There are several ways to generalise the hyperbolic plane and its isometry group to objects in higher dimensions. Perhaps the most familiar is (real) hyperbolic three space, popularised by the work of Thurston [14]. The Poincaré disc and half plane models of the hyperbolic plane naturally come with a complex structure and it is natural to generalise them to complex hyperbolic space in higher co...

A domain G in a Banach space is said to be δ-hyperbolic if it is a Gromov δ-hyperbolic space in the quasihyperbolic metric. Then G has the Gromov boundary ∂∗G and the norm boundary ∂G. We show that the following properties are quantitatively equivalent: (1) G is C-uniform. (2) G is δ-hyperbolic and there is a natural bijective map G ∪ ∂∗G → G ∪ ∂G, which is η-quasimöbius rel ∂∗G. (3) G is δ-hyp...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

in terms of their trace and determinant are foundational in the real hyperbolic geometry. The counterpart of this characterization for isometries of H C was given by Giraud [8] and Goldman [9]. In this paper we offer algebraic characterization for the isometries of H H. The methods we follow carry over to the complex hyperbolic space, and yields an alternative characterization of the isometries...