Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X . Define I : M(X) → R by I(μ) = ∫

Let S be the unit sphere in the Euclidean space R, and let e be the standard metric on S induced from R. Suppose that (u, ρ) are the spherical coordinates in R, where u ∈ S, ρ ∈ [0,∞). By choosing the smooth function φ(ρ) := sinh ρ on [0,∞) we can define a Riemannian metric h on the set {(u, ρ) : u ∈ S, 0 ≤ ρ < ∞} as follows h = dρ + φ(ρ)e. This gives the space form R(−1) which is the hyperboli...

A subset M of a topological space 5 is said to have a convex metric (even though S may have no metric) if the subspace M of 5 has a convex metric. It is known [5 J that a compact continuum is locally connected if it has a convex metric. The question has been raised [5] as to whether or not a compact locally connected continuum M can be assigned a convex metric. Menger showed [5] that M is conve...

Let (M, g) be an (n+1)-dimensional space time, with bounded curvature, with respect to a bounded framing. If (M, g) is vacuum, or satisfies a mild condition on the stress-energy tensor, then we show that (M, g) locally admits coordinate systems in which the Lorentz metric g is wellcontrolled in the (space-time) Sobolev space L, for any p <∞.

Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M . With respect to that metric, the volume of a rectifiable section σ : M → B is the mass of the image σ(M) as a rectifiable n-current in B. For any homology class of sections of B, there is a mass-minimizing Cartesian curr...

In this paper, we continue our study of the Weil-Petersson geometry as in the previous paper [10], in which we have proved the boundedness of the Weil-Petersson volume, among the other results. The main results of this paper are that the volume and the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space are rational numbers. In particular, the Ricci curvature define...

In this paper, we continue our study of the Weil-Petersson geometry as in the previous paper [10], in which we have proved the boundedness of the Weil-Petersson volume, among the other results. The main results of this paper are that the volume and the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space are rational numbers. In particular, the Ricci curvature define...

Let L ≥ 1, > 0 be real numbers, (M,d) be a finite metric space and (N, ρ) be a metric space (Rudin 1976). The metric space (M,d) is said to be Lbilipschitz embeddable into (N, ρ) if there is an injective function f :M → N with 1/L · d(x, y) ≤ ρ(f(x), f(y)) ≤ L · d(x, y) for all x, y ∈ N (Farb & Mosher 1999, David & Semmes 2000, Croom 2002). In this paper, we also say that (M,d) is -far from bei...

Many authors have introduced the concept of fuzzy metric spaces in different ways.In this paper, we get a special structure for some separable fuzzy metric and M-fuzzy metric spaces, where the metric space has been introduced by George and Veermani.

We prove a related fixed point theorem for n mappings which arenot necessarily continuous in n fuzzy metric spaces using an implicit relationone of them is a sequentially compact fuzzy metric space which generalizeresults of Aliouche, et al. [2], Rao et al. [14] and [15].