Abstract

The purpose of the paper is to provide a geometrization of the $k$-Lagrangians of second order as a framework of a variational problem.

Keywords: $k$-Lagarnge space.

MSC: 53B40

Abstract

The area of the autoroulette of a closed curve is double the area of the roulette of that curve on a straight base line taken with respect to the same generating point. This is an equivalent to Steiner's theorem for pedals. Also, an extension of Steiner's theorem is asserted.

Keywords: Pedal, autoroulettes, barycenter of curvatures .

MSC: 53A04

Abstract

In the present paper we define a geodesic mapping of two nonsymmetrical affine connexion spaces and obtain necessary and sufficient conditions that a mapping of two such spaces be geodesic (\S 1). Particularly we study a geodesic mapping of two generalized Riemannian spaces (\S 2). Finally, we generalize the notion of Thomas's projective parameters as an invariant of geodesic mappings (\S 3).

Keywords: Geodesic mapping, general affine connexion space, generalized Riemannian space, Thomas's parameters.

MSC: 53B05

Abstract

We consider a hyperbolic Kaehlerian space with vanishing conformal invariant. We prove a theorem which is fully analogous to results for Riemannian and Kaehlerian spaces with vanishing conformal invariants. Also, we prove two theorems which are valid in some special cases.

Keywords: Almost constant curvature, total holomorphic sectional curvature, cross sectional curvature, separated basis.

MSC: 53A30

Abstract

In this paper the $F$-structure, satisfying $F^{3}+F=0$ on the Lagrangian space, is examined. The construction of this structure is given as the prolongation of $f_{v}$-structure defined on $T_{V}(E)$ using the almost product or almost complex structure on $T(E)$. Moreover, the metric tensor $G$, with respect to which $F$ is an isometry, is constructed as well as the connection compatible with such structures.

Keywords: $F(3,1)$-structure, Lagrangian space.

MSC: 53B40, 53C60

Abstract

This paper is devoted to a study of the second order infinitesimal bendings of a class of toroids generated by a simple polygonal meridian. A necessary and sufficient condition for the existence of the second order infinitesimal bendings is determined.

Keywords: Second order infinitesimal bendings, toroid.

MSC: 53A05

Abstract

One possibility to classify hyperbolic sapce groups is to look for their fundamental domains. For simplicial domains are combinatorialy classified face pairing identifications, but the space of realization is not known. In this paper two series of fundamental simplices are investigated, which have three equivalence classes for edges and two for vertices. Three edges in the first class belong to the same face and vertices of that face are in the same class. Those simplices are hyperbolic, mainly with outer vertices. If so, then truncated simplex tilings are also investigated and classified with their metric data and other conditions.

Keywords: Tilings, Zhuk simplices.

MSC: 51M20, 52C22, 20H15

Abstract

Following Poincar\`e's geometric method, we construct two new nonorientable noncompact hyperbolic space forms by the regular octahedron in Fig\.~1. The construction is motivated by Thurston's example [6], discussed also by Apansov [1] in details. Our new space forms will be denoted by $$ \tilde D_1= H^3/G_1\quad\text{and}\quad \tilde D_2= H^3/G_2, $$ where $\tilde D_1$ and $\tilde D_2$ are obtained by pairing faces of $D$ via isometries of groups $G_1$ and $G_2$, respectively, acting discontinuously and freely on the hyperbolic 3-space $H^3$ (Fig\.~2, Fig\.~3). These groups are defined by generators and relations in Sect\.~3. The complete computer classification of possible space forms by our octahedron will be discussed in [4], where it turns out that our two space forms are isometric, i.e\. $G_1$ and $G_2$ are conjugated by an isometry $\varphi$ of $H^3$, i.e\. $G_2=\varphi^{-1}G_1\varphi$, $$ \align G_1&=(g_1,g_2,\bar g_1,\bar g_2\;\raise2pt\vbox{\hrule width.5cm}\;g_1\bar g_1^{-1}g_2\bar g_2^{-1}=g_1g_1g_2g_2=\bar g_1\bar g_1\bar g_2\bar g_2=1),\\ G_2&=(t_1,t_2,\bar g_1,\bar g_2\;\raise2pt\vbox{\hrule width.5cm}\;t_1\bar g_1^{-1}t_2^{-1}\bar g_2=t_1t_2t_1^{-1}t_2^{-1}=\bar g_1\bar g_1\bar g_2\bar g_2=1). \endalign $$

Keywords: Noncompact hyperbolic space, regular octahedron.

MSC: 51N20, 52C22