Related papers: A note on flat metric connections with antisymmetr…
We obtain a series of new results on the problem of irreducibility of commuting varieties associated with symmetric pairs or, in other words, $Z_2$-graded simple Lie algebras. In particular, we present many examples of reducible commuting…
We pursuit the research line proposed in \cite{YZ-Gflat} about the classification of Hermitian manifolds whose $s$-Gauduchon connection $\nabla^s =(1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ is flat, where $s \in \mathbb{R}$ and…
The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A…
We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of…
These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate…
We introduce the notion of a manifold admitting a simple compact Cartan 3-form $\om^3$. We study algebraic types of such manifolds specializing on those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form…
We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural…
Recently many papers on cone metric spaces have been appeared, and main topological properties of such spaces have been obtained. A cone metric space is Hausdorff, and first countable, so the topology of it coincides with a topology induced…
We discuss the general structure of metric geometries, and how metricity implies the complete antisymmetry of Cartan tensor; an application in the frame of Lie group theory is given. Interpretations of the completely antisymmetric torsion…
A subclassification of stationary spacetimes, endowed with one timelike and one spacelike Killing vectors, i.e., Petrov $G{_2}I$ on $T_2$ spaces, is proposed. Special attention deserves the Collison's theorem [1] and the branch of metrics…
Type IIB S-folds of the form $\textrm{AdS}_{4} \times \textrm{S}^1 \times \textrm{S}^5$ have been shown to contain axion-like deformations parameterising flat directions in the 4D scalar potential and corresponding to marginal deformations…
For the Riemannian manifold $M^{n}$ two special connections on the sum of the tangent bundle $TM^{n}$ and the trivial one-dimensional bundle are constructed. These connections are flat if and only if the space $M^{n}$ has a constant…
We consider a smooth fibration equipped with a flat complex vector bundle and a hypersurface cutting the fibration into two pieces. Our main result is a gluing formula relating the Bismut-Lott analytic torsion form of the whole fibration to…
We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an…
For a differentiable manifold $M$, a pair $(M, \nabla)$ is called an affine manifold if $\nabla$ is a flat and torsion-free connection on the tangent bundle $TM\rightarrow M$. A Riemannian metric $g$ on $M$ is said to be a Hessian metric on…
In this paper, we investigate analytic and geometric properties of obstruction flatness of strongly pseudoconvex CR hypersurfaces of dimension $2n-1$. Our first two results concern local aspects. Theorem 3.2 asserts that any strongly…
The moduli space of flat SU(2) connections on a punctured surface, having prescribed holonomy around the punctures, is a compact smooth manifold if the prescription is generic. This paper gives a direct, elementary proof that the trace of…
The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion…
We prove that the deformation space of geodesic triangulations of a flat torus is homotopy equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983, in the case of flat tori. A key tool of the proof is a…
We consider an almost complex manifold with Norden metric (i. e. a metric with respect to which the almost complex structure is an anti-isometry). On such a manifold we study a linear connection preserving the almost complex structure and…