Narrow Results

In this article we prove some properties of the weakly hyperbolic spaces. Moreover, a convergence-extension theorem for analytic hypersurfaces in weakly hyperbolic spaces is given.

Using the geometry of geodesics, we discuss the global aspects of complete real hypersurfaces in hyperbolic spaces of constant holomorphic sectional curvature over any division algebra 𝕂. Our assumption is that the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. Note that we do not assume constancy of the principal curvatures. Under this assumption, we give a complete global classification of such hypersurfaces. Since the argument is purely geometric, we need not vary the argument for different base algebras. The foliations of with totally geodesic leaves called 𝕂-lines play an important role.

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space ${ℍ}^4,$

In this paper, we study classification of magnetic curves corresponding to Killing vector fields of ${ℍ}^3$. First, we solve the geodesic equation analytically. Then we calculate the trajectories generated by all the six Killing vector fields, which are considered as magnetic field vectors, by using perturbation method up to first-order with respect to the strength of the magnetic field. We present a comparison of our solution with the numerical solution for one case. We also prove that 3-dimensional $(\alpha )$-Kenmotsu manifolds cannot have any magnetic vector field in the direction of their Reeb vector fields.

A proof of the rigidity of hyperbolic spaces is given for a class of conformally compact manifolds whose conformal infinity is the standard sphere. The method is based on quasilocal mass type theorems in general relativity.

The axiomatic approach to metric convexity goes back to the pioneering work of Karl Menger in 1928. This is an overview of this concept and the role it plays in metric fixed point theory especially in conjunction with spaces possessing a “hyperbolic” type structures. These include the CAT(0) spaces, hyperconvex metric spaces, and $ℝ$-trees. Much of the discussion involves the existence of “approximate” fixed point sequences for mappings satisfying weak contractive conditions. Applications of a well-known fixed point theorem due to Caristi are also included. These involve fixed and approximate fixed points for mappings satisfying local “directional” contractive and non-expansive conditions. Convexity plays a role in this part of the discussion as well. While the paper is semi-expository in nature, some detailed proofs appear here for the first time. Also the concept of a weak $Z$-directional contraction introduced in Sec. 8 appears to be new. Several suggestions for further research are also discussed.

In this paper, we prove some fixed point results for relatively nonexpansive mappings using Thakur iteration in uniformly convex $W-$ Hyperbolic spaces. Our results generalize some of the existing results in the literature. We illustrate our results with an example and shown the convergence of Thakur iteration to the fixed points of relatively nonexpansive mappings using Matlab code.

Let X_p = G_p/K_p be homogeneous spaces with compact subgroups K_p of locally compact groups G_p with some dimension parameter p such that the double coset spaces G_p//K_p can be identified with some fixed locally compact space X. For the projections T_p : X_p → X, and for a given probability measure ν ϵ M¹(X) there exist unique "radial", i.e. K_p-invariant measures ν_p ϵ M¹(X_p) with T_p(ν_p) = ν as well as associated radial random walks on the homogeneous spaces X_p. We generally ask for limit theorems for the random variables on X for n,p → ∞. In particular we give a survey about existing results for the Euclidean spaces X_p = ℝ^p with K_p = SO(_p) and X = [0, ∞[ as well as to some matrix extension of this rank one setting. Moreover, we derive a new central limit theorem for the hyperbolic spaces X_p of dimensions p over the skew fields 𝔽 = ℝ, ℂ, ℍ.