### 4-Dimensional (Para)-Kähler-Weyl Structures

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold $(M,g)$ onto a complete Riemannian manifold $(\widehat{M},\widehat{g})$ in terms of developing geodesic triangles of $M$ onto $\widehat{M}$. More precisely, we show that if ${A}_{0}:T{{|}_{{x}_{0}}M\to T|}_{{\widehat{x}}_{0}}\widehat{M}$ is some isometric map between the tangent spaces and if for any two geodesic triangles $\gamma $, $\omega $ of $M$ based at ${x}_{0}$ the development through ${A}_{0}$ of the composite path $\gamma \xb7\omega $ onto $\widehat{M}$ results in a closed path based at ${\widehat{x}}_{0}$, then there exists a Riemannian covering map...

If $(M,\nabla )$ is a manifold with a symmetric linear connection, then ${T}^{*}M$ can be endowed with the natural Riemann extension $\overline{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\overline{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $\mathcal{P}$ on $({T}^{*}M,\overline{g})$ and prove that $\mathcal{P}$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\overline{g}$ reduces to the...

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].

We study the complex hypersurfaces $f:{M}^{\left(n\right)}\to {\u2102}^{n+1}$ which together with their transversal bundles have the property that around any point of M there exists a local section of the transversal bundle inducing a ∇-parallel anti-complex shape operator S. We give a class of examples of such hypersurfaces with an arbitrary rank of S from 1 to [n/2] and show that every such hypersurface with positive type number and S ≠ 0 is locally of this kind, modulo an affine isomorphism of ${\u2102}^{n+1}$.

For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on...

To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$, we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M\times (-1,0)$. We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice of contact...

We study $G$-almost geodesic mappings of the second type $\underset{\theta}{\to}{\pi}_{2}\left(e\right)$, $\theta =1,2$ between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider $e$-structures that generate mappings of type $\underset{\theta}{\to}{\pi}_{2}\left(e\right)$, $\theta =1,2$. For a mapping $\underset{\theta}{\to}{\pi}_{2}(e,F)$, $\theta =1,2$, we determine the basic equations which generate them.

We define the concept of a bi-Legendrian connection associated to a bi-Legendrian structure on an almost -manifold ${M}^{2n+r}$. Among other things, we compute the torsion of this connection and prove that the curvature vanishes along the leaves of the bi-Legendrian structure. Moreover, we prove that if the bi-Legendrian connection is flat, then the bi-Legendrian structure is locally equivalent to the standard structure on ${\mathbb{R}}^{2n+r}$.