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

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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 ${\u2102}^{n+1}$.

We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville [Vil] related with the first variation of the volume on a compact Einstein manifold.

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.