On the completeness of total spaces of horizontally conformal submersions

In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

Metric spaces related to Abelian groups

<p>When working with a metric space, we are dealing with the additive group (R, +). Replacing (R, +) with an Abelian group (G, ∗), offers a new structure of a metric space. We call it a G-metric space and the induced topology is called the G-metric topology. In this paper, we are studying G-metric spaces based on L-groups (i.e., partially ordered groups which are lattices). Some results in G-metric spaces are obtained. The G-metric topology is defined which is further studied for its topological properties. We prove that if G is a densely ordered group or an infinite cyclic group, then every G-metric space is Hausdorff. It is shown that if G is a Dedekind-complete densely ordered group, (X, d) a G-metric space, A ⊆ X and d is bounded, then f : X → G with f(x) = d(x, A) := inf{d(x, a) : a ∈ A} is continuous and further x ∈ cl_XA if and only if f(x) = e (the identity element in G). Moreover, we show that if G is a densely ordered group and further a closed subset of R, K(X) is the family of nonempty compact subsets of X, e &lt; g ∈ G and d is bounded, then d′ (A, B) &lt; g if and only if A ⊆ N_d(B, g) and B ⊆ N_d(A, g), where N_d(A, g) = {x ∈ X : d(x, A) &lt; g}, d_B(A) = sup{d(a, B) : a ∈ A} and d′ (A, B) = sup{d_A(B), d_B(A)}.</p>

Metric topology on the moduli space

<p>We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.</p>

The Metric Topology

The chapter is an extensive account of the metric topology and is a prerequisite for all the subsequent chapters. The leading sections develop the basic metric properties such as closure and interior, continuity and equivalent metrics, separation properties, product spaces, and countability axioms. This is followed by a detailed study of completeness, compactness, local compactness, and function spaces. Chapter applications include contraction mappings, continuous nowhere differentiable functions, space-filling curves, closed convex subsets of ?n, and a number of approximation results. The chapter concludes with a detailed section on orthogonal polynomials and Fourier series of continuous functions, which, together with section 3.7, provides an excellent background for Hilbert spaces. The study of sequence and function spaces in this chapter leads up gradually into Banach spaces.

New topologies between the usual and Niemytzki

We use the technique of Hattori to generate new topologies on the closed upper half plane which lie between the usual metric topology and the Niemytzki topology. We study some of their fundamental properties and weaker versions of normality.

Topological n-cells and Hilbert cubes in inverse limits

<p>It has been shown by S. Mardešić that if a compact metrizable space X has dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then X must contain an arc. We are going to prove that if X = (|K_a|,p^b_a,(A,)<a href="http://www.codecogs.com/eqnedit.php?latex=\preceq" target="_blank"><img title="\preceq" src="http://latex.codecogs.com/gif.latex?\preceq" alt="" /></a>)is an inverse system in set theory of triangulated polyhedra|K_a|with simplicial bonding functions p^b_a and X = lim X, then there exists a uniquely determined sub-inverse system X_X= (|L_a|, p^b_a|L_b|,(A,<a href="http://www.codecogs.com/eqnedit.php?latex=\preceq" target="_blank"><img title="\preceq" src="http://latex.codecogs.com/gif.latex?\preceq" alt="" /></a>)) of X where for each a, L_a is a subcomplex of K_a, each p^b_a|L_b|:|L_b| → |L_a| is surjective, and lim X_X = X. We shall use this to generalize the Mardešić result by characterizing when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain a topological n-cell and do the same in the case of an inverse system of finite triangulated polyhedra with simplicial bonding maps. We shall also characterize when the inverse limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps must contain an embedded copy of the Hilbert cube. In each of the above settings, all the polyhedra have the weak topology or all have the metric topology(these topologies being identical when the polyhedra are finite).</p>