…

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the…

We obtain upper bounds for the eigenvalues of the Schr\"odinger operator $L=\Delta_g+q$ depending on integral quantities of the potential $q$ and a conformal invariant called the min-conformal volume. Moreover, when the Schr\"odinger…

We continue to study left-invariant pseudo-Riemannian metrics on Lie groups being in the null cone of the $O(p,q)$-action using the moving bracket approach. In particular, the Lie algebra being in the null cone implies that the…

Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically…

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

We find the index of symmetry for all solvable three-dimensional Lie groups with a left-invariant metric. When combined with the work of Reggiani on unimodular three-dimensional Lie groups, the index of symmetry is then known for all…