Journal•ISSN: 0024-6093
Bulletin of The London Mathematical Society
Wiley-Blackwell
About: Bulletin of The London Mathematical Society is an academic journal published by Wiley-Blackwell. The journal publishes majorly in the area(s): Series (mathematics) & Bounded function. It has an ISSN identifier of 0024-6093. Over the lifetime, 5543 publications have been published receiving 183148 citations.
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5,201 citations
TL;DR: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp.
Abstract: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp., £55.00, isbn 0-19-850254-1 (Clarendon Press, Oxford, 2000).
1,904 citations
TL;DR: The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston as mentioned in this paper, who has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods.
Abstract: The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods. The basic aim of this article is to discuss the various geometries which arise and explain their significance for the theory of 3-manifolds. The idea is that many 3-manifolds admit 'nice' metrics which give one new insight into properties of the manifolds. For the purposes of this article, the nicest metrics are those of constant curvature. An observer in a manifold with a constant curvature metric will see the same picture wherever he stands and in whichever direction he looks. Such manifolds have special topological properties. However, we will also need to consider nice metrics which are not of constant curvature. In this article, I will explain what is meant by a 'nice' metric and describe their classification in dimension three which is due to Thurston. Then I will discuss some of the 3-manifolds which admit these nice metrics and the relationship between their geometric and topological properties. In this introduction all manifolds and metrics will be assumed to be smooth so that the objects of interest are all Riemannian manifolds. It has been known since the nineteenth century that in dimension two there is a very close relationship between geometry and topology. I will start by describing some basic facts about closed surfaces. I will discuss these in more detail in §1. Each
1,677 citations
1,492 citations
TL;DR: In this article, the shape-space l. k m whose points represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric.
Abstract: The shape-space l. k m whose points a represent the shapes of not totally degenerate /c-ads in IR m is introduced as a quotient space carrying the quotient metric. When m = 1, we find that Y\ = S k ~ 2 ; when m ^ 3, the shape-space contains singularities. This paper deals mainly with the case m = 2, when the shape-space I* ca n be identified with a version of CP*~ 2 . Of special importance are the shape-measures induced on CP k ~ 2 by any assigned diffuse law of distribution for the k vertices. We determine several such shape-measures, we resolve some of the technical problems associated with the graphic presentation and statistical analysis of empirical shape distributions, and among applications we discuss the relevance of these ideas to testing for the presence of non-accidental multiple alignments in collections of (i) neolithic stone monuments and (ii) quasars. Finally the recently introduced Ambartzumian density is examined from the present point of view, its norming constant is found, and its connexion with random Crofton polygons is established.
1,468 citations