Related papers: Canonical connections attached to generalized quat…

Given a non-degenerate $(0,2)$-tensor field $h$ on a smooth manifold $M$, we consider a natural generalized complex and a generalized product structure on the generalized tangent bundle $TM\oplus T^*M$ of $M$ and we show that they are…

We consider a family of $\alpha$-connections defined by a pair of generalized dual quasi-statistical connections $(\hat{\nabla},\hat{\nabla}^*)$ on the generalized tangent bundle $(TM\oplus T^*M, \check{h})$ and determine their curvature,…

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a…

The canonical-type connection on the almost contact manifolds with B-metric is constructed. It is proved that its torsion is invariant with respect to a subgroup of the general conformal transformations of the almost contact B-metric…

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent…

The notion of generalized almost paracontact structure on the generalized tangent bundle $TM\oplus T^*M$ is introduced and its properties are investigated. The case when the manifold $M$ carries an almost paracontact metric structure is…

We review the theory of quaternionic Kahler and hyperkahler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure.…

Let $M$ be a smooth manifold, let $TM$ be its tangent bundle and $T^{*}M$ its cotangent bundle. This paper investigates integrability conditions for generalized metrics, generalized almost para-complex structures, and generalized Hermitian…

We give a new normalization condition for connections on sub-Riemannian manifolds with constant symbols. The condition is formulated in terms of Cartan connections and depends only on the first degree of homogeneity of the curvature. The…

We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading.…

Let (M,I,J,K) be a compact hypercomplex manifold admitting an HKT-metric. Assume that the canonical bundle of (M,I) is trivial as a holomorphic line bundle. We show that the holonomy of Obata connection on M is contained in SL(n,H). In…

A recent canonical divergence, which is introduced on a smooth manifold $\mathrm{M}$ endowed with a general dualistic structure $(\mathrm{g},\nabla,\nabla^*)$, is considered for flat $\alpha$-connections. In the classical setting, we…

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In…

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the…

Motivated by generalized geometry, we discuss differential geometric structures on the total space $\mathfrak{T}M$ of the bundle $TM\oplus T^*M$, where $M$ is a differentiable manifold; $\mathfrak{T}M$ is called a big-tangent manifold. The…

The works of Commichau--Grauert and Hirschowitz showed that a formal equivalence between embeddings of a compact complex manifold is convergent, if the embeddings have sufficiently positive normal bundles in a suitable sense. We show that…

A set of canonical parahermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the…

The goal of this paper is to introduce the lifting theory that has an important role in geometry. Therefore, using the lifts of differential geometric structures we show that tangent bundle TM of paracomplex manifold M admits para-complex…

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. We completely compute these…

In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to a lc pair is big as soon as there exists…