The spectral problem of the three-dimensional manifolds M3 A admitting Sol-geometry in Thurston’s sense is investigated. Topologically M3 A are the torus bundles over a circle with a hyperbolic glueing map A. The eigenfunctions of the corresponding Laplace-Beltrami operators are described in terms of the modified Mathieu functions. It is shown that the multiplicities of the eigenvalues do not d...

This paper studies hypersurfaces admitting a locally symmetric connection which is induced by the Gauß conormal map in affine geometry. It is known that the rank of the shape operator is of importance to this topic. In dimension two we give new results for an arbitrary shape operator. In the case of a nondegenerate shape operator hypersurfaces with locally symmetric conormal connection can be t...

The Einstein-Maxwell equations in D-dimensions admitting (D−3) commuting Killing vector fields have been investigated. The existence of the electric, magnetic and twist potentials have been proved. The system is formulated as the harmonic map coupled to gravity on three-dimensional base space generalizing the Ernst system in the four-dimensional stationary Einstein-Maxwell theory. Some classes ...

We analyze parametrized families of multimodal 1D maps that arise as singular limits of parametrized families of rank one maps. For a generic 1-parameter family of such maps that contains a Misiurewicz-like map, it has been shown that in a neighborhood of the Misiurewicz-like parameter, a subset of parameters of positive Lebesgue measure exhibits nonuniformly expanding dynamics characterized by...

We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. show that if is semisimple and arbitrary, then we have in the sense must be isomorphic. The proof uses result on decomposition as direct vector space sum two subalgebras. hence isomorphic to text]. This solves some open existence structures prove additional results text], where comp...

Special geometries defined by a class of differential forms on manifolds are again in the center of interests of geometers. These interests are motivated by the fact that such a setting of special geometries unifies many known geometries as symplectic geometry and geometries with special holonomy [Joyce2000], as well as other geometries arised in the M-theory [GMPW2004], [Tsimpis2005]. A series...

Let S be a hypersurface in Riemannian manifold M . One of the ways for deforming S consists in shifting points, which constitute S, along trajectories of some Newtonian dynamical system. Such situation arises in describing the propagation of electromagnetic wave (light) in non-homogeneous media in the limit of geometric optics. Hypersurface S models wave front set (the set of points with consta...

Let $\overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3)$ be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider }$ to the associated $3$-torsion of principally polarized abelian surface $A/\mathbf {Q}$. We prove that moduli space $\mathcal {A}_2(\overline })$ surfaces $B/\mathbf {Q}$ admitting symplectic isomorphism $B...