Related papers: A new example of a compact ERP G2-structure

A $\mathrm{G}_2$-structure on a $7$-manifold $M$ is called a $\mathrm{G}_2T$-structure if $M$ admits a $\mathrm{G}_2$-connection $\nabla^T$ with totally skew-symmetric torsion $T_\varphi$. If furthermore, $T_\varphi$ is closed then it is…

We show that the compact quotient $\Gamma\backslash\mathrm{G}$ of a seven-dimensional simply connected Lie group $\mathrm{G}$ by a co-compact discrete subgroup $\Gamma\subset\mathrm{G}$ does not admit any exact $\mathrm{G}_2$-structure…

Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G_2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow…

We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an…

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-K\"ahler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $\varphi$…

We study the automorphism group of a compact 7-manifold $M$ endowed with a closed non-parallel G$_2$-structure, showing that its identity component is abelian with dimension bounded by min$\{6,b_2(M)\}$. This implies the non-existence of…

We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also…

We give a complete classification of left invariant para-K\"ahler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection…

In this paper, we study the evolution of $L^2$ one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the $L^2$ norm of a smooth one form with compact support is non-increasing…

In this paper we study backward Ricci flow of locally homogeneous geometries of $4$-manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near…

We construct harmonic morphisms on the compact simple Lie group G2. The construction uses eigenfamilies in a representation theoretic scheme.

We consider the Laplacian coflow of a $\mathrm{G}_2$-structure on warped products of the form $M^7= M^6 \times_f S^1$ with $M^6$ a compact 6-manifold endowed with an $\mathrm{SU}(3)$-structure. We give an explicit reinterpretation of this…

Geometrical applications of the non-compact form of Cartan's exceptional Lie group G(2) is considered. This group generates specific rotations of 7-dimensional Minkowski-like space with three extra time-like coordinates and in some limiting…

In this paper, we investigate left invariant Riemannian metrics on Lie groups with one and two-dimensional commutator subgroups. We explicitly provide the Levi-Civita connection, sectional curvature, and Ricci curvature, and we give…

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$…

We show that Lorentzian manifolds whose isometry group is of dimension at least $\frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be…

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time…

We observed in our previous paper that all the complex structures on four-dimensional compact solvmanifolds, including tori, are left-invariant. In this paper we will give an example of a six-dimensional compact solvmanifold which admits a…