Geometry IV: Non-regular Riemannian Geometry
Gespeichert in:
1. Verfasser: | |
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Format: | Elektronisch E-Book |
Sprache: | English |
Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1993
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Schriftenreihe: | Encyclopaedia of Mathematical Sciences
70 |
Schlagworte: | |
Online-Zugang: | Volltext |
Beschreibung: | The book contains a survey of research on non-regular Riemannian geome try, carried out mainly by Soviet authors. The beginning of this direction oc curred in the works of A. D. Aleksandrov on the intrinsic geometry of convex surfaces. For an arbitrary surface F, as is known, all those concepts that can be defined and facts that can be established by measuring the lengths of curves on the surface relate to intrinsic geometry. In the case considered in differential is defined by specifying its first geometry the intrinsic geometry of a surface fundamental form. If the surface F is non-regular, then instead of this form it is convenient to use the metric PF' defined as follows. For arbitrary points X, Y E F, PF(X, Y) is the greatest lower bound of the lengths of curves on the surface F joining the points X and Y. Specification of the metric PF uniquely determines the lengths of curves on the surface, and hence its intrinsic geometry. According to what we have said, the main object of research then appears as a metric space such that any two points of it can be joined by a curve of finite length, and the distance between them is equal to the greatest lower bound of the lengths of such curves. Spaces satisfying this condition are called spaces with intrinsic metric. Next we introduce metric spaces with intrinsic metric satisfying in one form or another the condition that the curvature is bounded |
Beschreibung: | 1 Online-Ressource (VII, 252 p) |
ISBN: | 9783662028971 9783642081255 |
ISSN: | 0938-0396 |
DOI: | 10.1007/978-3-662-02897-1 |
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isbn | 9783662028971 9783642081255 |
issn | 0938-0396 |
language | English |
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series2 | Encyclopaedia of Mathematical Sciences |
spelling | Reshetnyak, Yu. G. Verfasser aut Geometry IV Non-regular Riemannian Geometry edited by Yu. G. Reshetnyak Berlin, Heidelberg Springer Berlin Heidelberg 1993 1 Online-Ressource (VII, 252 p) txt rdacontent c rdamedia cr rdacarrier Encyclopaedia of Mathematical Sciences 70 0938-0396 The book contains a survey of research on non-regular Riemannian geome try, carried out mainly by Soviet authors. The beginning of this direction oc curred in the works of A. D. Aleksandrov on the intrinsic geometry of convex surfaces. For an arbitrary surface F, as is known, all those concepts that can be defined and facts that can be established by measuring the lengths of curves on the surface relate to intrinsic geometry. In the case considered in differential is defined by specifying its first geometry the intrinsic geometry of a surface fundamental form. If the surface F is non-regular, then instead of this form it is convenient to use the metric PF' defined as follows. For arbitrary points X, Y E F, PF(X, Y) is the greatest lower bound of the lengths of curves on the surface F joining the points X and Y. Specification of the metric PF uniquely determines the lengths of curves on the surface, and hence its intrinsic geometry. According to what we have said, the main object of research then appears as a metric space such that any two points of it can be joined by a curve of finite length, and the distance between them is equal to the greatest lower bound of the lengths of such curves. Spaces satisfying this condition are called spaces with intrinsic metric. Next we introduce metric spaces with intrinsic metric satisfying in one form or another the condition that the curvature is bounded Mathematics Global differential geometry Differential Geometry Mathematik https://doi.org/10.1007/978-3-662-02897-1 Verlag Volltext |
spellingShingle | Reshetnyak, Yu. G. Geometry IV Non-regular Riemannian Geometry Mathematics Global differential geometry Differential Geometry Mathematik |
title | Geometry IV Non-regular Riemannian Geometry |
title_auth | Geometry IV Non-regular Riemannian Geometry |
title_exact_search | Geometry IV Non-regular Riemannian Geometry |
title_full | Geometry IV Non-regular Riemannian Geometry edited by Yu. G. Reshetnyak |
title_fullStr | Geometry IV Non-regular Riemannian Geometry edited by Yu. G. Reshetnyak |
title_full_unstemmed | Geometry IV Non-regular Riemannian Geometry edited by Yu. G. Reshetnyak |
title_short | Geometry IV |
title_sort | geometry iv non regular riemannian geometry |
title_sub | Non-regular Riemannian Geometry |
topic | Mathematics Global differential geometry Differential Geometry Mathematik |
topic_facet | Mathematics Global differential geometry Differential Geometry Mathematik |
url | https://doi.org/10.1007/978-3-662-02897-1 |
work_keys_str_mv | AT reshetnyakyug geometryivnonregularriemanniangeometry |