Related papers: Metrisability of Manifolds
In this paper we give a brief introduction to criteria for metrisability of a manifold and to some aspects of non-metrisable manifolds. Bias towards work currently being done by the author and his colleagues at the University of Auckland…
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…
We gather in this note results and examples about collared or non-collared boundaries of non-metrisable manifolds. Almost everything is well known but a bit scattered in the literature, and some of it is apparently not published at all.
We exhibit a family of metrizable manifolds such that any finite group appears as the fundamental group of one of them. These spaces are especially interesting as they can be easily visualized, as opposed to classical examples of spaces…
We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with…
We establish geometric criteria to decide whether a contact manifold is overtwisted. Starting with the original definition, we first relate the different overtwisted disks in each dimension and show that a manifold is overtwisted if the…
Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations. This note generalizes those methods by…
The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.
To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.
A semi-computable set S in a computable metric space need not be computable. However, in some cases, if S has certain topological properties, we can conclude that S is computable. It is known that if a semi-computable set S is a compact…
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…
Certain notions of convergence of sequences functions such as pointwise convergence and (uniform) convergence on compact or bounded sets come from suitable topological function spaces; see [1]. Under certain conditions these topologies…
The conventional definition of a topological metric over a space specifies properties that must be obeyed by any measure of "how separated" two points in that space are. Here it is shown how to extend that definition, and in particular the…
We examine sufficient conditions for the dual of a topological group to be metrizable and locally compact.
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
P.J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension > 1 is metrizable, and proved it is if one assumes the consistency of a supercompact cardinal, and, in addition, that the manifold is…
We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To…
In this note we relate some selection principles to metrisability and separability of a manifold. In particular we show that $\sf{S}_{fin}(\mathcal K,\mathcal O)$, $\sf{S}_{fin}(\Omega,\Omega)$ and $\sf{S}_{fin}(\Lambda,\Lambda)$ are each…
We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric…
This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?