We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate t...

In this work, we introduce the concept of a hyperbolic valued dislocated metric space, and show that fixed point self-mappings in such spaces is exist unique by using Banach, Kannan Chatterjea type contractions. Furthermore, discuss an illustrative example to substantiate applicability usefulness our main result.

In this article, we study the geometry of an infinite-dimensional hyperbolic space. We will consider group isometries Hilbert ball equipped with Carathéodory metric and learn about some special subclasses group. also find unitary equivalence condition compute cardinalities.