Related papers: Torsion-free connections on $G$-structures

We consider a family of Riemannian manifolds M such that for each unit speed geodesic gamma of M there exists a distinguished bijective correspondence L between infinitesimal translations along gamma and infinitesimal rotations around it.…

Suppose we are given a compact Riemannian manifold (Q,g)with completely integrable geodesic flow. Let G be a compact connected Lie group acting freely on Q by isometries. The natural question arises: will the geodesic flow on Q/G equipped…

We show that torsion-free four-dimensional $\mathrm{GL}(2)$-structures are flat up to a coframe transformation with a mapping taking values in a certain subgroup $H\subset\mathrm{SL}(4,\mathbb{R})$ which is isomorphic to a semidirect…

A novel gravity theory based on Poisson Generalized Geometry is investigated. A gravity theory on a Poisson manifold equipped with a Riemannian metric is constructed from a contravariant version of the Levi-Civita connection, which is based…

By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan…

Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics on $L(M)$, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi--Civita connection and…

Let G be a connected reductive group and X be a smooth curve over an algebraically closed field of characteristic zero. We show that every meromorphic G-connection on X admits a possibly degenerate oper structure; in particular, every…

A general theorem on the existence of natural torsion-free affine connections on a complete family of compact complex submanifolds in a complex manifold is proved. Applications to twistor theory are discussed.

Let G be one of the Ricci-flat holonomy groups SU(n), Sp(n), Spin(7) or G_2, and M a compact manifold of dimension 2n, 4n, 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G-structures on M to the…

Let G be a n-dimensional Lie group (n>2) with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can…

We examine geometric representability results for various classes of equiaffine curvature operators. We show every Ricci flat algebraic curvature operator is geometrically realizable by a Ricci flat torsion free connection on the tangent…

A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature it is widely open whether a similar…

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in…

Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is…

Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In…

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually…

Let $G$ be a generalized Baumslag-Solitar group and $\mathcal{C}$ be a class of groups containing at least one non-unit group and closed under taking subgroups, extensions, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where…

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with…

This note is a continuation of the author's paper \cite{Li}. We prove that if the metric $g$ of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with…

Given a (semi-Riemannian) generalised metric $\mathcal G$ and a divergence operator $\mathrm{div}$ on an exact Courant algebroid $E$, we geometrically construct a canonical generalised Levi-Civita connection $D^{\mathcal G, \mathrm{div}}$…