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Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and…

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H where G is a metrizable topological group and H is a closed…

We consider the space $\mathcal{M}$ of Euclidean similarity classes of framed loops in $\mathbb{R}^3$. Framed loop space is shown to be an infinite-dimensional K\"{a}hler manifold by identifying it with a complex Grassmannian. We show that…

Compact hyperbolic 3-manifolds are used in cosmological models. Their topology is characterized by their homotopy group $\pi_1(M)$ whose elements multiply by path concatenation. The universal covering of the compact manifold $M$ is the…

We study isometric immersions $f: M^n \rightarrow \mathbb{H}^{n+1}$ into hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of dimension $n$ on which a compact connected group of intrinsic isometries acts with principal…

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics…

We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the…

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to…

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured…

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings. The construction of such mappings comes from our construction of non-trivial compact…

Let $M$ be a compact hyperkahler manifold with maximal holonomy (IHS). The group $H^2(M, R)$ is equipped with a quadratic form of signature $(3, b_2-3)$, called Bogomolov-Beauville-Fujiki (BBF) form. This form restricted to the rational…

We use the generalized Gauss-Bonnet formula for Riemannian polyhedra discovered by Allendoerfer, Weil and Chern to show that hyperbolic space of dimension $n$ has no isometric immersion into Euclidean space of dimension $2n-1$.

Utilizing the framework of quaternionic contact geometry, we define a sequence of Riemannian metrics $\{g_L\}$ on the quaternionic Heisenberg group $\mathfrak{H}_{\mathbb{H}}$ by rescaling the vertical directions. By analyzing the limit of…

It is well-known that the $L^2$ metric on the moduli space of hyperbolic monopoles, defined using the Coulomb gauge-fixing condition, diverges. This article shows that an alternative gauge-fixing condition inspired by supersymmetry cures…

We classify six-dimensional homogeneous nearly K\"{a}hler manifolds and give a positive answer to Gray and Wolf's conjecture: every homogeneous nearly K\"{a}hler manifold is a Riemannian 3-symmetric space equipped with its canonical almost…

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown…

Let G be the group preserving a nondegenerate sesquilinear form on a vector space V, and H a symmetric subgroup of G of the type G1 x G2. We explicitly parameterize the H-orbits in the Grassmannian of r-dimensional isotropic subspaces of V…

For a canonical formulation of quantum gravity, the superspace of all possible 3-geometries on a Cauchy hypersurface of a 3+1-dimensional Lorentzian manifold plays a key role. While in the analogous 2+1-dimensional case the superspace of…

Hyperbolic geometry, a Riemannian manifold endowed with constant sectional negative curvature, has been considered an alternative embedding space in many learning scenarios, \eg, natural language processing, graph learning, \etc, as a…

The subject of this paper is six-dimensional nearly (para-)K\"ahler geometry with pseudo-Riemannian metrics. Firstly, we derive the analogue of the well-known exterior differential system characterising a nearly K\"ahler manifold and prove…