Link to this article: http://journals.cambridge.org/abstract_S0960129509007439 How to cite this article: M. ALI-AKBARI, B. HONARI, M. POURMAHDIAN and M. M. REZAII (2009). The space of formal balls and models of quasi-metric spaces.In this paper we study quasi-metric spaces using domain theory. Our main objective in this paper is to study the maximal point space problem for quasi-metric spaces. Here we prove that quasi-metric spaces that satisfy certain completeness properties, such as Yoneda and Smyth completeness, can be modelled by continuous dcpo's. To achieve this goal, we first study the partially ordered set of formal balls (BX, ) of a quasi-metric space (X, d). Following Edalat and Heckmann, we prove that the order properties of (BX, ) are tightly connected to topological properties of (X, d). In particular, we prove that (BX, ) is a continuous dcpo if (X, d) is algebraic Yoneda complete. Furthermore, we show that this construction gives a model for Smyth-complete quasi-metric spaces. Then, for a given quasi-metric space (X, d), we introduce the partially ordered set of abstract formal balls (BX, , ≺). We prove that if the conjugate space (X, d −1 ) of a quasi-metric space (X, d) is right K-complete, then the ideal completion of (BX, , ≺) is a model for (X, d). This construction provides a model for any Yoneda-complete quasi-metric space (X, d), as well as the Sorgenfrey line, Kofner plane and Michael line.

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Let \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^n=(M,F)$\end{document}Fn=(M,F) be a Finsler manifold and G be the Sasaki–Matsumoto metric on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘. Bejancu and Farran [“Finsler geometry and natural foliations on the tangent bundle,” Rep. Math. Phys. 58, 131 (2006)] proved that \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^n=(M,F)$\end{document}Fn=(M,F) is a Riemannian manifold if and only if the Sasaki–Matsumoto metric G on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘ is bundlelike for the vertical foliation. Let \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_1+n_2}=(M_1\times _fM_2,F)$\end{document}Fn1+n2=(M1×fM2,F) be the warped product Finsler manifold. In this paper the warped Sasaki–Matsumoto metric \documentclass[12pt]{minimal}\begin{document}${}^*\mathbf {G}$\end{document}*G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then \documentclass[12pt]{minimal}\begin{document}${}^*\mathbf {G}$\end{document}*G on \documentclass[12pt]{minimal}\begin{document}$TM^\circ$\end{document}TM∘ is bundlelike for the warped vertical foliation \documentclass[12pt]{minimal}\begin{document}$\mathcal {V}^*(TM^\circ )$\end{document}V*(TM∘) if and only if \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_1}_1=(M_1,F_1)$\end{document}F1n1=(M1,F1) and \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}^{n_2}_2=(M_2,F_2)$\end{document}F2n2=(M2,F2) are Riemannian manifolds.

Let 𝔽1 = (M1,F1) and 𝔽2 = (M2,F2) be two Finsler manifolds and let M = M1 × M2 and S is a spray in M. Also 𝔽 = (M1 × f M2, F) is a warped product Finsler manifolds, such that the function f : M1 → ℝ+ is not constant. In this paper, we define a non-linear connection on warped product 𝔽, and finally, we have presented some necessary and sufficient conditions under which the spray manifold (M1 × M2, S) is projectively equivalent to the warped product Finsler manifolds (M1 × f M2, F).

In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M, K) is studied. It is shown that geometry of these foliations are closely related to the geometry of the Cartan space (M, K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.