Characterization of manifolds of constant curvature by spherical curves

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the coefficients that dictate the RM frame motion (da Silva, da Silva in Mediterr J Math 15:70, 2018). Here, we shall prove the converse, i.e., we show that if all geodesic spherical curves on a Riemannian manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical and, consequently, the given manifold has constant sectional curvature. We also furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of three-dimensional manifolds. Finally, we also show that the same results are valid for semi-Riemannian manifolds of constant sectional curvature.Comment: To appear in Annali di Matematica Pura ed Applicat

Spatial patterns of random walkers under evolution of the attractiveness: persistent nodes, degree distribution, and spectral properties

In this paper we explore the features of a graph generated by random walkers with nodes that have evolutionary attractiveness and Boltzmann-like transition probabilities that depend both on the euclidean distance between the nodes and on the ratio ($\beta$) of the attractiveness between them. We show that persistent nodes, i.e., nodes that never been reached by random walker in asymptotic times are possible in the stationary case differently from the case where the attractiveness is fixed and equal to one for all nodes ($\beta =1$). Simultaneously, we also investigate the spectral properties and statistics related to the attractiveness and degree distribution of the evolutionary network. Finally, we study a crossover between persistent phase and no persistent phase and we also show the existence of a special type of transition probability that leads to a power law behaviour for the time evolution of the persistence.Comment: 21 pages, 8 figure

Anisotropic simplicial minisuperspace model

The computation of the simplicial minisuperspace wavefunction in the case of anisotropic universes with a scalar matter field predicts the existence of a large classical Lorentzian universe like our own at late timesComment: 19 pages, Latex, 6 figure