We establish well-posedness of a class of first order Hamilton-Jacobi equation in geodesic metric spaces. The result is then applied to solve a Hamilton-Jacobi equation in the Wasserstein space of probability measures, which arises from the variational formulation of a compressible Euler equation.

We use cross ratios to describe second real continuous bounded cohomology for locally compact σ-compact topological groups. We also investigate the second continuous bounded cohomology group of a closed subgroup of the isometry group Iso(X) of a proper hyperbolic geodesic metric space X and derive some rigidity results for Iso(X)-valued cocycles.

Let X be a geodesic metric space with H1(X) uniformly generated. If X has asymptotic dimension one then X is quasi-isometric to an unbounded tree. As a corollary, we show that the asymptotic dimension of the curve graph of a compact, oriented surface with genus g ≥ 2 and one boundary component is at least two.

We study the geodesic flow on the global holomorphic sections of the bundle π : TS → S induced by the neutral Kähler metric on the space of oriented lines of R, which we identify with TS. This flow is shown to be completely integrable when the sections are symplectic and the behaviour of the geodesics is described.

The method based on the Horský-Mitskievitch conjecture is applied to the Levi-Civita vacuum metric. It is shown, that every Killing vector is connected with a particular class of Einstein-Maxwell fields and each of those classes is found explicitly. Some of obtained classes are quite new. Radial geodesic motion in constructed space-times is discussed and graphically illustrated in the Appendix.

A supersymmetric extension of the Hunter-Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric. AMS Subject Classification (2000): 37K10, 17A70.

in this paper, we generalize fuzzy banach contraction theorem establishedby v. gregori and a. sapena [fuzzy sets and systems 125 (2002) 245-252]using notion of altering distance which was initiated by khan et al. [bull. austral.math. soc., 30(1984), 1-9] in metric spaces.

The Teichmüller space of surfaces of genus g > 1 with the Teichmüller metric is not nonpositively curved, in the sense that there are distinct geodesic rays from a point that always remain within a bounded distance of each other ([Ma1].) Despite this phenomenon, Teichmüller space and its quotient, Moduli space, share many properties with spaces of negative curvature: for instance, most convergi...

We study the Teichmüller metric on the Teichmüller space of a surface of finite type, in regions where the injectivity radius of the surface is small. The main result is that in such regions the Teichmüller metric is approximated up to bounded additive distortion by the sup metric on a product of lower dimensional spaces. The main technical tool in the proof is the use of estimates of extremal ...

At the very least, a geometric theory of manifolds would include a notion of distance, which could be expected to take the form of a metric that generates the topology of the manifold’s underlying topological space. In an interesting geometric theory this metric would be a path metric, meaning that there is a notion of the length of a path, and that the distance between two points is the infimu...