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Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=d p (x,y) and aquasi-axis if lim infd(x,y)/d p (x,y)>0 asx, y range over the vertex set of p andd p (x,y)?8. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if a is a translation whose directionD(a) is a thin end, then there exists an axis inD(a) andD(a-1) invariant under ar for somer not exceeding the maximum number of disjoint rays inD(a).The thinness ofD(a) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.

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117 Pallagi J., Schultz B., Szirmai J. Visualization of geodesic curves, spheres and equidistant surfaces in S 2 ×R space, KoG , Vol. 14, 2010, pp. 35

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196 218 249 Cohen, M. , and Lustig, M. , Paths of geodesics and geometric intersection numbers, I, in: Combinatorial group theory and

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and related fields. In this paper, we study three centrality measures that are based on geodesics (shortest paths) between nodes: global Q-measures, local Q-measures, and betweenness centrality. Informally, these measures characterize different aspects

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Abstract  

We characterize totally geodesic invariant submanifolds of C- and S-manifolds. We prove that the second fundamental form is parallel if and only if the submanifold of an S-manifold is totally geodesic. We show that this is not true for the C-manifolds.

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In this paper we discuss the deformation retract of Buchdahi space into itself and onto a geodesics in Buchdahi space, we discuss also the deformation retract of Minkowski space into itself and onto a geodesics by using Lagrangian equations. We study also the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been discussed.

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Acta Mathematica Hungarica
Authors: Alfonso Carriazo, Luis M. Fernández, and María Belén Hans-Uber

Summary We study some special types of slant submanifolds of S-manifolds related to the second fundamental form of the immersion: totally f-geodesic and f-umbilical, pseudo-umbilical and austere submanifolds. We also give several examples of such submanifolds.

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