Search results for "hyperbolic space"

2016

AbstractThe different factors involved in the growth process of complex networks imprint valuable information in their observable topologies. How to exploit this information to accurately predict structural network changes is the subject of active research. A recent model of network growth sustains that the emergence of properties common to most complex systems is the result of certain trade-offs between node birth-time and similarity. This model has a geometric interpretation in hyperbolic space, where distances between nodes abstract this optimisation process. Current methods for network hyperbolic embedding search for node coordinates that maximise the likelihood that the network was pro…

The latent geometry of the human protein interaction network

2017

Abstract Motivation A series of recently introduced algorithms and models advocates for the existence of a hyperbolic geometry underlying the network representation of complex systems. Since the human protein interaction network (hPIN) has a complex architecture, we hypothesized that uncovering its latent geometry could ease challenging problems in systems biology, translating them into measuring distances between proteins. Results We embedded the hPIN to hyperbolic space and found that the inferred coordinates of nodes capture biologically relevant features, like protein age, function and cellular localization. This means that the representation of the hPIN in the two-dimensional hyperboli…

On nonimmersibility of compact hypersurfaces into a ball of a simply connected space form

1996

We give a nonimmersibility theorem of a compact manifold with nonnegative scalar curvature bounded from above into a geodesic ball of a simply connected space form.

Total curvatures of convex hypersurfaces in hyperbolic space

1999

We give sharp upper estimates for the difference circumradius minus inradius and for the angle between the radial vector (respect to the center of an inball) and the normal to the boundary of a compact $h$-convex domain in the hyperpolic space. We apply these estimates to get the limit at the infinity for the quotients Volume/Area and (Total $k$-mean curvature)/Area of a family of $h$-convex domains which expand over the whole space. The theorem for the first quotient gives an extension to arbitrary dimension of a result of Santalo and Yanez for the hyperbolic plane.

THE HOROSPHERICAL GEOMETRY OF SUBMANIFOLDS IN HYPERBOLIC SPACE

2005

Some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic -space are studied as an application of the theory of Legendrian singularities.

Curvature locus and principal configurations of submanifolds of Euclidean space

2017

We study relations between the properties of the curvature loci of a submanifold M in Euclidean space and the behaviour of the principal configurations of M, in particular the existence of umbilic and quasiumbilic fields. We pay special attention to the case of submanifolds with vanishing normal curvature. We also characterize local convexity in terms of the curvature locus position in the normal space.

Dyakonons in hyperbolic metamaterials

2013

We have analyzed surface-wave propagation that takes place at the boundary between an isotropic medium and a semi-infinite metal-dielectric periodic medium cut normally to the layers. In the range of frequencies where the periodic medium shows hyperbolic space dispersion, hybridization of surface waves (dyakonons) occurs. At low to moderate frequencies, dyakonons enable tighter confinement near the interface in comparison with pure SPPs. On the other hand, a distinct regime governs dispersion of dyakonons at higher frequencies. Full Text: PDF References Z. Ruan, M. Qiu, "Slow electromagnetic wave guided in subwavelength region along one-dimensional periodically structured metal surface", Ap…

Hadamard-type theorems for hypersurfaces in hyperbolic spaces

2006

Abstract We prove that a bounded, complete hypersurface in hyperbolic space with normal curvatures greater than −1 is diffeomorphic to a sphere. The completeness condition is relaxed when the normal curvatures are bounded away from −1. The diffeomorphism is constructed via the Gauss map of some parallel hypersurface. We also give bounds for the total curvature of this parallel hypersurface.

Non-immersion theorem for a class of hyperbolic manifolds

1998

Abstract It is proved that a non-simply-connected complete hyperbolic manifold cannot be isometrically immersed in a Euclidean space with a flat normal connection. In particular, the complete hyperbolic manifold M n with π 1 ( M ) ≠ 0 cannot be isometrically immersed in R 2 n − 1 .

The horospherical Gauss-Bonnet type theorem in hyperbolic space

2006

We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space and show that totally umbilic hypersurfaces with vanishing cur- vatures are only horospheres. We also show that the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space. + (i1) by using the model in Minkowski space. We introduced the notion of hyperbolic Gauss indicatrices slightly modified the definition of hyperbolic Gauss maps. The notion of hyperbolic indicatrices is independent of the choice of the model of hyperbolic space. Using the hyperbolic Gauss indicatrix, we defined the principal hyperbolic curv…