Related papers: A Quantitative Guessing Geodesics Theorem

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique…

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients…

We analyze in the context of geometrothermodynamics a Legendre invariant metric structure in the equilibrium space of an ideal gas. We introduce the concept of thermodynamic geodesic as a succession of points, each corresponding to a state…

Hyperbolic space is a geometry that is known to be well-suited for representation learning of data with an underlying hierarchical structure. In this paper, we present a novel hyperbolic distribution called \textit{pseudo-hyperbolic…

Several years ago the so-called quantum geometrodynamics in extended phase space was proposed. The main role in this version of quantum geometrodynamics is given to a wave function that carries information about geometry of the Universe as…

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…

A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. We prove existence and uniqueness results for geodesic bicombings satisfying different convexity conditions. In combination with recent work…

We establish central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. Our techniques allow us to remove the usual assumptions of properness and smoothness of the…

Let (X_i,d_i), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a ``hyperbolic product'' X_1{times}_h X_2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple…

Let $(M, \partial M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that the boundary is smooth and strictly convex. We show that the induced…

We prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. Specifically, in our main theorem we prove that for $\gamma$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i)…

We start to develop the quantization formalism in a hyperbolic Hilbert space. Generalizing Born's probability interpretation, we found that unitary transformations in such a Hilbert space represent a new class of transformations of…

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated…

We derive geometric formulas for the mass of asymptotically hyperbolic manifolds using coordinate horospheres. As an application, we obtain a new rigidity result of hyperbolic space: if a complete asymptotically hyperbolic manifold has…

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We…

In this paper, the notion of the catenary curve in the sphere and in the hyperbolic plane is introduced. In both spaces, a catenary is defined as the shape of a hanging chain when its potential energy is determined by the distance to a…

We introduce a new technique for proving the classical Stable Manifold theorem for hyperbolic fixed points. This method is much more geometrical than the standard approaches which rely on abstract fixed point theorems. It is based on the…

In this article, we use the second intrinsic volume to define a metric on the space of homothetic classes of Gaussian bounded convex bodies in a separable real Hilbert space. Using kernels of hyperbolic type, we can deduce that this space…

We develop a kind of quantum formalism (Hilbert space probabilistic calculus) for measurements performed over cognitive (in particular, conscious) systems. By using this formalism we could predict averages of cognitive observables.…