1998 | OriginalPaper | Chapter
Riemannian Symmetric Spaces
Author : Armand Borel
Published in: Semisimple Groups and Riemannian Symmetric Spaces
Publisher: Hindustan Book Agency
Included in: Professional Book Archive
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
Let M be a connected Riemannian manifold. The distance function d(x, y) (x, y ∈ M), which is by definition the inf limit of the lengths of the rectifiable arcs joining x to y, defines a metric compatible with the topology of M. By the Hopf-Rinow theorem, the following conditions are equivalent: (i)Every geodesic can be indefinitely extended, that is, the Levi-Cività connection of M is complete (III,3.1).(ii)M is complete as a metric space.(iii)Every bounded set is relatively compact.