Related papers: Closed G2-structures with conformally flat metric

In this article, we develop foundational theory for geometries of the space of closed $G_2$-structures in a given cohomology class as an infinite-dimensional manifold. We introduce Sobolev-type metrics, construct their Levi-Civita…

On a manifold $(\mathbb{R}^n, e^{2u} |dx|^2)$, we say $u$ is normal if the $Q$-curvature equation that $u$ satisfies $(-\Delta)^{\frac{n}{2}} u = Q_g e^{nu}$ can be written as the integral form $u(x)=\frac{1}{c_n}\int_{\mathbb…

We review a recent series of $G_2$ manifolds constructed via solvable Lie groups obtained in math.DG/0409137. They carry two related distinguished metrics, one negative Einstein and the other in the conformal class of a Ricci-flat metric.

A Riemannian manifold is said to be almost positively curved if the sets of points for which all $2$-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$ admits a…

Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the `bundle of conformal blocks', a representation of the mapping class groupoid of…

Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let $P = P_{g,k}…

We derive all possible causality conditions for conformally flat Lorentzian metrics on the two-dimensional cylinder.

Given a smooth function $f(x)$ on $\mathbb{R}^n$ which is positive somewhere and satisfies $f(x)=O(|x|^{-l})$ for any $l>\frac{n}{2}$, we show that there exists a complete and conformal metric $g=e^{2u}|dx|^2$ with finite total Q-curvature…

A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of $2$ forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that…

Let $R$ be a commutative ring. Roughly speaking, we prove that an $R$-module $M$ is flat iff it is a direct limit of $R$-module affine algebraic varieties, and $M$ is a flat Mittag-Leffler module iff it is the union of its $R$-submodule…

We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Then we give a list of possibilities…

G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a…

The conformal properties of metrics are meaningful in Riemannian and Finsler geometry, and cubic metrics are useful in physics and biology. In this paper, we study the conformally flat cubic metrics with weakly isotropic scalar curvature.…

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere. The conformal structure of generic conformally flat…

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…

Let $R$ by a right coherent ring and $R$-Mod denote the category of left $R$-modules. We show that there is an abelian model structure on $R$-Mod whose cofibrant objects are precisely the Gorenstein flat modules. Employing a new method for…

We give an answer to the following question: for which metric in an abstract lattice the completion as a metric space coincides with the completion as a lattice. We obtain the answer for inductive limits of lattices which are complete in…

The geometry of conformal minimal two-spheres immersed in G(2,6;R) is studied in this paper by harmonic maps. Then in most cases, we determine the linearly full reducible conformal minimal immersions from S^2 to G(2,8;R) identified with the…

The equations for a G_2-structure with torsion on a product M^7 = N^6 x S^1 are studied in relation to the induced SU(3)-structure on N^6. All solutions are found in the case when the Lee-form of the G_2-structure is non-zero and N^6 is a…

We show obstructions to the existence of a coclosed $G_2$-structure on a Lie algebra $\mathfrak g$ of dimension seven with non-trivial center. In particular, we prove that if there exist a Lie algebra epimorphism from $\mathfrak g$ to a…